Extension of M-Convexity and L-Convexity to Polyhedral Convex Functions

Advances in Applied Mathematics 25, 352–427 (2000) doi:10.1006/aama.2000.0702, available online at http://www.idealibrary.com on

Extension of M-Convexity and L-Convexity to Polyhedral Convex Functions1 Kazuo Murota Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan E-mail: [email protected]

and Akiyoshi Shioura Department of Mechanical Engineering, Sophia University, 7-1 Kioi-cho, Chiyoda-ku, Tokyo 102-8554, Japan E-mail: [email protected] Received March 30, 1999; accepted May 24, 2000

The concepts of M-convex and L-convex functions were proposed by Murota in 1996 as two mutually conjugate classes of discrete functions over integer lattice points. M/L-convex functions are deeply connected with the well-solvability in nonlinear combinatorial optimization with integer variables. In this paper, we extend the concept of M-convexity and L-convexity to polyhedral convex functions, aiming at clarifying the well-behaved structure in well-solved nonlinear combinatorial optimization problems in real variables. The extended M/L-convexity often appears in nonlinear combinatorial optimization problems with piecewise-linear convex cost. We investigate the structure of polyhedral M-convex and L-convex functions from the dual viewpoint of analysis and combinatorics and provide some properties and characterizations. It is also shown that polyhedral M/L-convex functions have nice conjugacy relationships. © 2000 Academic Press Key Words: combinatorial optimization; matroid; base polyhedron; convex analysis; polyhedral convex function.

1 This work is supported by Grant-in-Aid of the Ministry of Education, Science, Sports and Culture of Japan.

352 0196-8858/00 $35.00 Copyright © 2000 by Academic Press All rights of reproduction in any form reserved.

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1. INTRODUCTION In the area of combinatorial optimization, there exist many “well-solved” problems, i.e., the problems which have nice combinatorial structure and which can be solved efficiently (see [5, 23, 39]). Many researchers have been trying to identify the well-behaved structure in combinatorial optimization problems. The concept of matroid, introduced by Whitney [47], plays an important role in the field of combinatorial optimization (see [46, 48–50]). Matroidal structure is closely related to the well-solvability of combinatorial optimization problems such as those on graphs and matroids and can be found in a fairly large number of efficiently solvable problems. Matroidal structure yields the tractability of problems in the following way: • Global optimality is equivalent to local optimality, which implies the success of the so-called greedy algorithm for the problem of optimizing a linear function over a single matroid. • A nice duality theorem, Edmonds’ intersection theorem [9], guarantees the existence of a certificate for the optimality in the matroid intersection problem in terms of dual variables. In 1970, Edmonds introduced the concept of polymatroid by extending that of matroid to sets of real vectors ([9], see also [46]). A polymatroid V P ⊆ R+ is a polyhedron given as 
  V  P = x ∈ R+  xw ≤ ρX ∀X ⊆ V  w∈V

by a submodular set function ρ 2V → R with certain additional conditions, where R+ denotes the set of nonnegative reals, and ρ is called submodular if ρX + ρY  ≥ ρX ∩ Y  + ρX ∪ Y 

∀X Y ⊆ V 

Polymatroids share nice combinatorial properties of matroids: for example, the greedy algorithm for matroids still works for polymatroids, and a duality holds for the polymatroid intersection problem. Fujishige, emphasizing the essential role of submodularity of ρ, generalized the concept of polymatroid to that of a submodular system [16, 17]. In recent years, nonlinear combinatorial optimization problems have been investigated more often due to theoretical interest and necessity in practical application. The nonlinear resource allocation problem (see [19, 22]) and the convex cost submodular flow problem (see [17, 21]) are examples of nonlinear combinatorial optimization problems. Both problems have nice combinatorial structures, which lead to efficient combinatorial algorithms. These results, however, do not completely fit in the framework of matroid, polymatroid, and submodular systems.

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The concepts of M-convex and L-convex functions, introduced by Murota [29, 30, 32], afford a nice framework for well-solved nonlinear combinatorial optimization problems. An M-convex function is a natural extension of the concept of valuated matroid introduced by Dress and Wenzel [7, 8] (see also [26–28, 36]) as well as a quantitative generalization of the set of integral points in an integral base polyhedron [17]. An L-convex function is an extension of the submodular set function. Let V be a finite set. A function f ZV → R ∪ +∞ is called M-convex if domZ f =  and it satisfies (M-EXC[Z]): (M-EXC[Z]) ∀x y ∈ domZ f , ∀u ∈ supp+ x − y, ∃v ∈ supp− x − y such that f x + f y ≥ f x − χu + χv  + f y + χu − χv  where domZ f = x ∈ ZV  −∞ < f x < +∞ supp+ x − y = w ∈ V  xw > yw supp− x − y = w ∈ V  xw < yw and χw ∈ 0 1V is the characteristic vector of w ∈ V . A function g ZV → R ∪ +∞ is called L-convex2 if domZ g =  and it satisfies (LF1[Z]) and (LF2[Z]):

λ ∈ Z,

(LF1[Z])

gp + gq ≥ gp ∧ q + gp ∨ q

∀p q ∈ domZ g,

(LF2[Z])

∃r ∈ R such that gp + λ1 = gp + λr ∀p ∈ domZ g

where p ∧ q, p ∨ q ∈ R V  denote the vectors with p ∧ qv = minpv qv, p ∨ qv = maxpv qv v ∈ V , and 1 ∈ R V  is the vector with each component being equal to one. M/L-convex functions have nice properties: • Local optimality is equivalent to global optimality. • M/L-convex functions can be extended to ordinary convex functions. • M/L-convex functions are conjugate to each other. • A (discrete) separation theorem and a Fenchel-type duality theorem hold for a pair of M-convex/M-concave (L-convex/L-concave) functions. 2 In the original definition [32], an L-convex function is assumed to be integer-valued. See Remark 6.1.

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The minimization of M/L-convex functions can be done in polynomial time [11, 43]. Application of M-convex functions can be found in system analysis through polynomial matrices [25, 31, 33, 36], and in mathematical economics [6]. M-convexity and L-convexity appear in various nonlinear combinatorial optimization problems with integer variables. Such nice combinatorial properties, however, are enjoyed not only by combinatorial optimization problems in integer variables but also by those in real variables. We dwell on this point by considering the minimum cost flow/tension problems. Let G = V A be a directed graph with a specified vertex subset T ⊆ V . Suppose we are given a family of piecewise-linear convex functions fa R → R ∪ +∞ a ∈ A, each of which represents the cost of flow on the arc a. A function ξ A → R is called a flow. The boundary ∂ξ V → R of a flow ξ is given by  ∂ξv = ξa  a∈ A leaves v  − ξa  a∈ A enters v v ∈ V  (1) Then, the cost function f R T → R ∪±∞ of the minimum cost flow that realizes a supply/demand vector x ∈ R T is defined by     ξ ∈ RA    f x = inf x ∈ R T  (2) fa ξa  ∂ξw = −xw w ∈ T  a∈A   ∂ξw = 0 w ∈ V \ T  Suppose we are given another family of piecewise-linear convex functions ga R → R ∪ +∞ a ∈ A, each of which represents the cost of tension on the arc a. Any function p V → R is called a potential. Given a potential p, its coboundary δp A → R is defined by δpa = pu − pv

a = u v ∈ A

(3)

Then, the cost function g R T → R ∪±∞ of the minimum cost tension that realizes a potential vector p ∈ R T is written as    p ∈ RV    gp  = inf p ∈ R T  (4) ga −δpa   pw = p w w ∈ T  a∈A It is well known that the minimum cost flow/tension problems with piecewise-linear convex cost can be solved efficiently by various combinatorial algorithms (see [1, 42]). It can be shown that both f and g are polyhedral convex functions (see Section 4 for the definition of a polyhedral convex function), which is a direct extension of results in Iri [20] and Rockafellar [42] for the case of T  = 2.

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We consider here the cost functions fZ and gZ for the integer version of the minimum cost flow/tension problems:    ξ ∈ ZA     fZ x = inf x ∈ ZT  fa ξa  ∂ξw = −xw w ∈ T   a∈A ∂ξw = 0 w ∈ V \ T     p ∈ ZV     gZ p  = inf p ∈ ZT  ga −δpa   w w ∈ T  pw = p a∈A It is shown in [32, 34, 35] that fZ satisfies (M-EXC[Z]) and gZ satisfies (LF1[Z]) and (LF2[Z]); i.e., fZ and gZ are M-convex and L-convex, respectively. These results indicate that the polyhedral convex functions f and g defined by (2) and (4) must have nice combinatorial properties like Mconvexity and L-convexity, respectively. As is shown later in Example 2.4, f satisfies the property (M-EXC): (M-EXC) ∀x y ∈ dom f , ∀u ∈ supp+ x − y, ∃v ∈ supp− x − y, ∃α0 > 0 such that f x + f y ≥ f x − αχu − χv  + f y + αχu − χv 

0 ≤ ∀α ≤ α0 

which is a generalization of (M-EXC[Z]), and g satisfies (LF1) and (LF2): (LF1) (LF2) ∀λ ∈ R ,

gp + gq ≥ gp ∧ q + gp ∨ q

∀p q ∈ dom g,

∃r ∈ R such that gp + λ1 = gp + λr

∀p ∈ dom g

which can be obtained by generalizing (LF1[Z]) and (LF2[Z]), where dom f = x ∈ R V  −∞ < f x < +∞ dom g = p ∈ R V  −∞ < gp < +∞ The observation above indicates the possibility of extending the concepts of M-convexity and L-convexity to polyhedral convex functions. This can be done in the following way. For a polyhedral convex function f R V → R ∪ +∞, we call f M-convex if dom f =  and f satisfies the property (M-EXC). Similarly, for a polyhedral convex function g R V → R ∪ +∞ we call g L-convex if dom g =  and g satisfies (LF1) and (LF2). The aim of this paper is to investigate the structures of polyhedral Mconvex and L-convex functions from the dual viewpoint of analysis and combinatorics and to provide a nice framework for well-solvable nonlinear combinatorial optimization problems in real variables. The organization of this paper is as follows.

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Section 2 provides some natural classes of polyhedral M/L-convex functions. We also prove the polyhedral M-convexity and L-convexity of the functions f and g defined in (2) and (4). To investigate polyhedral M/L-convex functions, we need to consider the set version of M/L-convexity. A polyhedron B ⊆ R V is called M-convex if it is nonempty and satisfies (B-EXC): (B-EXC) such that

∀x y ∈ B, ∀u ∈ supp+ x − y, ∃v ∈ supp− x − y, ∃α0 > 0

x − αχu − χv  ∈ B y + αχu − χv  ∈ B

0 ≤ ∀α ≤ α0 

As is explained later in Theorem 3.3, an M-convex polyhedron is nothing but the base polyhedron of a submodular system [17]. Similarly, a polyhedron D ⊆ R V is called L-convex if it is nonempty and satisfies (LS1) and (LS2): (LS1)

p q ∈ D ⇒ p ∧ q p ∨ q ∈ D,

(LS2)

p ∈ D ⇒ p + λ1 ∈ D ∀λ ∈ R .

We investigate the structure of M/L-convex polyhedra in Section 3. Section 4 shows fundamental properties of polyhedral M/L-convex functions. We present various equivalent axioms for polyhedral M/L-convex functions and give some properties on local structure of polyhedral M/L-convex functions such as directional derivatives, subdifferentials, minimizers, etc. In Section 4, we also investigate positively homogeneous polyhedral M/L-convex functions, which are important subclasses of polyhedral M/L-convex functions. It is shown that positively homogeneous polyhedral M/L-convex functions have one-to-one correspondences with certain set functions, and also with L/M-convex polyhedra. For a function f R V → R ∪ +∞, its conjugate function f • R V → R ∪±∞ is defined by f • p = sup p x − f x x∈R V

p ∈ R V 

where p x = pvxv  v ∈ V . It is shown in [32, 35] that there is a conjugacy relationship between integer-valued M/L-convex functions over the integer lattice. In Section 5, we show that the conjugacy relationship also exists for polyhedral M/L-convex functions. Section 5 also provides various characterization of polyhedral M/L-convex functions by local structures such as directional derivative, the set of minimizers, and subdifferentials. As is mentioned above, the concepts of M/L-convexity were originally introduced for functions defined over the integer lattice [29, 30, 32]. In Section 6, we clarify the relationship of M/L-convexity over the integer lattice and polyhedral M/L-convexity discussed in this paper.

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Finally, some duality theorems on polyhedral M/L-convex functions are presented in Section 7. Although duality theorems in this section can be obtained by application of the existing theorems in convex analysis to polyhedral M/L-convex functions, they are worth stating in their own right in connection to integrality properties. We also explain a more general and stronger result on the transformation of polyhedral M/L-convex function called “network induction.” The validity of network induction is proved by using the duality for polyhedral M/L-convex functions. In this paper, we focus on polyhedral convex functions with M/Lconvexity. The concepts of M/L-convexity can be further extended to general (not necessarily polyhedral) convex functions by considering the property (M-EXC) and properties (LF1), (LF2), respectively. Results in this direction will be reported elsewhere. 2. EXAMPLES OF POLYHEDRAL M/L-CONVEX FUNCTIONS Polyhedral M-convex and L-convex functions have various examples. Let V be a nonempty finite set. Example 2.1 (Affine Functions). Let B ⊆ R V be an M-convex polyhedron, p0 ∈ R V , and α ∈ R . Then the function f R V → R ∪ +∞ such that dom f = B

f x = p0  x + α

x ∈ B

is polyhedral M-convex with equality in (M-EXC), which is an immediate consequence of (B-EXC) for B. Let D ⊆ R V be an L-convex polyhedron, x0 ∈ R V , and ν ∈ R . Then the function g R V → R ∪ +∞ such that dom g = D

gp = p x0  + ν

p ∈ D

is polyhedral L-convex with equality in the submodular inequality (LF1) and r = 1 x0  in (LF2). We denote by
1 the class of one-dimensional piecewise-linear convex functions with a nonempty effective domain, i.e.,
1 = ϕ  ϕ R → R ∪ +∞ piecewise-linear convex dom ϕ = 

(5)

A piecewise-linear convex function (in this paper) is nothing but a onedimensional polyhedral convex function. Note that the effective domain dom ϕ of a function ϕ ∈
1 is a closed interval on R .

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Example 2.2. For ϕ ∈
1 , the function f R 2 → R ∪ +∞ defined by
ϕx1  x1 + x2 = 0, f x1  x2  = +∞ (otherwise) is polyhedral M-convex. For ψ ∈
1 , the function g R 2 → R ∪ +∞ defined by gp1  p2  = ψp1 − p2 

p1  p2  ∈ R 2 

is polyhedral L-convex. Furthermore, f and g are conjugate to each other if ϕ and ψ are conjugate to each other. Example 2.3 (Separable-Convex Functions). Let B ⊆ R V be an Mconvex polyhedron. For a family of functions fw ∈
1 w ∈ V , the function f R V → R ∪ +∞ defined by  fw xw x ∈ B, f x = w∈V +∞ x ∈ B is polyhedral M-convex if dom f = . Let D ⊆ R V be an L-convex polyhedron. For functions guv ∈
1 indexed by u v ∈ V , the function g R V → R ∪ +∞ defined by   guv pv − pu p ∈ D, gp = u v∈V +∞ p ∈ D is polyhedral L-convex with r = 0 in (LF2) if dom g = . Example 2.4 (Minimum Cost Flow/Tension Problems). In the Introduction, we have defined the cost functions f and g of the minimum cost flow/tension problems by (2) and (4), respectively. We here state the following properties of f and g: (i) f is polyhedral M-convex if f x0  is finite for some x0 ∈ R T . (ii) g is polyhedral L-convex if gp0  is finite for some p0 ∈ R T . (iii) Suppose that fa and ga are conjugate to each other for all a ∈ A. Then, f and g are conjugate to each other if one of the following conditions holds: (a) f x0  is finite (i.e., −∞ < f x0  < +∞) for some x0 ∈ R T , (b) gp0  is finite (i.e., −∞ < gp0  < +∞) for some p0 ∈ R T , (c) f x0  < +∞, gp0  < +∞ for some x0 ∈ R T , p0 ∈ R T . In the following, we prove (M-EXC) for f and (LF1), (LF2) for g. We omit the proofs of (iii) and the following properties: • if f x0  is finite for some x0 ∈ R T , then dom f =  and f > −∞,

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since their proofs are just a straightforward extension of the results in [20, 42] for the case T  = 2. [(M-EXC) for f ] Let x y ∈ dom f and u ∈ supp+ x − y. Also, let ξ η ∈ R A be flows such that

−xw w ∈ T , −yw w ∈ T , ∂ξw = ∂ηw = 0 w ∈ V \ T , 0 w ∈ V \ T ,   f x = fa ξa f y = fa ηa a∈A

a∈A

Note that the existence of such ξ and η is assured by the polyhedral convexity of each fa . By a standard augmenting path argument we see that there exist π A → 0 ±1 and v ∈ supp− x − y ⊆ T  such that supp+ π ⊆ supp+ ξ − η

supp− π ⊆ supp− ξ − η ∂π = χv − χu

Put α0 = minξa − ηa  a ∈ A πa = 0 > 0 Let α be any value with 0 ≤ α ≤ α0 . Then, we have ∂ξ − απ = −x + αχu − χv  ∂η + απ = −y − αχu − χv  fa ξa − απa + fa ηa + απa ≤ fa ξa + fa ηa

a ∈ A

which implies the inequality f x − αχu − χv  + f y + αχu − χv   ≤ fa ξa − απa + fa ηa + απa a∈A

≤



fa ξa + fa ηa = f x + f y

a∈A

Thus, f satisfies (M-EXC). [(LF1) for g] Let p  q ∈ dom g, and p q ∈ R V be vectors with pw = p w w ∈ T   ga −δpa = gp 

a∈A

qw = q w w ∈ T   ga −δqa = gq 

a∈A

Note that the existence of such p and q is assured by the polyhedral convexity of each ga . For a = u v ∈ A, we have ga −δp ∧ qa + ga −δp ∨ qa ≤ ga −δpa + ga −δqa

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by the convexity of ga . It is obvious that p ∧ qw = p ∧ q w Hence, we have gp ∧ q  + gp ∨ q  ≤

 a∈A

≤



a∈A

p ∨ qw = p ∨ q w w ∈ T 

ga −δp ∧ qa + ga −δpa +

 a∈A

 a∈A

ga −δp ∨ qa

ga −δqa = gp  + gq 

[(LF2) for g] Let 1T (resp. 1V ) be the vector in R T (resp. R V ) with each component being equal to one. For p ∈ R T and λ ∈ R , we have gp + λ1T    = inf ga −δpa ˜ a∈A



  p˜ ∈ R V    pw ˜ = p w + λw ∈ T 

  p ∈ RV   = inf ga −δp + λ1V a  pw + λ = p w + λw ∈ T  a∈A    p ∈ RV    = inf = gp  ga −δpa  pw = p ww ∈ T  a∈A 

3. M-CONVEX AND L-CONVEX POLYHEDRA In this section, we introduce two classes of polyhedra, called M-convex polyhedra and L-convex polyhedra. In fact, M-convex and L-convex polyhedra are familiar objects in combinatorial optimization, and this section is a recapitulation of almost known facts on these polyhedra from our point of view. The results in this section will be the basis of argument on polyhedral M-convex and L-convex functions beginning in Section 4. 3.1. Definitions and Notation We denote by R the set of reals and by Z the set of integers. Also, denote by R+ the set of nonnegative reals. Throughout this paper, we assume that V is a nonempty finite set. For any finite set X, its cardinality is denoted by X. The characteristic vector of a subset X ⊆ V is denoted by χX ∈ 0 1V , i.e.,
1 w ∈ X, χX w = 0 w ∈ V \ X. In particular, we use the notation 0 = χ , 1 = χV .

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Let x = xw  w ∈ V  ∈ R V . We define suppx = v ∈ V  xv = 0 +

supp x = v ∈ V  xv > 0  !x!1 = xv p x =

 v∈V

v∈V

pvxv p ∈ R V 

supp− x = v ∈ V  xv < 0 !x!∞ = max xv v∈V

xX =



xv X ⊆ V 

v∈X

We sometimes write 0V , 1V , and p xV to indicate that 0, 1, p, and x are vectors in R V . For ε > 0, we define N∞ x ε = y ∈ R V  !y − x!∞ < ε For any p q ∈ R V , p ∧ q and p ∨ q denote the vectors in R V such that p ∧ qw = minpw qw p ∨ qw = maxpw qw w ∈ V  For a V → R ∪−∞ and b V → R ∪+∞ with av ≤ bv v ∈ V , we define the interval "a b# ⊆ R V  by "a b# = x ∈ R V  av ≤ xv ≤ bv v ∈ V  For any two vectors p q ∈ R V , we denote Box"p q# = "p ∧ q p ∨ q#. Let S ⊆ R V . The set S is called convex if 1 − αx + αy ∈ S holds for any x y ∈ S and any α ∈ "0 1#, and called conic if αx ∈ S holds for any x ∈ S and any α > 0. A conic set is also called a cone. The convex hull of S, denoted by convS, is the smallest convex set containing S; that is, 
k  k     convS = λi xi k ≥ 1 xi ∈ S λi ≥ 0 λi = 1 i=1

i=1

Note that S is convex if and only if S = convS. For any set S ⊆ R V , the convex closure of S, denoted by $ S, is the smallest closed convex set containing S. Hence, N∞ x ε denotes a closed neighborhood y ∈ R V  !y − x!∞ ≤ ε. A set S ⊆ R V is called polyhedral if it is represented as the intersection of a finite number of closed half-spaces, i.e., there exist some pi ki=1 ⊆ R V  and αi ki=1 ⊆ R  k ≥ 0 such that S = x ∈ R V  pi  x ≤ αi ∀i = 1  k

(6)

A polyhedral set is also called a polyhedron. Obviously, any polyhedron is a closed convex set.

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We use the following convention involving ±∞: infα  α ∈  = +∞ supα  α ∈  = −∞ +∞ ≤ +∞ −∞ ≤ −∞ 0 × +∞ = +∞ × 0 = 0 × −∞ = −∞ × 0 = 0 In this paper, we never use expressions such as +∞ − ∞ or −∞ + ∞. 3.2. M-Convex Polyhedra A polyhedron B ⊆ R V is called M-convex if it is nonempty and satisfies (B-EXC): (B-EXC) such that

∀x y ∈ B, ∀u ∈ supp+ x − y, ∃v ∈ supp− x − y, ∃α0 > 0

x − αχu − χv  ∈ B

y + αχu − χv  ∈ B

0 ≤ ∀α ≤ α0 

We denote by 0 the family of M-convex polyhedra, i.e., 0 = B ⊆ R V  B M-convex polyhedron It is well known as a folklore that what we call an “M-convex polyhedron” is nothing but the base polyhedron of a submodular system [17] (see also Theorem 3.3). We use the term “M-convex polyhedron” for denotational symmetry to “L-convex polyhedron.” 3.2.1. Properties of M-Convex Polyhedra For any S ⊆ R V and x ∈ S, the exchange capacity c˜S x v u u v ∈ V  is defined by c˜S x v u = supα  α ∈ R+  x + αχv − χu  ∈ S where the subscript S to c˜ may be omitted when there is no ambiguity. The property (B-EXC) can be rewritten in terms of exchange capacity, as follows: (B-EXC ) ∀x y ∈ B, ∀u ∈ supp+ x − y, ∃v ∈ supp− x − y such that c˜B x v u > 0, c˜B y u v > 0. We also consider a one-sided exchange property: (B-EXC− ) ∀x y ∈ B, ∀u ∈ supp+ x − y, ∃v ∈ supp− x − y such that c˜B x v u > 0. The exchange property (B-EXC− ), apparently weaker than (B-EXC), is in fact equivalent to (B-EXC). Theorem 3.1. For a polyhedron B ⊆ R V , (B-EXC) ⇐⇒ (B-EXC− ). Proof.

Proof is given in Subsection 3.2.3.

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Fundamental properties of M-convex polyhedra are shown below. An Mconvex polyhedron is contained in a hyperplane x ∈ R V  xV  = r for some r ∈ R . Theorem 3.2. Let B ⊆ R V be a polyhedron with (B-EXC− ). Then, xV  = yV  ∀x y ∈ B. Proof. Assume, to the contrary, that x0 V  > y0 V  holds for some x0  y0 ∈ B. Let x∗ ∈ B be a vector with !x∗ − y0 !1 = inf!x − y0 !1  x ∈ B xV  = x0 V  Note that supp+ x∗ − y0  = . Let u ∈ supp+ x∗ − y0 . By (B-EXC− ), there exist some v ∈ supp− x∗ − y0  and a sufficiently small α > 0 such that x = x∗ − αχu − χv  ∈ B. It, however, is a contradiction to the choice of x∗ since x V  = x0 V  and !x − y0 !1 = !x∗ − y0 !1 − 2α. We shall show that an M-convex polyhedron is described by a submodular set function. Let ρ 2V → R ∪ +∞ and denote dom ρ = X ⊆ V  ρX < +∞. A function ρ is said to be submodular if it satisfies ρX + ρY  ≥ ρX ∩ Y  + ρX ∪ Y 

∀X Y ⊆ V 

(7)

We denote the class of (normalized) submodular set functions by  = ρ 2V → R ∪ +∞  ρ submodular, ρ = 0 ρV  < +∞ A set function µ 2V → R ∪−∞ is called supermodular if −µ is submodular. For any nonempty B ⊆ R V , define ρB 2V → R ∪ +∞ by ρB X = sup xX x∈B

X ⊆ V 

(8)

For a set function ρ 2V → R ∪ +∞, we define Bρ ⊆ R V by V Bρ = x ∈ R  xX ≤ ρX X ⊆ V  xV  = ρV 

(9)

Note that the exchange capacity associated with x ∈ Bρ can be written as [17]: c˜Bρ x v u = minρX − xX  X ⊆ V u ∈ X v ∈ X

u v ∈ V  (10) The following fact has been known to experts (cf. [4, 9, 46, Chap. 18]), but the precise statement cannot be found in the literature. Theorem 3.3. (i) For B ∈ 0 , we have ρB ∈  and BρB  = B. (ii) For ρ ∈  , we have Bρ ∈ 0 and ρBρ = ρ. (iii) The mappings B (→ ρB B ∈ 0  and ρ (→ Bρ ρ ∈   provide a one-to-one correspondence between 0 and  and are the inverse of each other.

polyhedral m/l-convex functions Proof.

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Proof is given later in Subsection 3.2.3.

For any sets S1  S2 ⊆ R V , we denote by S1 + S2 ⊆ R V  the Minkowskisum of S1 and S2 , i.e., S1 + S2 = p1 + p2  pi ∈ Si i = 1 2 Theorem 3.4 [17]. For B1  B2 ∈ 0 , the set B1 + B2 is M-convex with B1 + B2 = BρB1 + ρB2 . Remark 3.5. The intersection of two M-convex polyhedra is not necessarily an M-convex polyhedron, as shown in the example below. Let V = a b c d and define B1  B2 ⊆ R V as B1 = conv1 1 0 0 1 0 1 0 0 0 1 1 0 1 0 1 B2 = conv1 1 0 0 1 0 1 0 0 0 1 1 0 1 1 0 1 0 0 1 Then, B1 and B2 are M-convex polyhedra, and the submodular functions ρi = ρBi i = 1 2 defined by (8) are ρi  = 0

ρi v = 1 for v ∈ V i = 1 2

ρi X = 2

for X ⊆ V with X ≥ 3 i = 1 2

ρ1 a d = ρ1 b c = 1 ρ1 a b = ρ1 c d = ρ1 a c = ρ1 b d = 2 ρ2 b d = 1 ρ2 a b = ρ2 a c = ρ2 a d = ρ2 b c = ρ2 c d = 2 We see that B1 ∩ B2 = conv1 1 0 0 1 0 1 0 0 0 1 1 which is not an M-convex polyhedron. Note that the associated set function ρ = ρB1 ∩B2 in (8) is ρa d = ρb c = ρb d = 1 ρa b = ρc d = ρa c = 2 ρX = ρ1 X = ρ2 X

if X = 2

which is not submodular. Separation theorems of the following form hold for submodular functions and M-convex polyhedra. The latter half of Theorem 3.7 is already shown in [34, Theorem 3.6].

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Theorem 3.6 (Frank [12]). Let ρ 2V → R ∪ +∞, µ 2V → R ∪ −∞ be a pair of functions with ρ −µ ∈  . If µX ≤ ρX ∀X ⊆ V , then there exists x∗ ∈ R V such that µX ≤ x∗ X ≤ ρX

∀X ⊆ V 

Moreover, if ρ and µ are integer-valued, then there exists such x∗ ∈ ZV . Theorem 3.7. Let B1  B2 ∈ 0 . If B1 ∩ B2 = , then there exists p∗ ∈ 0 1V ∪ −χV  such that inf p∗  x − sup p∗  x > 0

x∈B1

x∈B2

(11)

Moreover, if both of B1 and B2 are integral polyhedra, i.e., Bi = Bi ∩ ZV i = 1 2, then “> 0” in (11) can be replaced by “≥ 1.” Proof. Put ρi = ρBi i = 1 2. If ρ1 V  = ρ2 V , then we can choose p∗ = χV or −χV . Hence, we may assume that ρ1 V  = ρ2 V . Define the functions ρ and µ by µX = ρ1 V  − ρ1 V \ X

ρX = ρ2 X X ⊆ V  V

Then ρ −µ ∈  and there is no x ∈ R with µX ≤ xX ≤ ρX ∀X ⊆ V . By Theorem 3.6, there exists some X0 ⊆ V such that µX0  > ρX0 . Since inf χX0  x = µX0  > sup χX0  x = ρX0 

x∈B1

x∈B2

we can choose p∗ = χX0 . The second claim follows from the integrality of B1  B2 . 3.2.2. M- -Convex Polyhedra Theorem 3.2 shows that an M-convex polyhedron is contained in a hyperplane of the form x ∈ R V  xV  = r for some r ∈ R . Therefore, any Mconvex polyhedron loses no information by the projection onto a (V −1)dimensional space. We call a polyhedron Q ⊆ R V M- -convex if the set  R V  defined by Q⊆  = x  x ∈ R V  x ∈ Q x = −xV  Q 0 0

(12)

 = v0  ∪ V . In fact, by the results is an M-convex polyhedron, where V of [15, 17], what we name M -convex polyhedron here is nothing but a generalized polymatroid in [13, 14], where a generalized polymatroid is a nonempty set Q ⊆ R V given by Q = x ∈ R V  µX ≤ xX ≤ ρX X ⊆ V 

(13)

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with a pair of submodular/supermodular functions ρ 2V → R ∪ +∞, µ 2V → R ∪−∞ satisfying the inequality ρX − µY  ≥ ρX \ Y  − µY \ X

X Y ⊆ V 

-

M -convex polyhedron is an essentially equivalent concept to M-convex polyhedron, while the class of M- -convex polyhedra properly contains that of M-convex polyhedra. Every property of M-convex polyhedra can be restated in terms of M- -convex polyhedra, and vice versa. M- -convex polyhedra can be characterized by various exchange properties, as follows (cf. [37, Remark 5.2; 45]). For Q ⊆ R V and u v ∈ V , we denote c˜Q x  u = supα  α ∈ R+  x − αχu ∈ Q c˜Q x v  = supα  α ∈ R+  x + αχv ∈ Q (G-EXC) ∀x y ∈ Q, ∀u ∈ supp+ x − y, either (i) or (ii) (or both) holds: (i) ∃v ∈ supp− x − y such that c˜Q x v u > 0 and c˜Q y u v > 0, (ii) c˜Q x  u > 0 and c˜Q y u  > 0. (G-EXC− ) ∀x y ∈ Q, ∀u ∈ supp+ x − y, either c˜Q x v u > 0 ∃v ∈ supp− x − y or c˜Q x  u > 0. Theorem 3.8. Let Q ⊆ R V be a nonempty polyhedron. Then, Q is M- -convex ⇐⇒ Q satisfies (G-EXC) ⇐⇒ Q satisfies (G-EXC− ) Proof.

Proof is given later in Subsection 3.2.3.

The following property, which seems more natural when described in terms of M- -convex polyhedron, is obvious from (13). Theorem 3.9. Let Q ⊆ R V be an M- -convex polyhedron. Then "x y# ⊆ Q holds for any x y ∈ Q with x ≤ y. 3.2.3. Proofs This section provides the proofs of Theorems 3.1, 3.3, and 3.8. We use the following results on submodular set functions. A family  ⊆ 2V is called a distributive lattice if X ∩ Y X ∪ Y ∈  holds for any X Y ∈ . Note that dom ρ of a submodular function ρ ∈  is a distributive lattice. Theorem 3.10. Let ρ ∈  . For any x ∈ Bρ, the family x = X ⊆ V  xX = ρX is a distributive lattice with  V  ⊆ x. We first prove a slightly stronger claim than Theorem 3.3 (i). A sequence Xi ki=0 k ≥ 0 of distinct elements in 2V with X0 ⊆ X1 ⊆ · · · ⊆ Xk is called a chain.

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Lemma 3.11. Let B ⊆ R V be a nonempty polyhedron with (B-EXC− ), and Xi ki=0 k ≥ 0 be a chain such that Xi ki=0 ⊆ dom ρB . Then, there exists x∗ ∈ B with x∗ Xi  = ρB Xi  i = 0 1  k. Proof. The claim is shown by induction on the integer k. Let xk ∈ B be a vector with xk Xk  = ρB Xk . By the inductive hypothesis, there exists a vector x ∈ B satisfying xXi  = ρB Xi  i = 0 1  k − 1. We may assume that x minimizes the value !x − xk !1 of all such vectors. Suppose that xXk  < ρB Xk . By (B-EXC− ), there exist u ∈ supp+ x − xk  \ Xk and v ∈ supp− x − xk  such that x = x − αχu − χv  ∈ B with a sufficiently small α > 0. Here v ∈ V \ Xk−1 holds since u ∈ V \ Xk−1 and x Xk−1  ≤ ρB Xk−1  = xXk−1 . The vector x satisfies x Xi  = ρB Xi  i = 0 1  k − 1 and !x − xk !1 = !x − xk !1 − 2α, a contradiction. Hence xXk  = ρB Xk . For any polyhedron S ⊆ R V , a vector x ∈ S is called an extreme point of S if there are no y1  y2 ∈ S \ x and α ∈ R with 0 < α < 1 such that x = αy1 + 1 − αy2 . Theorem 3.12 [17, Theorem 3.22]. Let ρ ∈  . Then, x ∈ R V is an V  extreme point of Bρ if and only if there exists a chain Xi i=0 such that V 

Xi i=0 ⊆ dom ρ xXi  = ρXi 

X0 = 

XV  = V

i = 0 1  V 

Lemma 3.13. For any nonempty polyhedron B ⊆ R V with (B-EXC− ), we have ρB ∈  and B = BρB . Proof. The outline of the proof is similar to that for [4, Lemma 5.2]. We first consider the case when B is bounded. Let X Y ∈ dom ρB . From Lemma 3.11, there exists x∗ ∈ B with x∗ X ∩ Y  = ρB X ∩ Y  and x∗ X ∪ Y  = ρB X ∪ Y , which implies the inequality ρB X + ρB Y  ≥ x∗ X + x∗ Y  = x∗ X ∩ Y  + x∗ X ∪ Y  = ρB X ∩ Y  + ρB X ∪ Y  Therefore, ρB ∈  . Lemma 3.11 and Theorem 3.12 imply that any extreme point of BρB  is contained in B, which shows that BρB  ⊆ B. The reverse inclusion is obvious. Next, assume that B is unbounded. Let x0 be any vector in B. For each nonnegative integer k, put Bk = x ∈ B  !x − x0 !∞ ≤ k and ρk = ρBk . Since each Bk is a bounded M-convex polyhedron, it holds that ρk ∈  and

polyhedral m/l-convex functions Bk = Bρk . Hence, we have BρB  =

∞

k=1

Bρk  =

∞

k=1

369 Bk = B, and

ρB X + ρB Y  = lim ρk X + ρk Y  k→∞

≥ lim ρk X ∩ Y  + ρk X ∪ Y  k→∞

= ρB X ∩ Y  + ρB X ∪ Y  for any X Y ∈ dom ρB , i.e., ρB ∈  . Next, we prove Theorem 3.3 (ii). Lemma 3.14 (cf. [17]). Let ρ ∈  , x ∈ Bρ, and u ∈ V . Put X0 = v ∈ V  cx ˜ v u = 0

Y0 = v ∈ V  cx ˜ u v > 0

Then we have u ∈ V \ X0 , xX0  = ρX0 , and u ∈ Y0 , xY0  = ρY0 . Note that the set Y0 in Lemma 3.14 is nothing but the dependence function depx u defined in [17]. We consider another one-sided exchange property: (B-EXC+ ) ∀x y ∈ B, ∀u ∈ supp+ x − y, ∃v ∈ supp− x − y such that c˜B y u v > 0. Lemma 3.15. For any ρ ∈  , the set Bρ satisfies (B-EXC− ) and (B-EXC+ ). Proof. We prove (B-EXC− ) only, since (B-EXC+ ) can be shown in the similar way. Let x y ∈ Bρ with x = y and u ∈ supp+ x − y. To the contrary assume cx ˜ v u = 0 for all v ∈ supp− x − y. Put X0 = v ∈ V  cx ˜ v u = 0. By Lemma 3.14 we have xX0  = ρX0  ≥ yX0 , whereas xX0  < yX0  since u ∈ V \ X0 and supp− x − y ⊆ X0 , a contradiction. Lemma 3.16. Let B ⊆ R V be a nonempty polyhedron with (B-EXC− ) and (B-EXC+ ). For any x y ∈ B, and v ∈ supp− x − y, there exists y  ∈ B such that y  v = xv y  w = yw

xw ≥ y  w ≥ yw w ∈ supp+ x − y w ∈ v ∪ supp+ x − y

Proof. Put supp+ x − y = u1  u2   uk  k ≥ 1, and y0 = y. For i = 1 2  k, we define αi ∈ R+ and yi ∈ B iteratively as αi = minyi−1 v − xv xui  − yi−1 ui  cy ˜ i−1  ui  v yi = yi−1 + αi χui − χv 

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Assume yk v > xv. By (B-EXC− ), there exists some ui ∈ supp− yk − x ⊆ supp+ x − y such that yˆ = yk − αχv − χui  ∈ B for a sufficiently small α > 0. Since ui ∈ supp+ yˆ − yi  and supp− yˆ − yi  = v, (B-EXC+ ) implies yi + βχui − χv  ∈ B for a sufficiently small β > 0, a contradiction to the fact that αi = cy ˜ i−1  ui  v. Lemma 3.17.

For any ρ ∈  , the set Bρ satisfies (B-EXC).

Proof. First note that Bρ satisfies (B-EXC− ) and (B-EXC+ ) by Lemma 3.15. Let x y ∈ Bρ with x = y and u ∈ supp+ x − y. Put ˜ u v > 0. Applying Lemma 3.16 repeatedly, we have Y0 = v ∈ V  cy some y  ∈ Bρ such that y  v = xv v ∈ supp− x − y \ Y0  xw ≥ y  w ≥ yw w ∈ supp+ x − y y  w = yw w ∈ supp− x − y \ Y0  ∪ supp+ x − y Moreover, y  w = yw holds for w ∈ supp+ x − y ∩ Y0 since otherwise y  Y0  > yY0  = ρY0  by Lemma 3.14, a contradiction. In particular, y  u = yu < xu. Applying (B-EXC− ) to x, y  , and u ∈ supp+ x − y  , we have v0 ∈ supp− x − y   with cx ˜ v0  u > 0. From supp− x − y   = − supp x − y ∩ Y0 follows cy ˜ u v0  > 0. Theorem 3.18 [17]. For any ρ ∈  we have Bρ =  and ρBρ = ρ. Theorem 3.3(iii) is an immediate corollary of Lemmas 3.13, 3.17, and Theorem 3.18. We then give the proof of Theorem 3.1. Proof of Theorem 3 1. The implication (B-EXC) ⇒ (B-EXC− ) is obvious. Let B ⊆ R V be a nonempty polyhedron with (B-EXC− ). Then, we have ρB ∈  and B = BρB , as shown in Lemma 3.13. Hence, Lemma 3.17 implies (B-EXC) for B. Finally, we prove Theorem 3.8. Since the implications “Q is M- -convex ⇒ (G-EXC) ⇒ (G-EXC− )” are easy to see, we show “(G-EXC− ) ⇒ Q is M- -convex” only. We first note that (G-EXC− ) for Q is equivalent to the following property  defined by (12): for Q  ∀u ∈ supp+ x − y \ v , ∃v ∈ supp− x − (G0 -EXC− ) ∀x y ∈ Q, 0 y ∪ v0  such that c˜Qx v u > 0. Using this property, we prove the submodularity of ρQ and the equation  = Bρ , where ρ  2V → R ∪ +∞ and Bρ  ⊆ R V are defined Q Q Q Q by (8) and (9), respectively. The proof is similar to that of Theorem 3.3(i).

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Lemma 3.19. Let Xi ki=0 k ≥ 0 be a chain such that Xi ki=0 ⊆  with x X  = ρ X  i = 0 1  k. dom ρQ. Then there exists x∗ ∈ Q ∗ i i Q Proof. The claim is shown by induction on the integer k. By the induc such tive hypothesis and the definition of ρQ, there exist vectors x y ∈ Q that xXi  = ρQXi 

i = 1 2  k

yX0  = ρQX0 

(14) (15)

 be a pair of vectors minimizing the value !x − y ! among Let x∗  y∗ ∈ Q ∗ ∗ 1  satisfying (14), (15), and xX  = maxx X   all pairs of vectors x y ∈ Q 0 k  14. We further assume that x and y maximize the value x ∈ Q ∗ ∗ y∗ v0  − x∗ v0  among all such vectors. Claim. y∗ w ≤ x∗ w ∀w ∈ V \ X0 ∪ v0 . Assume, to the contrary, that there exists some w0 ∈ V \ X0 ∪ v0  with y∗ w0  > x∗ w0 . Then, (G0 -EXC− ) implies either (a-1), (a-2), or (a-3): (a-1) ∃v ∈ X0 such that c˜Qy∗  v w0  > 0, (a-2) ∃v ∈ supp− y∗ − x∗  \ X0 such that c˜Qy∗  v w0  > 0, (a-3) v0 ∈ V \ X0 , y∗ v0  ≥ x∗ v0 , and c˜Qy∗  v w0  > 0 for v = v0 .  where α is a sufficiently small positive Put y  = y∗ − αχw0 − χv  ∈ Q, number. In either case we have a contradiction: (a-1) ⇒ y  X0  > ρQX0  (a-2) ⇒ y  X0  = ρQX0  !y  − x∗ !1 < !y∗ − x∗ !1  (a-3) ⇒ y  X0  = ρQX0  !y  − x∗ !1 = !y∗ − x∗ !1  y  v0  − x∗ v0  > y∗ v0  − x∗ v0  This ends the proof of the claim. Suppose that either v0 ∈ V \ X0 and x∗ v0  ≥ y∗ v0 , or v0 ∈ X0 holds. Then, for i = 1 2  k, we have  − y∗ V  \ Xi  y∗ Xi  = y∗ V  − x∗ V  \ Xi  = x∗ Xi  = ρ Xi  ≥ x∗ V Q implying y∗ Xi  = ρQXi . It remains to consider the case where v0 ∈ V \ X0 

x∗ v0  < y∗ v0 

(16)

Assume, to the contrary, that x∗ X0  < ρQX0  = y∗ X0 . Then there exists some u ∈ supp+ x∗ − y∗  \ X0 . Hence, (G0 -EXC− ) implies either (b-1) or (b-2):

372

murota and shioura (b-1) ∃v ∈ supp− x∗ − y∗  ∩ X0 such that c˜Qx∗  v u > 0, (b-2) c˜Qx∗  v u > 0 for v = v0 .

 where α is a sufficiently small positive numPut x = x∗ + αχv − χu  ∈ Q, ber. In case of (b-1), we have u ∈ X1 \ X0 since x∗ satisfies (14). Therefore, x also satisfies (14) and x X0  > x∗ X0 , a contradiction. In case of (b-2), we have x X0  = x∗ X0  and !x − y∗ !1 < !x∗ − y∗ !1 by (16), a contradiction. In the same way as the proof of Lemma 3.13, we can show that ρQ ∈   = Bρ  by using Lemma 3.19 and Theorem 3.12. and Q Q 3.3. L-Convex Polyhedra A polyhedron D ⊆ R V is called L-convex if it is nonempty and satisfies (LS1) and (LS2): (LS1) (LS2)

p q ∈ D ⇒ p ∧ q p ∨ q ∈ D, p ∈ D ⇒ p ± λ1 ∈ D ∀λ ∈ R .

We denote by 0 the family of L-convex polyhedra, i.e.,

0 = D ⊆ R V  D L-convex polyhedron

3.3.1. Properties of L-Convex Polyhedra The properties (LS1) and (LS2) can be characterized by local properties: (LS1loc -a)

∀p ∈ D, ∃ε > 0 such that

p1  p2 ∈ D ∩ N∞ p ε ⇒ p1 ∧ p2  p1 ∨ p2 ∈ D (LS1loc -b)

∀p q ∈ D, 0 < ∃ε ≤ 1 such that

1 − εp + εp ∧ q ∈ D (LS2loc )

1 − εp + εp ∨ q ∈ D

∀p ∈ D, ∃ε > 0 such that p + λ1 ∈ D ∀λ ∈ "0 ε#.

Theorem 3.20. Let D ⊆ R V be a closed set. (i) If D is convex, then (LS1) ⇐⇒ (LS1loc -a) ⇐⇒ (LS1loc -b). (ii) (LS2) ⇐⇒ (LS2loc ). Proof. (i) Since (LS1) ⇒ (LS1loc -a) is obvious, we show (LS1loc -a) ⇒ (LS1loc -b) ⇒ (LS1). Assume (LS1loc -a). Let p q ∈ D, and ε be the value in (LS1loc -a) associated with p. The convexity of D implies 1 − ε p + ε q ∈ D ∩ N∞ p ε for a sufficiently small ε > 0. Hence, we have p ∧ 1 − ε p + ε q = 1 − ε p + ε p ∧ q ∈ D p ∨ 1 − ε p + ε q = 1 − ε p + ε p ∨ q ∈ D i.e., (LS1loc -b) holds.

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Next assume (LS1loc -b) for D to prove (LS1). Let p q ∈ D. We show p ∧ q ∈ D only, since p ∨ q ∈ D can be shown similarly. Set α∗ = supα  0 ≤ α ≤ 1 1 − αp + αp ∧ q ∈ D p∗ = 1 − α∗ p + α∗ p ∧ q ∈ D Assume α∗ < 1. By (LS1loc -b), there exists some ε with 0 < ε ≤ 1 such that 1 − εp∗ + εp∗ ∧ q = 1 − ε − α∗ + εα∗ p + ε + α∗ − εα∗ p ∧ q ∈ D a contradiction to the definition of α∗ . Hence α∗ = 1 and p ∧ q ∈ D hold. (ii) We show the direction (LS2loc ) ⇒ (LS2) only. Let p ∈ D and λ∗ = supλ  λ ∈ R+  p + λ1 ∈ D. To the contrary assume that λ∗ < +∞. Then we have p∗ = p + λ∗ 1 ∈ D. However, (LS2loc ) implies p∗ + ε1 ∈ D for some ε > 0, a contradiction to the definition of λ∗ . Thus we have p + λ1 ∈ D for all λ ≥ 0. Similarly, we have p + λ1 ∈ D for all λ ≤ 0. We show the system of inequalities which describes the polyhedral structure of L-convex polyhedra. A function γ V × V → R ∪ +∞ with γv v = 0 ∀v ∈ V  is called a distance function. For a distance function γ, define the set Dγ ⊆ R V by V Dγ = p ∈ R  pv − pu ≤ γu v u v ∈ V 

(17)

V

Given a nonempty set D ⊆ R , the function γD V × V → R ∪+∞ is defined by γD u v = suppv − pu p∈D

(18)

Note that γD is a distance function, and D ⊆ DγD  holds in general. Given a distance function γ, we denote by Gγ the directed graph with the vertex set V and the arc set Aγ = u v  u v ∈ V u = v γu v < +∞, where the value γu v is regarded as the length of the arc u v ∈ Aγ . For u v ∈ V , let $ γ u v be the length of a shortest path from u to v in Gγ , i.e.,
    $ γ u v = inf γu  v   P ⊆ Aγ  P path from u to v u v ∈P

It is well known that $ γ > −∞ if and only if Gγ has no negative cycle. We consider the triangle inequality γv1  v2  + γv2  v3  ≥ γv1  v3 

∀v1  v2  v3 ∈ V 

(19)

for a distance function, and let  be the family of distance functions with triangle inequality, i.e.,  = γ  γ V × V → R ∪ +∞ γv v = 0 v ∈ V  γ satisfies (19) Note that for any distance function γ V × V → R ∪ +∞, if $ γ > −∞ then $ γ ∈  , and if γ ∈  then $ γ = γ.

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Lemma 3.21. Let γ V × V → R ∪ +∞ be a distance function. (i) (ii) (iii)

Dγ =  ⇐⇒ Gγ has no negative cycle. γ . Dγ =  ⇒ Dγ = D$ Dγ =  ⇒ Dγ ∈ 0 .

(iv)

If γ ∈  , then Dγ =  and γ = γDγ . (v) If γu v ∈ Z ∪ +∞ ∀u v ∈ V , then Dγ is an integral polyhedron, i.e., Dγ = Dγ ∩ ZV . Proof.

(i), (ii)

These are well-known facts. See, e.g., [20, 23].

(iii) Condition (LS2) for Dγ is obvious. To show (LS1) for Dγ, we take p q ∈ Dγ. Then, it is easy show that p ∧ qv − p ∧ qu ≤ γu v

p ∨ qv − p ∨ qu ≤ γu v

for any u v ∈ V , i.e., p ∧ q p ∨ q ∈ D. (iv) Since γ ∈  , the graph Gγ has no negative cycle and therefore Dγ =  by (i). We have γu v = $ γ u v = sup pv − pu = γDγ u v p∈Dγ

where the second equality is by a fundamental result concerning the shortest path problem. (v) The claim is obvious from the total unimodularity of the coefficient matrix of the system of inequalities in (17). Lemma 3.22. For any nonempty polyhedron D ⊆ R V , we have γD ∈  . Moreover, if D ∈ 0 , then DγD  = D. Proof. The first claim is clear from the following inequality: for any v1  v2  v3 ∈ V , we have γD v1  v2  + γD v2  v3  = suppv2  − pv1  + suppv3  − pv2  p∈D

p∈D

≥ suppv3  − pv1  = γD v1  v3  p∈D

Suppose D ∈ 0 . We show that DγD  ⊆ D. Let q ∈ DγD . For any u v ∈ V there exists puv ∈ D with puv v − puv u ≥ qv − qu, where we may assume that puv u  = qu and puv v ≥ qv by (LS2). For each u ∈ V , the vector pu = v∈V p uv ∈ D satisfies pu u = qu, pu v ≥ qv ∀v ∈ V . Therefore, q = u∈V pu ∈ D. Theorem 3.23. (i) (ii)

For D ∈ 0 , we have γD ∈  and DγD  = D.

For γ ∈  , we have Dγ ∈ 0 and γDγ = γ.

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(iii) The mappings D (→ γD D ∈ 0  and γ (→ Dγ γ ∈   provide a one-to-one correspondence between 0 and  and are the inverse of each other. Proof.

This is immediate from Lemma 3.21(iii), (iv), and Lemma 3.22.

As a special case of L-convex polyhedra, we investigate the class of L-convex cones, which are conic L-convex polyhedra. This amounts to restricting the function values of γ to 0 +∞. We show below the relationship of L-convex cones with “transitive” directed graphs, and with distributive lattices. A directed graph G = V A is called transitive if for any u v v w ∈ A we have u w ∈ A. Given a set of vectors D ⊆ R V , we define the directed graph GD = V AD  by AD = u v  u v ∈ V pu ≥ pv ∀p ∈ D For a directed graph G = V A, the set DG ⊆ R V  is defined by DG = p ∈ R V  pu ≥ pv ∀u v ∈ A The next theorem is an immediate consequence of Theorem 3.23. Theorem 3.24. (i) For any L-convex cone D ⊆ R V , GD = V AD  is transitive and DGD = D.

(ii) For any transitive directed graph G = V A, DG ⊆ R V  is an L-convex cone and GDG = G. (iii) The mappings D (→ GD and G (→ DG provide a one-to-one correspondence between L-convex cones D ⊆ R V  and transitive directed graphs G = V A and are the inverse of each other. On the other hand, Birkhoff’s representation theorem [2] yields a oneto-one correspondence between transitive directed graphs G = V A and distributive lattices  ⊆ 2V  with  V  ⊆  . This fact, together with Theorem 3.24, provides a one-to-one correspondence between L-convex cones and distributive lattices, as follows. Given a set of vectors D ⊆ R V , we define D ⊆ 2V by D = X ⊆ V  χX ∈ D For any family of subsets  ⊆ 2V with  V  ⊆  , define D ⊆ R V by
  λX χX  λX ≥ 0 X ∈  \ V  D = X∈

Theorem 3.25. (i) For an L-convex cone D ⊆ R V , D is a distributive lattice with  V  ⊆ D , and DD = D.

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(ii) For a distributive lattice  ⊆ 2V with  V  ⊆  , D is an L-convex cone and D =  . Moreover, for any p ∈ D there exists a chain 0 with  V  ⊆  0 ⊆  and λX ∈ R X ∈ 0  such that λX ≥ 0 ∀X ∈ 0 \ V  and p = X∈0 λX χX . (iii) The mappings D (→ D and  (→ D provide a one-to-one correspondence between L-convex cones D ⊆ R V  and distributive lattices  ⊆ 2V  with  V  ⊆  and are the inverse of each other. The intersection of L-convex polyhedra is also L-convex. Theorem 3.26. Let D1  D2 ∈ 0 . Define γ V × V → R ∪ +∞ by γu v = minγD1 u v γD2 u v u v ∈ V . Then, (i)

D1 ∩ D2 = Dγ,

(ii)

D1 ∩ D2 =  ⇐⇒ Gγ has no negative cycle,

(iii)

D1 ∩ D2 =  ⇒ D1 ∩ D2 ∈ 0 .

Proof. The claim (i) follows from D1 ∩ D2 = DγD1  ∩ DγD2  = Dγ. The claims (ii) and (iii) are by (i) and (iii) of Lemma 3.21, respectively. Remark 3.27. The Minkowski-sum of two L-convex polyhedra is not necessarily an L-convex polyhedron, as shown below. Put V = a b c d. Let γ1  γ2 V × V → 0 1 be such that γ1 u v = 1 ⇐⇒ u v ∈ c a c b d a d b d c γ2 u v = 1 ⇐⇒ u v ∈ b a b c d a d b d c Since γ1  γ2 ∈  , we have Dγ1  Dγ2  ∈ 0 by Lemma 3.21. However, D = Dγ1  + Dγ2  is not L-convex. Let p1 = 1 1 0 0

p2 = q1 = 0 0 0 0

q2 = 1 0 1 0

Then, it holds that pi  qi ∈ Dγi  i = 1 2. Hence, we have p1 + p2  q1 + q2 ∈ D, but p1 + p2  ∧ q1 + q2  = 1 0 0 0 ∈ D. Note that p ∈ Dγi   pd = 0 = convSi 

i = 1 2

p ∈ D  pd = 0 = convS1 + S2  where S1 = 0 0 0 0 1 1 0 0 1 1 1 0 S2 = 0 0 0 0 1 0 1 0 1 1 1 0 Separation theorems of the following form hold for distance functions with triangle inequality and for L-convex polyhedra.

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Theorem 3.28. Let γ V × V → R ∪ +∞, ν V × V → R ∪−∞ be functions such that γ −ν ∈  . Suppose that k  i=1

νv2i+1  v2i  ≤

k  i=1

γv2i−1  v2i 

holds for any distinct elements vi 2k i=1 ⊆ V with v2k+1 = v1 . Then, there exists p∗ ∈ R V with νu v ≤ p∗ v − p∗ u ≤ γu v

∀u v ∈ V 

(20)

Moreover, if γ and ν are integer-valued, then there exists such p∗ ∈ ZV . Proof. The existence of p∗ ∈ R V with (20) is equivalent to Dν 0  ∩ Dγ = , where ν 0 V × V → R ∪ +∞ is defined by ν 0 u v = −νv u u v ∈ V . Define η V × V → R ∪ +∞ by ηu v = minγu v ν 0 u v u v ∈ V . By Theorem 3.26 (ii), it suffices to show that Gη has no negative cycle. Suppose, to the contrary, that v1  v2   vh  ⊆ V h ≥ 1 forms a negative cycle. Since γ and ν 0 satisfy the triangle inequality, we may assume that h = 2k k ≥ 1 and ηv2i−1  v2i  = γv2i−1  v2i 

ηv2i  v2i+1  = −νv2i+1  v2i  i = 1 2  k

where v2k+1 = v1 . This implies ki=1 νv2i+1  v2i  > ki=1 γv2i−1  v2i , a contradiction. The integrality assertion follows from the argument above and Lemma 3.21 (v). Remark 3.29. The assumption in Theorem 3.28 cannot be replaced by the weaker condition νu v ≤ γu v

∀u v ∈ V 

as shown in the following example. Put V = a b c d, and define γ V × V → R , ν V × V → R as γ a b c d

a 0 2 1 1

b c d 0 1 −1 0 1 1 −1 0 0 1 2 0

a b c ν a 0 0 −1 b 0 0 −1 c −1 −1 0 0 d −1 −1

d −1 −1 0 0

where it reads as γa d = −1, γb a = 2, etc. We can easily check that γ −ν ∈  and νu v ≤ γu v ∀u v ∈ V . However, there is no p∗ ∈ R V

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satisfying (20). Suppose, to the contrary, that such p∗ ∈ R V exists. Then we have p∗ a − p∗ b ≤ minγb a −νa b = 0 p∗ b − p∗ c ≤ minγc b −νb c = −1 p∗ c − p∗ d ≤ minγd c −νc d = 0 p∗ d − p∗ a ≤ minγa d −νd a = −1 Addition of these inequalities yields 0 ≤ −2, a contradiction. Note that the assumption of Theorem 3.28 is not satisfied since νa b + νc d = 0 > −2 = γc b + γa d. The latter part of the following theorem is already shown in [34, Theorem 5.6]. Theorem 3.30. Let D1  D2 ∈ 0 . If D1 ∩ D2 = , then there exists x∗ ∈ 0 ±1V such that inf p x∗  − sup p x∗  > 0

p∈D1

p∈D2

(21)

Moreover, if D1 and D2 are integral polyhedra, i.e., Di = Di ∩ ZV i = 1 2, then “> 0” in (21) can be replaced by “≥ 1.” Proof. Put γi = γDi i = 1 2. Since Dγ1  ∩ Dγ2  = , it follows from Theorem 3.28 that there distinct elements

kexists a sequence of k vi 2k ⊆ V k ≥ 1 such that γ v  v  + γ 2i i=1 1 2i−1 i=1 2 v2i  v2i+1  < 0, i=1 where v2k+1 = v1 . Let x∗ ∈ 0 ±1V be a vector such that x∗ v2i  = −1, x∗ v2i−1  = 1 i = 1  k, and x∗ w = 0 otherwise. Then we have inf p x∗  − sup p x∗ 

p∈D1

p∈D2

= inf

p∈D1

≥−

k  i=1

k  i=1

pv2i−1  − pv2i  − sup

γ1 v2i−1  v2i  −

k 

p∈D2 i=1

k  i=1

pv2i+1  − pv2i 

γ2 v2i  v2i+1  > 0

The integrality assertion is obvious from the inequality above.

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3.3.2. L- -Convex Polyhedra Due to the property (LS2), an L-convex polyhedron loses no information when restricted to a hyperplane p ∈ R V  pv = 0 for any v ∈ V . We  call a polyhedron P ⊆ R V L- -convex if the set P˜ ⊆ R V defined by P˜ = λ p + λ1  p ∈ P λ ∈ R   = v0  ∪ V . We see that L- -convex is an L-convex polyhedron, where V polyhedra are essentially the same as L-convex polyhedra, while the class of L- -convex polyhedra properly contains that of L-convex polyhedra. Intervals are the only polyhedra that are both L- - and M- -convex. Theorem 3.31 (cf. [38, Lemma 5.7]). A polyhedron S ⊆ R V is both polyhedral M- -convex and polyhedral L- -convex if and only if S is an interval; i.e., it is represented as S = "a b# for some a V → R ∪−∞ and b V → R ∪+∞ with av ≤ bv v ∈ V . Proof. Consider the polyhedral description of S in terms of linear inequalities of the form p x ≤ α. For an L- -convex set, p = χv − χu or p = ±χv for some elements u v ∈ V by Theorem 3.23 (i), while for M- -convex set, p = ±χX for some subset X ⊆ V by (13). Therefore, S is an interval. Remark 3.32. There exists no polyhedron which is both M-convex and L-convex, i.e., 0 ∩ 0 =  (Proof. If S ∈ 0 ∩ 0 , then Theorem 3.2 implies xV  = yV  for any x y ∈ S, whereas (LS2) implies that x + λ1 ∈ S for any λ ∈ R .) 4. POLYHEDRAL M-CONVEX AND L-CONVEX FUNCTIONS 4.1. Review of Fundamental Results on Polyhedral Convex Functions This section is devoted to a summary of the relevant results on polyhedral convex functions. See [41, 44] for more accounts. Let f R V → R ∪ ±∞ be a function. The epigraph of f , denoted by epi f , is the set epi f = x α  x ∈ R V  α ∈ R  α ≥ f x

⊆ R V × R 

The effective domain of f , denoted by dom f , is the set dom f = x ∈ R V  −∞ < f x < +∞ We call f convex if epi f is a convex set in R V × R . A function f is called concave if −f is convex. When f > −∞, f is convex if and only if f αx + 1 − αy ≤ αf x + 1 − αf y

∀x y ∈ R V  ∀α ∈ "0 1#

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A convex function is said to be polyhedral if its epigraph is polyhedral. Let f R V → R ∪±∞. The (convex) conjugate f • R V → R ∪±∞ of f is defined by f • p = sup p x − f x x∈R V

p ∈ R V 

(22)

Theorem 4.1 [41, Theorem 19.2]. (i) For a polyhedral convex function f R V → R ∪ +∞ with dom f = , its conjugate function f • is also polyhedral convex with dom f • = , f • > −∞, and f •• = f . (ii) The mapping f (→ f • induces a symmetric one-to-one correspondence in the class of polyhedral convex functions f R V → R ∪ +∞ with dom f = . A function f R V → R ∪ +∞ is called positively homogeneous if f αx = αf x for any x ∈ R V and α > 0. Note that f 0 = 0 if dom f = . is a posiTheorem 4.2 [41, Corollary 4.7.1]. If f R V → R ∪ +∞

tively homogeneous convex function, then f  ki=1 αi xi  ≤ ki=1 αi f xi  for any xi ∈ R V and αi ≥ 0 i = 1  k. For any nonempty set S ⊆ R V , we define the support function δ∗S R V → R ∪ +∞ of S by δ∗S p = supp x x∈S

p ∈ R V 

(23)

For any positively homogeneous function f R V → R ∪ +∞, we define the set Sf ⊆ R V by Sf = x ∈ R V  p x ≤ f p ∀p ∈ R V 

(24)

Theorem 4.3 [41, Sect. 13]. (i) For any nonempty polyhedron S ⊆ R V , δ∗S is a positively homogeneous polyhedral convex function with dom δ∗S = , and S = Sδ∗S . (ii) For any positively homogeneous polyhedral convex function f R V → R ∪ +∞ with dom f = , Sf is a nonempty polyhedron, and f = δ∗Sf . (iii) The mappings S (→ δ∗S and f (→ Sf provide a one-to-one correspondence between nonempty polyhedra S ⊆ R V  and positively homogeneous polyhedral convex functions f R V → R ∪ +∞ with dom f =  and are the inverse of each other.

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Let f R V → R ∪±∞. For any x ∈ dom f and y ∈ R V , we define the (one-sided) directional derivative of f at x with respect to y by f  x+ y = lim α↓0

f x + αy − f x α

if the limit exists. Note that if f is polyhedral convex, then f  x+ y is well-defined for any x ∈ dom f and y ∈ R V by f  x+ y = f x + αy − f x/α with a sufficiently small α > 0. The subdifferential ∂f x of f at x is defined by ∂f x = p ∈ R V  f y ≥ f x + p y − x ∀y ∈ R V  and each vector in ∂f x is called a subgradient of f at x. Theorem 4.4 [41, Theorem 23.10]. Let f R V → R ∪ +∞ be a polyhedral convex function with dom f = , and x ∈ dom f . Then, ∂f x is a nonempty polyhedral convex set, and f  x+ · is a positively homogeneous polyhedral convex function such that dom f  x+ · = , f  x+ · > −∞, and f  x+ · = δ∗∂f x . Let f R V → R ∪ +∞ be any function. We denote by arg min f the set of minimizers of f , i.e., arg min f = x ∈ R V  f x ≤ f y ∀y ∈ R V  Note that arg min f can be empty. For any p ∈ R V , the function f "p# R V → R ∪ +∞ is defined by f "p#x = f x + p x

x ∈ R V 

(25)

Theorem 4.5 [41, Theorem 23.5]. Let f R V → R ∪ +∞ be a polyhedral convex function with dom f = . For any x ∈ R V and p ∈ R V , we have p ∈ ∂f x ⇐⇒ x ∈ arg min f "−p# ⇐⇒ x ∈ ∂f • p ⇐⇒ p ∈ arg min f • "−x# For fi R V → R ∪ +∞ i = 1 2, the infimum convolution (or simply the convolution) f1 f2 R V → R ∪ ±∞ is defined by    x1  x2 ∈ R V   x ∈ R V  (26) f1 f2 x = inf f1 x1  + f2 x2    x1 + x2 = x It is easy to verify that if f1 domf1

f2 does not take the value −∞, then f2  = dom f1 + dom f2

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Theorem 4.6 [41, Sects. 16, 19, and 20]. Let fi R V → R ∪ +∞ i = 1 2 be polyhedral convex functions with dom fi = . Then, f1 f2 is also polyhedral convex and f1

f2 • = f1• + f2•

Moreover, if dom f1 ∩ dom f2 =  then it holds that f1 + f2 • = f1•

f2•

We consider the special case of one-dimensional functions. A function f R → R ∪ ±∞ is called piecewise-linear if dom f is a closed set and can be divided into a finite number of subintervals on each of which f is linear. We denote by
1 the class of piecewise-linear convex functions (cf. (5)). Note that a piecewise-linear convex function is nothing but a one-dimensional polyhedral convex function. For f R → R ∪ ±∞ and x ∈ dom f , we denote f+ x = f  x+ 1, f− x = −f  x+ −1. If f is piecewise-linear, then the values f+ x and f− x are well-defined for any x ∈ dom f and lim f+ y = lim f− y = f+ x y↓x

y↓x

lim f+ y = lim f− y = f− x y↑x

y↑x

4.2. Polyhedral M-Convex Functions A polyhedral convex function f R V → R ∪+∞ is called M-convex if dom f =  and f satisfies (M-EXC): (M-EXC) ∀x y ∈ dom f , ∀u ∈ supp+ x − y, ∃v ∈ supp− x − y, ∃α0 > 0 such that f x + f y ≥ f x − αχu − χv  + f y + αχu − χv 

0 ≤ ∀α ≤ α0 

We denote  = f  f R V → R ∪ +∞ polyhedral M-convex  0

= f  f R V → R ∪ +∞ positively homogeneous polyhedral M-convex 

It may be obvious from the definition that polyhedral M-convex functions are a quantitative extension of M-convex polyhedra. Theorem 4.7. (i) ⇐⇒ dom f ∈ 0 . (ii)

For a function f R V → 0 +∞, we have f ∈ 

For f ∈ , we have dom f ∈ 0 .

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4.2.1. Axioms for Polyhedral M-Convex Functions We consider two slightly different exchange axioms, where the former is weaker and the latter is stronger than (M-EXC). (M-EXCw ) ∀x y ∈ dom f , ∀u ∈ supp+ x − y, ∃v ∈ supp− x − y, ∃α > 0 such that f x + f y ≥ f x − αχu − χv  + f y + αχu − χv  (M-EXCs ) ∀x y ∈ dom f , ∀u ∈ supp+ x − y, ∃v ∈ supp− x − y such that f x + f y ≥ f x − αχu − χv  + f y + αχu − χv  for any α ∈ R with 0 ≤ α ≤ xu − yu/2k, where k =  supp− x − y. Theorem 4.8. For a polyhedral convex function f R V → R ∪ +∞ with dom f = , (M-EXC) ⇐⇒ (M-EXCw ). Proof. We show (M-EXCw ) ⇒ (M-EXC) only. Let x y ∈ dom f and u ∈ supp+ x − y. Then, there exist v ∈ supp− x − y, α0 > 0 such that f x + f y ≥ f x − αχu − χv  + f y + αχu − χv 

(27)

for α = α0 . The convexity of f implies (27) for any α ∈ "0 α0 #. Lemma 4.9. Let f ∈ , x y ∈ dom f be distinct vectors, u ∈ supp+ x − y, and supp− x − y = v1  v2   vk  k =  supp− x − y. Then, there exist sequences xi ki=0  yi ki=0 ⊆ dom f  and αi ki=1 ⊆ R+  satisfying

k i=1 αi = xu − yu/2 and the conditions x0 = x y0 = y xi = xi−1 − αi χu − χvi  yi = yi−1 + αi χu − χvi  i = 1  k (28) f xi−1  + f yi−1  ≥ f xi  + f yi 

i = 1  k

(29)

Proof. Put x0 = x, y0 = y. For i = 1 2  k, we define αi ∈ R and xi  yi ∈ R V iteratively by the equations αi = supα ∈ R+  α ≤ minxi−1 u − yi−1 u yi−1 vi  − xi−1 vi /2 f xi−1 − αχu − χvi  + f yi−1 + αχu − χvi  ≤ f xi−1  + f yi−1  xi = xi−1 − αi χu − χvi 

yi = yi−1 + αi χu − χvi 

Then, the sequences xi ki=0 , yi ki=0 , and αi ki=1 satisfy the conditions (28)

and (29). In the following, we show ki=1 αi = xu − yu/2.

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Assume, to the contrary, that ki=1 αi < xu − yu/2. Since u ∈ supp+ xk − yk , there exist some vi ∈ supp− xk − yk  ⊆ supp− x − y and α0 > 0 such that f xk  + f yk  ≥ f xk − αχu − χvi  + f yk + αχu − χvi 

(30)

for any α ∈ "0 α0 #. Here, vi = vk holds by the choice of xk and yk . By Theorem 4.14 to be shown later, we have f xi − αχu − χvi  − f xi  + f yi + αχu − χvi  − f yi  ≤ f xk − αχu − χvi  − f xk  + f yk + αχu − χvi  − f yk  ≤ 0 for any α ∈ "0 α0 #, where the second inequality is by (30). This contradicts the definitions of xi and yi . Theorem 4.10. For a polyhedral convex function f R V → R ∪ +∞ with dom f = , (M-EXC) ⇐⇒ (M-EXCs ). Proof. It suffices to prove (M-EXC) ⇒ (M-EXCs ). Assume (M-EXC) for f . Let x y ∈ dom f be distinct vectors, and u ∈ supp+ x − y. By Lemma 4.9, there exist sequences xi ki=0  yi ki=0 ⊆ dom f  and

αi ki=1 ⊆ R+  satisfying ki=1 αi = xu − yu/2, (28), and (29). Let i∗ be an integer with 1 ≤ i∗ ≤ k such that αi∗ = max αi ≥ xu − yu/2k 1≤i≤k

From Theorem 4.14 to be shown later, we obtain the inequalities f x − αi∗ χu − χvi  − f x + f y + αi∗ χu − χvi  − f y ∗

∗

≤ f xi∗  − f xi∗ −1  + f yi∗  − f yi∗ −1  ≤ 0 where the second inequality is by (29). From the convexity of f follows f x + f y ≥ f x − αχu − χvi  ∗

+ f y + αχu − χvi  ∗

∀α ∈ "0 αi∗ #

We can rewrite (M-EXC) in terms of directional derivatives. For a polyhedral convex function f R V → R ∪ +∞ with dom f =  and x ∈ dom f , define f  x+ · · V × V → R ∪ +∞ by f  x+ v u = f  x+ χv − χu 

u v ∈ V 

Note that f x − αχu − χv  − f x = αf  x+ v u for sufficiently small α > 0. We consider (M-EXC ):

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(M-EXC ) ∀x y ∈ dom f , ∀u ∈ supp+ x − y, ∃v ∈ supp− x − y such that f  x+ v u + f  y+ u v ≤ 0 Theorem 4.11. For a polyhedral convex function f R V → R ∪+∞ with dom f = , (M-EXC) ⇐⇒ (M-EXC  ). 4.2.2. Fundamental Properties of Polyhedral M-Convex Functions In this section, we show various properties of polyhedral M-convex functions. Global optimality of a polyhedral M-convex function is characterized by local optimality. Theorem 4.12. Let f ∈  and x ∈ dom f . Then, f x ≤ f y ∀y ∈ R V  ⇐⇒ f  x+ v u ≥ 0 ∀u v ∈ V . Proof. We show the ⇐ direction only. Assume, to the contrary, that f x0  < f x holds for some x0 ∈ R V . Put S = y ∈ R V  f y ≤ f x0 . Let x∗ ∈ S be a vector such that !x∗ − x!1 = inf y∈S !y − x!1 . By (M-EXC) applied to x x∗ and some u ∈ supp+ x − x∗ , there exists v ∈ supp− x − x∗  and a sufficiently small α > 0 such that f x∗  − f x∗ + αχu − χv  ≥ f x − αχu − χv  − f x ≥ αf  x+ v u ≥ 0 Hence, we have f x∗ + αχu − χv  ≤ f x∗  ≤ f x0 , which contradicts the choice of x∗ since !x∗ + αχu − χv  − x!1 = !x∗ − x!1 − 2α. A polyhedral M-convex function has supermodularity when projected onto the hyperplane x ∈ R V  xw0  = 0 w0 ∈ V , as stated in Theorem 4.14. Lemma 4.13. Let f ∈ , w0 ∈ V , and x ∈ R V . Then, for any u v ∈ V and λ µ ≥ 0 we have f x + λχu − χw0  + f x + µχv − χw0  ≤ f x + f x + λχu − χw0  + µχv − χw0 

(31)

Proof. Fix u v ∈ V and λ µ ≥ 0. For λ  µ ∈ R+ , we denote xλ  µ  = x + λ χu − χw0  + µ χv − χw0 . We may assume that x0 0 xλ µ ∈ dom f . Then, it follows from Theorem 3.9 that xλ  0 xλ  µ ∈ dom f

0 ≤ ∀λ ≤ λ

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Define the functions f1  f2 "0 λ# → R by f1 λ  = f xλ  0

f2 λ  = f xλ  µ 0 ≤ λ ≤ λ

Let γ be any value with 0 < γ ≤ λ. For any β ∈ "0 γ, (M-EXC  ) yields f1 + β − f2 − γ ≤ 0 since supp+ xγ µ − xβ 0 = u v and supp− xγ µ − xβ 0 = w0 . This implies f2 − f1 − γ = f2 − γ − f1 − γ = f2 − γ − limf1 + β ≥ 0 β↑γ

for any γ with 0 < γ ≤ λ. Therefore, the piecewise-linear function f2 − f1 is nondecreasing on "0 λ#, from which we have f xλ µ − f xλ 0 = f2 λ − f1 λ ≥ f2 0 − f1 0 = f x0 µ − f x0 0 i.e., the inequality (31) holds. Theorem 4.14. Let f ∈ , w0 ∈ V , and x ∈ R V . For any X Y ⊆ V \ w0  with X ∩ Y =  and λw ≥ 0 w ∈ X ∪ Y , we have 

f x +

w∈X

λw χw − χw0  + f x +

≤ f x + f x +

 w∈X∪Y

 w∈Y

λw χw − χw0 

λw χw − χw0 

(32)

Proof. We show the inequality (32) by induction on the cardinality of the set X ∪ Y . It suffices to consider the case when X =  and Y = . Note that the case when X = Y  = 1 is already shown in Lemma 4.13. Therefore, we may

assume that Y  ≥ 2. Let v ∈ Y . We assume that x ∈ dom f and x + w∈X∪Y λw χw − χw0  ∈ dom f . Then we have x = x + λv χv − χw0  ∈ dom f by Theorem 3.9. Hence, the inductive assumption yields f x +

 w∈X

λw χw − χw0  − f x

≤ f x +

 w∈X∪v

≤ f x +

 w∈X∪Y

λw χw − χw0  − f x + λv χv − χw0 

λw χw − χw0  − f x +

 w∈Y

λw χw − χw0 

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Directional derivative functions and subdifferentials of a polyhedral M-convex function have nice structures such as M/L-convexity, and they can be explicitly described by certain distance functions with triangle inequality (cf. Theorem 4.15(i)). For any distance function γ V × V → R ∪ +∞ (i.e., γv v = 0 for all v ∈ V ), we define fγ R V → R ∪ ±∞ by        λuv χv − χu  = x   x ∈ R V  (33) λ γu v  uv∈V fγ x = inf  uv∈V uv  λ ≥ 0 u v ∈ V  uv

Theorem 4.15. Let f ∈  and x ∈ dom f . (i)

The function γx V × V → R ∪ +∞ defined by γx u v = f  x+ v u

u v ∈ V 

(34)

satisfies γx v v = 0 ∀v ∈ V  and the triangle inequality (19), i.e., γx ∈  . (ii) We have f  x+ · = fγx and f  x+ · ∈ 0 . Proof.

(i)

For u v w ∈ V and a sufficiently small α > 0, we have

αγx u v + αγx v w = f x + αχv − χu  − f x + f x + αχw − χv  − f x ≥ f x + αχw − χu  − f x = αγx u w where the inequality is by Lemma 4.13. Obviously, γx is a distance function. (ii) The proof is postponed after Theorem 4.19. L-convexity appears in subdifferentials of a polyhedral M-convex function. Theorem 4.16. Let f ∈  and x ∈ dom f . (i)

∂f x ∈ 0 and ∂f x is represented as

∂f x = Dγx  = p ∈ R V  pv − pu ≤ f  x+ v u u v ∈ V  (35) where γx V × V → R ∪ +∞ is given by (34). (ii) For any y ∈ R V we have f y − f x ≥ supp∈∂f x p y − x = fγx y − x. Proof. Equation (35) follows from Theorems 4.5 and 4.12. The L-convexity of ∂f x is immediate from (35) and Theorem 4.15(i). The claim (ii) is immediate from (i) and the linear programming duality.

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The next theorem shows that each face of the epigraph of a polyhedral M-convex function is an M-convex polyhedron when it is projected to R V . The proof is obvious and therefore omitted. Theorem 4.17. inf f "−p# > −∞.

For f ∈  and p ∈ R V , we have arg min f "−p# ∈ 0 if

The class of polyhedral M-convex functions is closed under various fundamental operations. Let f R V → R ∪ +∞ be a function. For any subset U ⊆ V , we define fU R U → R ∪ +∞ by fU y = f y 0V \U 

y ∈ R U 

(36)

Theorem 4.18. Let f f1  f2 ∈ . (1) For p ∈ R V , the function f "−p# R V → R ∪ +∞ given by (25) is polyhedral M-convex. (2) For a ∈ R V and ν > 0, the functions ν · f a − x and ν · f a + x are polyhedral M-convex in x. (3) For any U ⊆ V , the function fU R U → R ∪ +∞ is polyhedral M-convex if dom fU = . (4) For ϕv ∈
1 v ∈ V , the function f˜ R V → R ∪ +∞ defined by  f˜x = f x + ϕv xv x ∈ R V  v∈V

is polyhedral M-convex if dom f˜ = . (5) For any a V → R ∪ −∞ and b V → R ∪ +∞ with a ≤ b, the restriction f"ab# R V → R ∪ +∞ of f defined by
f x x ∈ "a b#, f"ab# x = (37) +∞ x ∈ "a b# is polyhedral M-convex if dom f ∩ "a b# = . (6) The convolution f1 f2 R V → R ∪ ±∞ defined by (26) is polyhedral M-convex, provided f1 f2 x0  is finite for some x0 ∈ R V . (7) For U ⊆ V , the function fˆ R × R U → R ∪ ±∞ defined by fˆy0  y = inff x  xv = yv v ∈ U y0 = xV \ U

(38)

is polyhedral M-convex if fˆy0  y   is finite (i.e., −∞ < fˆy0  y   < +∞) for some y0  y   ∈ R × R U . Proof. Parts (1) to (5) are easy to prove. The proofs of (6) and (7) are given in Section 5 and subsection 7.3, respectively. A more general and stronger transformation, called “network induction,” is explained in subsection 7.3.

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4.2.3. Positively Homogeneous Polyhedral M-Convex Functions This section clarifies the relationship of positively homogeneous polyhedral M-convex functions to distance functions with triangle inequalities, and also to L-convex polyhedra. For a positively homogeneous polyhedral convex function f R V → R ∪ +∞ with 0 ∈ dom f , define γf V × V → R ∪ +∞ by γf u v = f  0+ v u = f χv − χu 

u v ∈ V 

(39)

Recall the definition of fγ in (33). Theorem 4.19. (ii)

(i) For f ∈ 0 , we have γf ∈  and fγf = f .

For γ ∈  , we have fγ ∈ 0  and γfγ = γ.

(iii) The mappings f (→ γf f ∈ 0  and γ (→ fγ γ ∈   provide a one-to-one correspondence between 0  and  and are the inverse of each other. Proof. (i) By Theorem 4.15 (i) and Theorem 4.16 (ii), we have γf ∈ 

and f ≥ fγf . For any x ∈ R V and λuv uv∈V satisfying uv∈V λuv χv − χu  = x and λuv ≥ 0 u v ∈ V , Theorem 4.2 implies that   λuv γf u v = λuv f χv − χu  u v∈V

u v∈V



≥f

 u v∈V

 λuv χv − χu  = f x

Hence, we have fγf = f . (ii) Polyhedral M-convexity of fγ can be shown as a special case of Example 2.4. To be more precise, we have fγ x = f x ∀x ∈ R V  for the function f in Example 2.4, where T = V , A = u v  u v ∈ V u = v, and
γu vξ ξ ≥ 0 fu v ξ = u v ∈ A +∞ ξ < 0 For any u v ∈ V , we have γfγ u v = fγ χv − χu , which is equal to the shortest path distance from u to v in the directed graph G = V A, where A = V × V and the length of an arc u v ∈ A is given by γu v. Since γ ∈  , it holds γfγ u v = γu v ∀u v ∈ V . (iii)

This is clear from (i) and (ii).

We now prove the polyhedral M-convexity of the directional derivative function f  x+ ·.

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Proof of Theorem 4.15(ii). By Theorems 4.15(i) and 4.19, it suffices to show f  x+ · = fγx , which can be done in the same way as the proof of Theorem 4.19(i). As immediate consequences of Theorems 3.23, 4.3, and 4.19, we obtain a one-to-one correspondence between positively homogeneous polyhedral M-convex functions and L-convex polyhedra. Recall the notation Sf and δ∗S in (23) and (24). Theorem 4.20. (i)

For f ∈ 0 , we have Sf ∈ 0 and δ∗Sf = f .

(ii) For D ∈ 0 , we have δ∗D ∈ 0  and Sδ∗D = D. (iii) The mappings f (→ Sf f ∈ 0  and D (→ δ∗D D ∈ 0  provide a one-to-one correspondence between 0  and 0 and are the inverse of each other. 4.2.4. Polyhedral M- -Convex Functions We introduce a variant of polyhedral M-convex functions, called polyhedral M- -convex functions. From Theorems 3.2 and 4.7(ii), we see that the effective domain of a polyhedral M-convex function is contained in a hyperplane x ∈ R V  xV  = r for some r ∈ R . Therefore, no information is lost when a polyhedral M-convex function is projected onto a (V −1)dimensional space. A function f R V → R ∪ +∞ is called polyhedral  M- -convex if the function f˜ R V → R ∪ +∞ defined by
f x x0 = −xV   f˜x0  x = x0  x ∈ R V  +∞ x0 = −xV   = v0  ∪ V . Polyhedral is a polyhedral M-convex function, where V M -convex functions are essentially equivalent to polyhedral M-convex functions, whereas the class of polyhedral M- -convex functions properly contains that of polyhedral M-convex functions. M- -convexity was originally introduced in [37] as a concept for functions defined over the integer lattice (see also [18]). Polyhedral M- -convex functions can be characterized by the following exchange property (cf. [37]): (M- -EXC) ∀x y ∈ dom f , ∀u ∈ supp+ x − y, either (i) or (ii) (or both) holds: (i) ∃v ∈ supp− x − y, ∃α0 > 0 such that f x + f y ≥ f x − αχu − χv  + f y + αχu − χv 

0 ≤ ∀α ≤ α0 

(ii) ∃α0 > 0 such that f x + f y ≥ f x − αχu  + f y + αχu 

0 ≤ ∀α ≤ α0 

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Theorem 4.21. Let f R V → R ∪ +∞ be a polyhedral convex function with dom f = . Then, f is M- -convex if and only if it satisfies (M- -EXC). Proof.

Proof is given in Section 5.

Example 4.22 (Separable-Convex Functions). Let  ⊆ 2V be a laminar family, i.e.,  is a family of subsets of V such that for any X Y ∈  at least one of X \ Y , Y \ X and X ∩ Y is empty. For fX ∈
1 X ∈ , the function f R V → R ∪ +∞ given by  f x = fX xX x ∈ R V  X∈

-

is polyhedral M -convex if dom f =  [6]. This can be proved similarly to Example 2.4. Let B ⊆ R V be an M- -convex polyhedron. For f0 ∈
1 and fv ∈
1 v ∈ V , the function f R V → R ∪ +∞ defined by   fv xv x ∈ B, f0 xV  + f x = v∈V +∞ x ∈ B is polyhedral M- -convex if dom f = . Another class of polyhedral M- -convex functions arises from the minimum cost flow problem, as in Example 2.4. From the definition of polyhedral M- -convex functions, every property of polyhedral M-convex functions can be restated in terms of polyhedral M- -convex functions, and vice versa. We present some properties below, which seem more natural when restated in terms of polyhedral M- -convex functions. For any function f R V → R ∪ +∞ and any subset U ⊆ V , we define the projection f U R U → R ∪ ±∞ of f to U as f U y = inf f y z z∈R V \U

y ∈ R U 

(40)

Theorem 4.23. For any polyhedral M- -convex function f R V → R ∪ +∞ and U ⊆ V , f U is polyhedral M- -convex if f U x0  is finite for some x0 ∈ R U . Proof.

This is obvious from Theorem 4.18(7).

Theorem 4.24. If f R V → R ∪ +∞ is polyhedral M- -convex, then it satisfies the supermodular inequality, f x + f y ≤ f x ∧ y + f x ∨ y Proof.

This is immediate from Theorem 4.14.

∀x y ∈ R V 

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A polyhedral convex function g R V → R ∪ +∞ is said to be L-convex if dom g =  and g satisfies (LF1) and (LF2): (LF1) gp + gq ≥ gp ∧ q + gp ∨ q ∀p q ∈ dom g, (LF2) ∃r ∈ R such that gp + λ1 = gp + λr ∀p ∈ dom g ∀λ ∈ R . We denote  = g  g R V → R ∪ +∞ polyhedral L-convex 0

= g  g R V → R ∪ +∞ positively homogeneous polyhedral L-convex

As is obvious from the definition, polyhedral L-convex functions are a quantitative generalization of L-convex polyhedra. Theorem 4.25. (i) For a function g R V → 0 +∞, g ∈  ⇐⇒ dom g ∈ 0 . (ii) For g ∈ , we have dom g ∈ 0 . 4.3.1. Axioms for Polyhedral L-Convex Functions We first show that (LF1) and (LF2) are equivalent to local properties (LF1loc ) and (LF2loc ): (LF1loc )

∀p ∈ dom g, ∃ε > 0 such that

gq1  + gq2  ≥ gq1 ∧ q2  + gq1 ∨ q2  (LF2loc )

∀q1  q2 ∈ N∞ p ε

∀p ∈ dom g, ∃ε > 0, ∃r ∈ R such that

q ∈ N∞ p ε q + λ1 ∈ N∞ p ε λ ∈ R  ⇒ gq + λ1 = gq + λr Theorem 4.26. Let g R V → R ∪ +∞ be a polyhedral convex function with dom g = . Then, (i) (LF1) ⇐⇒ (LF1loc ). (ii) (LF2) ⇐⇒ (LF2loc ). Proof. (i) We show (LF1loc ) ⇒ (LF1) only. Assume (LF1loc ) for g. Then, Theorem 3.20 (i) implies (LS1) for dom g. For p ∈ dom g, we denote by εp the value ε in (LF1loc ) associated with p. Let p p ∈ dom g and recall the definition of Box"p p # in subsection 3.1. Since Box"p p # ∩ dom g is a compact set, there exist q1  q2   qm ∈ Box"p p # ∩ dom g m ≥ 1 such that Box"p p # ∩ dom g ⊆ m i=1 N∞ qi  εqi . We show the submodular inequality gp + gp  ≥ gp ∧ p  + gp ∨ p 

(41)

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by induction on the integer m. We may assume m ≥ 2, since otherwise (41) holds immediately. For λ µ ∈ "0 1#, put pλ µ = p + λp ∨ p  − p + µp ∧ p  − p. Then, for w ∈ V we have   1 − λpw + λp w w ∈ supp+ p − p, pλ µw = pw w ∈ V \ suppp − p, (42)   1 − µpw + µp w w ∈ supp− p − p. Furthermore, for any λ µ ∈ "0 1# we have pλ µ ∈ convp p  p ∧ p  p ∨ p  ⊆ Box"p p # ∩ dom g We may assume that p ∈ N∞ q1  εq1 . Put p∗ = pλ∗  µ∗ , where λ∗ = supλ ∈ "0 1#  pλ 0 ∈ N∞ q1  εq1  µ∗ = supµ ∈ "0 1#  p0 µ ∈ N∞ q1  εq1  Then, we have p∗ ∈ N∞ q1  εq1 , implying gp + gp∗  ≥ gp ∧ p∗  + gp ∨ p∗ 

(43)

In the following, we assume λ∗ < 1 and µ∗ < 1, since the other cases can be shown similarly and more easily. Then, we have Box"p ∨ p∗  p ∨ p∗ # ∩ dom g ⊆ Box"p ∧ p∗  p ∧ p∗ # ∩ dom g ⊆ Box"p∗  p # ∩ dom g ⊆

m  i=2

m  i=2 m  i=2

N∞ qi  εqi  N∞ qi  εqi 

N∞ qi  εqi 

Hence, the inductive hypothesis implies that gp ∨ p∗  + gp ∨ p∗  ≥ gp∗  + gp ∨ p  gp ∧ p∗  + gp ∧ p∗  ≥ gp ∧ p  + gp∗  gp∗  + gp  ≥ gp∗ ∧ p  + gp∗ ∨ p  where we note that p ∨ p∗  ∧ p ∨ p∗  = p∗ 

p ∨ p∗  ∨ p ∨ p∗  = p ∨ p 

p ∧ p∗  ∧ p ∧ p∗  = p ∧ p 

p ∧ p∗  ∨ p ∧ p∗  = p∗

Combining the above inequalities with (43), we obtain the submodular inequality (41).

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(ii) We show (LF2loc ) ⇒ (LF2) only. Assume (LF2loc ) for g. Then, dom g satisfies (LS2) by Theorem 3.20(ii). Let xi ∈ R V and λi ∈ R i = 1  k be such that gp = max 1≤i≤k p xi  + λi  p ∈ dom g. As is shown later, the value 1 xi  is the same for all i. Thus, g satisfies (LF2). We now prove that 1 xi  = 1 xj  for any i j. Let p0 be any vector in dom g and define ψ R → R by ψµ = gp0 + µ 1 µ ∈ R . Then, ψ is piecewise-linear convex. Moreover, we have ψ+ µ+  = max 1≤i≤k 1 xi  for a sufficiently large value of µ+ , and ψ+ µ−  = min1≤i≤k 1 xi  for a sufficiently small value of µ− . By (LF2loc ), we see that ψ is a linear function. Hence, we have max 1≤i≤k 1 xi  = ψ+ µ+  = ψ+ µ−  = min1≤i≤k 1 xi . Theorem 4.27. Let a V → R ∪−∞ and b V → R ∪ +∞ be such that a ≤ b. Also, let g R V → R ∪ +∞ be a function with dom g = "a b#. have

(i)

g is submodular ⇐⇒ for any p ∈ R V , any u v ∈ V u = v, we gp + λχu  + gp + µχv  ≥ gp + gp + λχu + µχv 

(ii)

∀λ µ ≥ 0

(44)

The condition (44) can be replaced by gp + λχu  + gp + µχv  ≥ gp + gp + λχu + µχv 

(45)

for any λ ∈ "0 pl−1 − pl # and µ ∈ "0 pm−1 − pm #, where p1 > p2 > · · · > pk are distinct values in pvv∈V , and pu = pl , pv = pm . Proof. We show the “if” part of (i) only, and the other proofs are omitted. Let p q ∈ "a b#. Note that p ∧ q p ∨ q ∈ "a b#. We show the submodular inequality for p q by induction on the cardinality of suppp − q. Without loss of generality, we assume  supp− p − q ≥ 2. Let v ∈ supp− p − q, λ = qv − pv. Since p ∧ q + λχv ∈ "a b#, the inductive assumption and (44) imply that gp − gp ∧ q ≥ gp + λχv  − gp ∧ q + λχv  ≥ gp ∨ q − gq

4.3.2. Fundamental Properties of Polyhedral L -Convex Functions In this section, we show various properties of polyhedral L-convex functions. Lemma 4.28. Let g ∈ . Then, for any p q ∈ dom g and λ ∈ R we have gp + gq ≥ gp + λ1 ∧ q + gp ∨ q − λ1 In particular, we have gp + gq ≥ gp + λχX  + gq − λχX 

(46)

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for any p q ∈ dom g and λ ∈ "0 λ1 − λ2 #, where λ1 = max qv − pv

X = v ∈ V  qv − pv = λ1  (47)

v∈V

λ2 = max qv − pv

(48)

v∈V \X

Proof.

The inequality (46) can be obtained as LHS of 46 = gp + gq − λ1 + λr ≥ gp ∧ q − λ1 + gp ∨ q − λ1 + λr = gp ∧ q − λ1 + λ1 + gp ∨ q − λ1 = RHS of 46

The second inequality follows from this inequality since p ∨ q − λ1 − λ 1 = p + λχX 

p + λ1 − λ 1 ∧ q = q − λχX

for λ ∈ "0 λ1 − λ2 #. Global optimality of a polyhedral L-convex function is characterized by local optimality. Theorem 4.29. Let g ∈  and p ∈ dom g. Then, gp ≤ gq ∀q ∈ R  ⇐⇒ g p+ χX  ≥ 0 ∀X ⊆ V  and g p+ χV  = r = 0, where r is in (LF2). V

Proof. We show the sufficiency by contradiction. Suppose that there exists some q ∈ R V with gq < gp and assume that q minimizes the number of distinct values in pv − qvv∈V of all such vectors. Define λ1  λ2 , and X ⊆ V  by (47) and (48). Since g p+ χV  = r = 0, we have X = V and λ2 > −∞. By Lemma 4.28, we obtain gp + gq ≥ gp + λ1 − λ2 χX  + gq − λ1 − λ2 χX  



(49)



Put q = q − λ1 − λ2 χX . Since g p+ χX  ≥ 0, (49) implies gq  ≤ gq < gp. This inequality, however, is a contradiction since the number of distinct values in pv − q vv∈V is less than that of pv − qvv∈V . by

Given a set function ρ 2V → R ∪ +∞, define gρ R V → R ∪ +∞ gρ p =

k−1  j=1

pj − pj+1 ρVj  + pk ρVk 

(50)

where p1 > p2 > · · · > pk are distinct values in pvv∈V , and Vj = v ∈ V  pv ≥ pj  j = 1  k. The function gρ is called the Lov´ asz extension of ρ.

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Theorem 4.30. If ρ ∈  , then gρ p = sup p x x∈Bρ

Proof.

∀p ∈ R V 

See [17].

Theorem 4.31 (Lov´asz [24]). Let ρ 2V → R ∪ +∞ be a function such that ρ = 0 and ρV  < +∞. Then, ρ ∈  ⇐⇒ gρ is convex. Directional derivative functions and subdifferentials of a polyhedral L-convex function have nice structures such as M/L-convexity, and they can be explicitly described by certain submodular functions (cf. Theorem 4.32(i)). Theorem 4.32. (i)

Let g ∈  and p ∈ dom g.

The function ρp 2V → R ∪ +∞ defined by ρp X = g p+ χX 

X ⊆ V 

(51)

satisfies ρp  = 0, −∞ < ρp V  < +∞, and the submodular inequality (7), i.e., ρp ∈  . (ii) We have g p+ · = gρp and g p+ · ∈ 0 . Proof. We show g p+ · ∈ 0  only. The claim (i) is an immediate corollary of (ii), and the equation g p+ · = gρp will be shown later in Theorem 4.36(i). Let p ∈ dom g. For q1  q2 ∈ R V and a sufficiently small µ > 0, we have g p+ q1  + g p+ q2  = gp + µq1  − gp/µ + gp + µq2  − gp/µ ≥ gp + µq1 ∧ q2  − gp/µ + gp + µq1 ∨ q2  − gp/µ = g p+ q1 ∧ q2  + g p+ q1 ∨ q2  Hence, (LF1) holds for g p+ ·. Condition (LF2) for g p+ · can be shown similarly. M-convexity appears in subdifferentials of a polyhedral L-convex function. Theorem 4.33. Let g ∈  and p ∈ dom g. (i)

∂gp ∈ 0 and ∂gp is represented as ∂gp = Bρp   = x ∈ R V  xX ≤ g p+ χX ∀X ⊆ V   xV  = g p+ χV  

where ρp 2V → R ∪ +∞ is defined by (51).

(52)

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397

For any q ∈ R V we have gp + q − gp ≥ sup q x = gρp q x∈∂gp

Proof. From Theorems 4.5 and 4.29 follows (52), which, together with Theorem 4.32(i), implies ∂gp ∈ 0 . The claim (ii) is immediate from (i) and Theorem 4.30. The next theorem shows that each face of the epigraph of a polyhedral L-convex function is an L-convex polyhedron when it is projected to R V . The proof is obvious and therefore omitted. Theorem 4.34. inf g"−x# > −∞.

For g ∈  and x ∈ R V , we have arg min g"−x# ∈ 0 if

The class of polyhedral L-convex functions is closed under various fundamental operations. Theorem 4.35. Let g g1  g2 ∈ . (1) For x ∈ R V , the function g"−x# R V → R ∪ +∞ given by (25) is polyhedral L-convex. (2) For a ∈ R V , β ∈ R , and ν > 0, the function ν · ga + βp is polyhedral L-convex in p. (3) For any U ⊆ V , the function gU R U → R ∪ ±∞ given by (40) is polyhedral L-convex if gU p0  is finite for some p0 ∈ R V . (4) For ψv ∈
1 v ∈ V , the function g˜ R V → R ∪ ±∞ defined by
  gp ˜ = inf gq + ψv pv − qv p ∈ R V  q∈R V

v∈V

is polyhedral L-convex if gp ˜ 0  is finite for some p0 ∈ R V . (5) For any γ ∈  , the restriction gDγ R V → R ∪ +∞ of g defined by
gp p ∈ Dγ, gDγ p = +∞ p ∈ Dγ is polyhedral L-convex if dom g ∩ Dγ = . (6)

g1 + g2 ∈  if dom g1 ∩ dom g2 = .

A more general and stronger transformation, called “network induction,” is explained in subsection 7.3.

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4.3.3. Positively Homogeneous Polyhedral L-Convex Functions This section clarifies the relationship of positively homogeneous polyhedral L-convex functions with submodular functions, and with M-convex polyhedra. For a positively homogeneous polyhedral convex function g R V → R ∪ +∞ with 0 ∈ dom g, define a set function ρg 2V → R ∪ +∞ by ρg X = g 0+ χX  = gχX 

X ⊆ V 

(53)

Recall the definition of the Lov´asz extension gρ in (50). Theorem 4.36.

For g ∈ 0 , we have ρg ∈  and gρg = g.

(i)

(ii) For ρ ∈  , we have gρ ∈ 0  and ρgρ = ρ. (iii) The mappings g (→ ρg g ∈ 0  and ρ (→ gρ ρ ∈   provide a one-to-one correspondence between 0  and  and are the inverse of each other. Proof. (i) Condition (LF1) for g yields ρg ∈  . We see from Theorem 4.33(ii) that g ≥ gρg . On the other hand, from Theorem 4.2 follows gρg p =

k−1  j=1

≥g

pj − pj+1 gχVj  + pk gχVk 

k−1   pj − pj+1 χVj + pk χVk = gp j=1

where pj and Vj j = 1  k are defined as in the Lov´asz extension (50). (ii) We show (LF1) for gρ only. The other claims are obvious. First assume that ρ < +∞. By Theorem 4.27, it suffices to show that (45) holds for any p ∈ R V and u v ∈ V u = v, where g = gρ . Define pj and Vj j = 1  k as in the Lov´asz extension (50). Let l, m be such that pu = pl , pv = pm , and λ ∈ "0 pl−1 − pl #, µ ∈ "0 pm−1 − pm #. From the definition of gρ we have gρ p + λχu  = gρ p + λρVl−1 ∪ u − ρVl−1  gρ p + µχv  = gρ p + µρVm−1 ∪ v − ρVm−1  (Case 1. l = m.) We may assume that l < m. Then, (45) holds with equality, gρ p + λχu + µχv  = gρ p + λρVl−1 ∪ u − ρVl−1  + µρVm−1 ∪ v − ρVm−1  = gρ p + λχu  + gρ p + µχv  − gρ p

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(Case 2. l = m.) We may assume that λ ≥ µ. Then gρ p + λχu + µχv  = gρ p + λρVl−1 ∪ u − ρVl−1  + µρVl−1 ∪ u v − ρVl−1 ∪ u ≤ gρ p + λρVl−1 ∪ u − ρVl−1  + µρVl−1 ∪ v − ρVl−1  = gρ p + λχu  + gρ p + µχv  − gρ p where the inequality is by the submodularity of ρ. Hence, (45) follows. Next, we consider the general case where ρ may take the value +∞. Let x ∈ Bρ. For each positive integer k, we define ρk 2V → R by ρk X = min ρX \ Y  + xY  + kY  Y ⊆X

X ⊆ V 

Note that ρk is the submodular function associated with y ∈ Bρ  yv ≤ xv + k v ∈ V , which is a bounded M-convex polyhedron. Hence, we have ρk < +∞ and gρk satisfies (LF1). Since ρX = limk→∞ ρk X ∀X ⊆ V , it holds that gρ p = limk→∞ gρk p p ∈ R V  and therefore gρ also satisfies (LF1). (iii)

This is clear from (i) and (ii).

From Theorem 4.36, we see that a polyhedral convex function is positively homogeneous polyhedral L-convex if and only if it is the Lov´asz extension of a submodular set function. Corollary 4.37.

0

= gρ  ρ ∈  .

As immediate consequences of Theorems 3.3, 4.3, and 4.36, we obtain a one-to-one correspondence between positively homogeneous polyhedral L-convex functions and M-convex polyhedra. Recall the notation Sg and δ∗S in (23) and (24). Theorem 4.38. (ii)

(i)

For g ∈ 0 , we have Sg ∈ 0 and δ∗Sg = g.

For B ∈ 0 , we have δ∗B ∈ 0  and Sδ∗B = B.

(iii) The mappings g (→ Sg g ∈ 0  and B (→ δ∗B B ∈ 0  provide a one-to-one correspondence between 0  and 0 , and are the inverse of each other.

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4.3.4. Polyhedral L- -Convex Functions Due to the property (LF2), a polyhedral L-convex function loses no information when restricted to a hyperplane p ∈ R V  pv = 0 for any v ∈ V . We call a polyhedral convex function g R V → R ∪ +∞ L- -convex if the  function g˜ R V → R ∪ +∞ defined by gp ˜ 0  p = gp − p0 1



p0  p ∈ R V 

(54)

 = v0  ∪ V . We see that polyhedral L- is polyhedral L-convex, where V convex functions are essentially the same as polyhedral L-convex functions, while the class of polyhedral L- -convex functions properly contains that of polyhedral L-convex functions. The concept of L- -convexity was originally introduced in [18] as a concept for functions defined over the integer lattice. Theorem 4.39. Let g R V → R ∪ +∞ be a polyhedral convex function with dom g = . Then g is L- -convex ⇐⇒ g satisfies (46) for any p q ∈ dom g and λ ≥ 0. Proof. The “only if” part is clear from the definition of L- -convex functions and Lemma 4.28. Hence, we show the “if” part only. It suf fices to show that g˜ R V → R ∪ +∞ defined by (54) fulfills (LF1). Let p0  p q0  q ∈ dom g, ˜ and without loss of generality assume p0 ≥ q0 . Then gp ˜ 0  p + gq ˜ 0  q = gp − p0 1 + gq − q0 1 ≥ gp − p0 1 +p0 − q0  1 ∧ q − q0 1 + gp − p0 1 ∨ q − q0 1 −p0 − q0  1 = gp ∧ q − q0 1 + gp ∨ q − p0 1 = gp ˜ 0 ∧ q0  p ∧ q + gp ˜ 0 ∨ q0  p ∨ q where the inequality is by (46). Hence, (LF1) holds for g. ˜ Example 4.40 (Separable-Convex Functions). Let D ⊆ R V be an L- convex polyhedron. For any convex functions gv ∈
1 v ∈ V , the function g R V → R ∪ +∞ defined by  gv pv p ∈ D, gp = v∈V +∞ p ∈ D is polyhedral L- -convex if dom g = . From the definition of polyhedral L- -convex functions, every property of polyhedral L-convex functions can be restated in terms of polyhedral L- convex functions, and vice versa. We show two properties below, which seem more natural when restated in terms of polyhedral L- -convex functions.

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Theorem 4.41. Let g R V → R ∪ +∞ be a polyhedral L- -convex function. (i) For any a V → R ∪−∞ and b V → R ∪ +∞, the restriction g"a b# R V → R ∪ +∞ of g to "a b# defined by (37) is polyhedral L- -convex if dom g ∩ "a b# = . (ii) For any U ⊆ V , the function gU R U → R ∪ +∞ given by (36) is polyhedral L- -convex if dom gU = . Proof. Part (i) is immediate from the translation of Theorem 4.35(5), and (ii) is a corollary of (i). Examples 4.22 and 4.40 show that any separable-convex function is polyhedral M- -convex as well as polyhedral L- -convex. The converse is also true, as shown in Theorem 4.42 below. Theorem 4.42 (cf. [38, Theorem 3.17]). Let f R V → R ∪ +∞ be a polyhedral convex function with dom f = . Then f is both M - -convex and L- -convex if and only if f is separable-convex. Proof. Let f be both M - -convex and L- -convex. Theorems 3.31, 4.7(ii), and 4.25 (ii) imply that dom f is an interval; i.e., there exist a V → R ∪−∞ and b V → R ∪+∞ with av ≤ bv v ∈ V  such that dom f = "a b#. Similarly, Theorems 3.31, 4.17, and 4.34 imply that arg min f "−p# is an interval for any p ∈ R V with inf f "−p# > −∞. Hence, kv for each v ∈ V there exist a sequence ai vi=0 ⊆ R ∪ ±∞ kv ≥ 0 such that • a0 v = av < a1 v < · · · < akv −1 v < akv v = bv, • for any iv = 1 2  kv v ∈ V , the function f is linear over the interval x ∈ R V  aiv −1 v ≤ xv ≤ aiv v v ∈ V . Due to these properties, we have f x + αχv  − f x = f y + αχv  − f y for any x y ∈ dom f with xv = yv and any α ∈ R . For all v ∈ V , put fv α = f x0 + α − x0 vχv  − f x0  where x0 ∈ dom f . Then, we have  f x = fv xv + f x0  v∈V

α ∈ R 

x ∈ R V 

Moreover, each fv is polyhedral convex since f itself is polyhedral convex. Therefore, f is a separable-convex function. Remark 4.43. There exists no function which is both polyhedral M-convex and L-convex, i.e.,  ∩  =  (see Remark 3.32).

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Polyhedral M-convex and L-convex functions are conjugate to each other. Theorem 5.1. For f ∈  and g ∈ , we have f • ∈  and g• ∈ . Moreover, the mappings f (→ f • f ∈  and g (→ g• g ∈  provide a oneto-one correspondence between  and  and are the inverse of each other. Polyhedral M/L-convex functions are characterized by local polyhedral structures such as directional derivative functions, subdifferentials, and the sets of minimizers. Theorem 5.2. Let f R V → R ∪ +∞ be a polyhedral convex function with dom f = . Then i f ∈  ⇐⇒ ii f  x+ · ∈ 0  ∀x ∈ dom f  ⇐⇒ iii ∂f x ∈ 0 ∀x ∈ dom f  ⇐⇒ iv arg min f "−p# ∈ 0 ∀p ∈ R V with inf f "−p# > −∞ Theorem 5.3. Let g R V → R ∪ +∞ be a polyhedral convex function with dom g = . Then i g ∈  ⇐⇒ ii g p+ · ∈ 0  ∀p ∈ dom g ⇐⇒ iii ∂gp ∈ 0 ∀p ∈ dom g ⇐⇒ iv arg min g"−x# ∈ 0 ∀x ∈ R V with inf g"−x# > −∞ In the following, we first prove Theorem 5.3, then Theorem 5.1, and finally Theorem 5.2. Proof of Theorem 5 3. The implication (i) ⇒ (ii) is by Theorem 4.32(ii). To show the reverse implication, we assume (ii) for g. For any p ∈ dom g, there exists ε > 0 such that gq − gp = g p+ q − p

∀q ∈ N∞ p ε

Hence, g satisfies (LF1loc ) and (LF2loc ), yielding (i) by Theorem 4.26. The equivalence (ii) ⇐⇒ (iii) is by Theorems 4.4 and 4.38. The implication (i) ⇒ (iv) follows from Theorem 4.34. We conclude the proof of Theorem 5.3 by showing that (iv) ⇒ (ii) holds. Lemma 5.4. Let g R V → R ∪ +∞ be positively homogeneous polyhedral convex with dom g = . Suppose that arg min g"−x# is an L-convex cone for any x ∈ R V with inf g"−x# > −∞. Then g ∈ 0 .

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Proof. Define ρg 2V → R ∪ +∞ by (53). Since 0 ∈ dom g, there exists some x ∈ R V with 0 ∈ arg min g"−x#. By (LS2) for arg min g"−x#, we have 1 ∈ arg min g"−x# ⊆ dom g, i.e., ρg V  = g1 < +∞. As is shown below, we have g = gρg , where gρg is defined by (50) with ρ = ρg . Thus, we have ρg ∈  by Theorem 4.31, which, together with Corollary 4.37, implies g ∈ 0 . We now prove g = gρg . Note that gp ≤ gρg p p ∈ R V  by Theorem 4.2. This implies that gp = gρg p = +∞ if p ∈ dom g. Therefore, we may assume that p ∈ dom g. Then there exists some x ∈ R V such that p ∈ arg min g"−x#. By Theorem 3.25, there exists some chain 0 ⊆ 2V  and λX  X ∈ 0  ⊆ R  such that  p= λ X χX  λX ≥ 0 ∀X ∈ 0 \ V  X∈0

0 ⊆ X ⊆ V  χX ∈ arg min g"−x# By the linearity of g over arg min g"−x# and by (50), we have  gp = λX gχX  = gρg p X∈0

We now prove (iv) ⇒ (ii). For any p ∈ dom g, we denote gp = g p+ · for notational simplicity. By Lemma 5.4, we have only to prove that arg min gp "−x# is an L-convex cone for any p ∈ dom g and any x ∈ R V with inf gp "−x# > −∞. Let p ∈ dom g, and x ∈ R V be such that inf gp "−x# > −∞. Then we have inf g"−x# > −∞ and arg min g"−x# ∈ 0 . Let γ ∈  be such that Dγ = arg min g"−x#. Since arg min gp "−x# is the tangent cone of arg min g"−x# at p, it can be represented as arg min gp "−x# = q ∈ R V  qv − qu ≤ 0 ∀u v ∈ Ap  where Ap = u v  u v ∈ V pv − pu = γu v. Since Ap is transitive, arg min gp "−x# is an L-convex cone by Theorem 3.24. Thus, Theorem 5.3 is proven. The following lemma is needed for the proof of Theorem 5.1. Lemma 5.5. Let g ∈ , and x y ∈ R V be such that inf g"−x# > −∞ and inf g"−y# > −∞. Then for any u ∈ supp+ x − y, there exists v ∈ supp− x − y such that pv − pu ≤ qv − qu ∀p ∈ arg min g"−x# ∀q ∈ arg min g"−y#

(55)

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Proof. First we note that xV  = yV  = r, where r ∈ R is the value in (LF2) for g. Let u ∈ supp+ x − y. We have arg min g"−x# arg min g"−y# ∈ 0 by Theorem 4.34. Hence, it suffices to show the following: there exists some v ∈ supp− x − y such that pv ≤ qv ∀p ∈ Dx  ∀q ∈ Dy , where Dx = p  p ∈ arg min g"−x# pu = 0 Dy = q  q ∈ arg min g"−y# qu = 0 Assume, to the contrary, that for any v ∈ supp− x − y, there exists a pair of vectors pv ∈ Dx , qv ∈ Dy such that pv v > qv v. Set   p∗ = pv  v ∈ supp− x − y q∗ = qv  v ∈ supp− x − y Then we have p∗ ∈ Dx , q∗ ∈ Dy , and p∗ v > q∗ v ∀v ∈ supp− x − y. Let λ > 0 be any value with λ ≤ p∗ v − q∗ v for all v ∈ supp+ p∗ − q∗ . Putting
p∗ v − λ v ∈ supp+ p∗ − q∗ , p = p∗ − λ1 ∨ q∗ = q∗ v v ∈ V \ supp+ p∗ − q∗ 
q∗ v + λ v ∈ supp+ p∗ − q∗ , q = p∗ ∧ q∗ + λ1 = p∗ v v ∈ V \ supp+ p∗ − q∗ , we have gp∗  + gq∗  ≥ gp  + gq 

(56)

by Lemma 4.28. Since supp− x − y ⊆ supp+ p∗ − q∗ , we obtain p  x + q  y − p∗  x − q∗  y  = λ yv − xv  v ∈ supp+ p∗ − q∗     + q∗ v − p∗ vxv − yvv ∈ V \ supp+ p∗ − q∗   ≥ λ yv − xv  v ∈ V \ u = λxu − yu > 0 Combining this inequality with (56), we have g"−x#p  + g"−y#q  < g"−x#p∗  + g"−y#q∗  which is a contradiction since p∗ ∈ arg min g"−x#, q∗ ∈ arg min g"−y#. Proof of Theorem 5 1. First recall Theorem 4.1. [g ∈  ⇒ g• ∈ ] Let x y ∈ dom g• . Then, inf g"−x# > −∞ and inf g"−y# > −∞. By Lemma 5.5, for any u ∈ supp+ x − y there exists v ∈ supp− x − y satisfying the inequality (55), which implies g•  x+ v u + g•  y+ u v = suppv − pu  p ∈ arg min g"−x# + supqu − qv  q ∈ arg min g"−y# ≤ 0

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where the equality is by Theorems 4.4 and 4.5. Hence, we have (M-EXC ) for g• and therefore (M-EXC) by Theorem 4.11. [f ∈  ⇒ f • ∈ ] If f ∈ , then Theorems 4.5 and 4.17 imply ∂f • p = arg min f "−p# ∈ 0

∀p ∈ dom f • 

which, together with Theorem 5.3, yields f • ∈ . Proof of Theorem 5 2. The equivalence (ii) ⇐⇒ (iii) is by Theorems 4.4 and 4.20. By Theorems 4.5 and 5.1, the equivalence (i) ⇐⇒ (iii) ⇐⇒ (iv) is rewritten as follows in terms of f • , f • ∈  ⇐⇒ arg min f • "−x# ∈ 0 ∀x ∈ R V with inf f • "−x# > −∞ ⇐⇒ ∂f • p ∈ 0 ∀p ∈ dom f •  which is already shown in Theorem 5.3. Proof of Theorem 4 186. From the assumption and Theorem 4.6, f1 f2 is a polyhedral convex function with f1 f2 x > −∞ ∀x ∈ R V . Hence, it follows from Theorem 4.1 that f1 f2 • is also a polyhedral convex function such that f1 f2 • x > −∞ ∀x ∈ R V  and domf1 f2 • = . Theorems 4.6, 4.35(6), and 5.1 imply f1 f2 • = f1• + f2• ∈ , which shows f1 f2 ∈  by Theorem 5.1. Proof of Theorem 4 21. Let f R V → R ∪ +∞ be a polyhedral convex function with dom f =  satisfying (M- -EXC). Let p ∈ R V be any vector with inf f "−p# > −∞. It follows from (M- -EXC) that arg min f "−p# satisfies (G-EXC), i.e., arg min f "−p# is an M- -convex polyhedron by Theorem 3.8. From the definition of polyhedral M- -convex functions and Theorem 5.2, the function f is polyhedral M- -convex. 6. RELATIONSHIP WITH M/L-CONVEX FUNCTIONS OVER THE INTEGER LATTICE The concepts of M-convexity and L-convexity were originally introduced for functions defined over the integer lattice [29, 30, 32]. In this section, we present the relationship of M/L-convexity over the integer lattice and polyhedral M/L-convexity. A function f ZV → R ∪ +∞ is called discrete M-convex if domZ f =  and it satisfies (M-EXC[Z]) ∀x y ∈ domZ f , ∀u ∈ supp+ x − y, ∃v ∈ supp− x − y such that f x + f y ≥ f x − χu + χv  + f y + χu − χv 

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where domZ f = x ∈ ZV  −∞ < f x < +∞. On the other hand, a function g ZV → R ∪ +∞ is called discrete L-convex if domZ g =  and it satisfies (LF1[Z]) gp+gq ≥ gp∧q+gp∨q ∀pq ∈ domZ g, (LF2[Z]) ∃r ∈ R such that gp + λ1 = gp + λr ∀p ∈ domZ g ∀λ ∈ Z. We denote by "Z# and "Z#, respectively, the classes of discrete M-convex and L-convex functions: "Z# = f  f ZV → R ∪ +∞ discrete M-convex "Z# = g  g ZV → R ∪ +∞ discrete L-convex In the following, we sometimes regard f ZV → R ∪ +∞ as a function defined over R V . In such a case, we have dom f ⊆ ZV . Remark 6.1. The concept of M-convex function over the integer lattice was first defined in [29, 30], where the effective domain is assumed to be bounded. The paper [32] discusses the case when the effective domain is not necessarily bounded, but assumes that functions take only integer values. The concept of L-convex function over the integer lattice appeared first in [32], where function values are assumed be integral. Hence, the definition of L-convex functions above is a slight extension of the original one. We also define the set version of discrete M/L-convexity as follows. A set B ⊆ ZV is called discrete M-convex if B =  and it satisfies that

(B-EXC[Z]) ∀x y ∈ B, ∀u ∈ supp+ x − y, ∃v ∈ supp− x − y such x − χu + χv ∈ B

y + χu − χv ∈ B

We call D ⊆ ZV discrete L-convex if D =  and it satisfies (LS1[Z]) p q ∈ D ⇒ p ∧ q p ∨ q ∈ D, (LS2[Z]) p ∈ D ⇒ p + λ1 ∈ D ∀λ ∈ Z. We denote 0 "Z# = B  B ⊆ ZV  discrete M-convex 0 "Z# = D  D ⊆ ZV  discrete L-convex In this section, we use some known results on discrete M/L-convexity. See [29, 30, 32, 35] as references. Discrete M/L-convex functions can be extended to ordinary convex functions by taking their convex closure. Moreover, convex extensions of discrete M/L-convex functions coincide with “local” convex extensions. For any x ∈ R V and any integer k ≥ 0, we define the set HCk x = y ∈ ZV  .xv/ − k ≤ yv ≤ 0xv1 + k v ∈ V 

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$ ∈ 0 and Theorem 6.2. For B ∈ 0 "Z#, we have B $  .xv/ ≤ yv ≤ 0xv1v ∈ V  B ∩ HC0 x = y ∈ B

x ∈ R V  (57)

Proof. Let ρ 2V → Z ∪+∞ be the submodular function such that B = Bρ ∩ ZV . Since Bρ is an integral polyhedron (cf. [17]), we have $ ∈ 0 . For any a b ∈ ZV the set B $ ∩ "a b# is also an integral Bρ = B polyhedron if it is nonempty. Thus (57) follows. $ ∈ 0 and Theorem 6.3. For D ∈ 0 "Z#, we have D $  .pv/ ≤ qv ≤ 0pv1 v ∈ V  D ∩ HC0 p = q ∈ D

p ∈ R V  (58)

Proof. Let γ V × V → Z ∪+∞ be the distance function with triangle inequality which is associated with D (see [32, Theorem 4.20]; [35, Theorem $= 2.28]), i.e., γ is such that D = Dγ ∩ ZV . Lemma 3.21 (v) implies D $ ∩ "a b# is also an integral Dγ ∈ 0 . Note that for any a b ∈ ZV the set D polyhedron if it is nonempty. Thus (58) follows. Let f ZV → R ∪ +∞. The convex closure f¯ R V → R ∪ ±∞ of f is given by x ∈ R V  f¯x = sup p x + α  p y + α ≤ f y y ∈ ZV  p∈R V  α∈R

We also define a function f˜ R V → R ∪ +∞ by f˜x = sup p x + α  p y + α ≤ f y y ∈ HC0 x

(59)

p∈R V  α∈R

x ∈ R V  (60) Note that f˜ is the local convex extension of f , i.e., the convex closure of the restriction of f to the integral points around x. It admits an alternative expression      λy y = x λy = 1   λy f y y∈HC0 x f˜x = inf y∈HC0 x  λ ≥ 0 y ∈ HC x y∈HC x 0

y

0

x ∈ R V  (61) by LP duality. From the definition, we have f˜x = f x ∀x ∈ ZV  f˜x ≥ f¯x ∀x ∈ R V 

(62)

Favati and Tardella [11] investigated the class of functions f ZV → R ∪ +∞ for which f¯ = f˜ holds. They call such a function integrally convex. It is easy to see that global optimality of integrally convex functions can be characterized by local optimality.

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Theorem 6.4 [11, Proposition 3.1]. Let f ZV → R ∪ +∞ be an integrally convex function and x ∈ dom f . Then f x ≤ f y for any y ∈ ZV if and only if f x ≤ f y for any y ∈ ZV with y − x∞ ≤ 1. Theorems 6.5 and 6.7 show that discrete M/L-convex functions are integrally convex. Theorem 6.5. For f ∈ "Z#, we have f¯x = f˜x ∀x ∈ R V  and f¯x = f x ∀x ∈ ZV . Proof. We prove the former only. The latter follows from the former and the equation in (62). We first consider the case when x ∈ domZ f . Define the conjugate f • of f by (22). Let ρ 2V → Z ∪ +∞ be the submodular function associated with the discrete M-convex set domZ f ∈ 0 "Z#, and p ∈ dom f • . If xV  = ρV , then   f¯x ≥ sup p + αχV  x − sup p + αχV  y − f y α∈R

y∈R V

= p x − f • p + sup αxV  − ρV  = +∞ α∈R

In the similar way, we can show that f¯x = +∞ when xX > ρX ∃X ⊂ V . By the inequality in (62), we have f˜x = f¯x = +∞ To show f¯x = f˜x for x ∈ domZ f , we consider the following dual pair of LP problems: LP1 maximize p x + α subject to p y + α ≤ f y y ∈ HC1 x p ∈ R V  α ∈ R  

LP2 minimize

y∈HC1 x

subject to



y∈HC1 x

λy f y λy y = x

 y∈HC1 x

λy = 1 λy ≥ 0 y ∈ HC1 x

Problem (LP2) has a feasible solution since x ∈ HC1 x ∩ domZ f . Let p∗  α∗  ∈ R V × R  and λ∗ = λ∗y  y ∈ HC1 x be optimal solutions of (LP1) and (LP2), respectively. Then, it holds that  f¯x ≤ p∗  x + α∗ = λ∗y f y ≤ f˜x (63) y∈HC1 x

We will show that both of the inequalities hold with equality. Put B = y ∈ HC1 x  p∗  y + α∗ = f y = arg min f "−p∗ #y y∈HC1 x

∈ 0 "Z#

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The complementary slackness condition implies y ∈ HC1 x  λ∗y > $. In particular, we have x ∈ B ∩ HC0 x by 0 ⊆ B, which shows x ∈ B Theorem 6.2 since B ∈ 0 "Z#. Hence, there is another optimal solution λ˜ = λ˜ y  y ∈ HC1 x of (LP2) such that if λ˜ y > 0 then y ∈ B ∩ HC0 x. Since    λ˜ y f y = λ˜ y f y ≥ f˜x λ∗y f y = y∈HC1 x

y∈HC1 x

y∈HC0 x

the second inequality in (63) holds with equality. Let y0 ∈ B ∩ HC0 x. Then we have f "−p∗ #y0 − χu + χv  ≥ f "−p∗ #y0  for any u v ∈ V . Since local optimality means global optimality for discrete M-convex functions [32, Theorem 4.6], we have α∗ = f "−p∗ #y0  ≤ f "−p∗ #y = −p∗  y + f y ∀y ∈ domZ f , i.e., p∗  y + α∗ ≤ f y ∀y ∈ domZ f . By (59), the first inequality in (63) holds with equality. The following theorem claims that we can choose a common optimal λ in (61) for two discrete M-convex functions. An implication of this fact is discussed later in Corollary 6.8(i). Theorem 6.6. For any f g ∈ "Z# and x ∈ R V , there exists λ = λy  y ∈ HC0 x such that   λy y = x λy = 1 λy ≥ 0 y ∈ HC0 x (64) y∈HC0 x

f¯x = f˜x =



y∈HC0 x

y∈HC0 x

λy f y

gx ¯ = gx ˜ =

 y∈HC0 x

λy gy (65)

Proof. We may assume that x ∈ domZ f ∩ domZ g, which implies both f˜x and gx ˜ are finite. By (60), there exist p α q β ∈ R V × R such that p y + α ≤ f y y ∈ HC0 x

p x + α = f˜x

q y + β ≤ gy y ∈ HC0 x

q x + β = gx ˜

Put Bf = y ∈ HC0 x  p y + α = f y

∈ 0 "Z#

Bg = y ∈ HC0 x  q y + β = gy

∈ 0 "Z#

Then we have x ∈ Bf ∩ Bg = Bf ∩ Bg . Therefore, there exists λ = λy  y ∈ HC0 x satisfying (64) and λy = 0 y ∈ Bf ∩ Bg . Such λ satisfies (65) by LP-duality.

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Theorem 6.7 [32, Theorem 4.18]. Let g ∈ "Z#. (1)

For q ∈ domZ g and p ∈ "0 1#V , we have gq ¯ + p = gq ˜ + p = gq +

k 

pj − pj+1 gq + χVj  − gq

(66)

j=1

where pj and Vj j = 1  k are as in the definition of the Lov´ asz extension (50) and pk+1 = 0. (2) gp ¯ = gp ˜ = gp ∀p ∈ ZV . (3) gp ¯ + λ1 = gp ¯ + λr ∀p ∈ R V  λ ∈ R , where r is in (LF2[Z]). (4) The equalities in (66) remain valid for q ∈ domZ g and p ∈ R V with max v∈V pv − minv∈V pv ≤ 1. As a corollary of Theorems 6.6 and 6.7, we obtain the following properties. For a function g ZV → R ∪ −∞, we define the concave closure  g R V → R ∪ ±∞ of g by  gx =

inf

p∈R V  α∈R

p x + α  p y + α ≥ gy y ∈ ZV 

x ∈ R V 

Note that  g = −−g. Corollary 6.8. (i) Let f , g be functions such that f −g ∈ "Z#. If f x ≥ gx for any x ∈ ZV , then we have f¯x ≥  gx for any x ∈ R V . (ii) Let f , g be functions such that f −g ∈ "Z#. If f p ≥ gp for any p ∈ ZV , then we have f¯p ≥  gp for any p ∈ R V . The following theorems show that the convex extension of discrete M/L-convex functions are closely related to polyhedral M/L-convexity. Theorem 6.9. Let f ∈ "Z#. If f¯ is polyhedral convex, then f¯ ∈ . In particular, f¯ ∈  if domZ f is bounded. Proof.

By Theorems 6.2 and 6.5 (see also [32, Theorem 4.11]), we have arg min f¯"−p# = arg min f "−p# ∈ 0

∀p ∈ R V 

which, together with Theorem 5.2, implies f¯ ∈ . Theorem 6.10. Let g ∈ "Z#. If g¯ is polyhedral convex, then g¯ ∈ . In particular, g¯ ∈  if p ∈ domZ g  pv = 0 is bounded for some v ∈ V . Proof.

By Theorems 6.3 and 6.7 (1), we have arg min g"−x# ¯ = arg min g"−x# ∈ 0

which, together with Theorem 5.3, implies g¯ ∈ .

∀x ∈ R V 

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Remark 6.11. For f ZV → R ∪ +∞, if domZ f is bounded then its convex closure f¯ is polyhedral convex, but in general f¯ is not necessarily polyhedral convex, even if f is discrete M/L-convex. For example, consider the function f Z2 → Z ∪ +∞ defined by
2 x x y ∈ Z2  x + y = 0, f x y = +∞ (otherwise). It is easy to see that f is discrete M-convex. However, the convex closure f¯ of f has infinite number of linear-pieces and is not a polyhedral convex function. We now introduce two new classes of M/L-convexity. We call a function integral M-convex (resp. integral L-convex) if it is represented as the convex closure of some integer-valued discrete M-convex (resp. discrete L-convex) function. We denote by int and int , respectively, the classes of integral M-convex and L-convex functions: int = f  f = fZ for some fZ ZV → Z ∪ +∞ fZ ∈ "Z# int = g  g = gZ for some gZ ZV → Z ∪ +∞ gZ ∈ "Z# Note that integral M/L-convex functions may have infinite number of linear pieces and therefore are not polyhedral convex in general if the effective domain is unbounded (see Remark 6.11). It was shown in [32] that integer-valued discrete M-convex and Lconvex functions are conjugate to each other under the integer Fenchel transformation. Theorem 6.12 [32, Theorem 4.24]. (i) For an integer-valued discrete M-convex function f ZV → Z ∪ +∞, the function g ZV → Z ∪ +∞ defined by gp = sup p x − f x x∈ZV

p ∈ ZV 

is integer-valued discrete L-convex. (ii) For an integer-valued discrete L-convex function g ZV → Z ∪ +∞, the function f ZV → Z ∪ +∞ defined by f x = sup p x − gp p∈Z

V

x ∈ ZV 

is integer-valued discrete M-convex. (iii) The mappings f (→ g and g (→ f defined in (i) and (ii), respectively, provide a one-to-one correspondence between the classes of integer-valued discrete M-convex and L-convex functions and are the inverse of each other.

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Using this conjugacy relationship, we can show that integral M-convex and L-convex functions are conjugate to each other. Lemma 6.13. Let f ZV → Z ∪ +∞ be an integrally convex function. For any p ∈ R V , if inf f "−p# is finite, then arg min f "−p# = . Proof.

Put

ε0 = min pS − pT  − 0pS − pT 1 + 1 ST ⊆V

> 0

We claim that for any x y ∈ domZ f with y − x∞ ≤ 1, if f "−p#y < f "−p#x then f "−p#y ≤ f "−p#x − ε0 . Since y − x∞ ≤ 1, there exist some S T ⊆ V such that y = x + χS − χT . If f "−p#y < f "−p#x then we have f y − f x < pS − pT  ≤ 0pS − pT 1, implying f y − f x ≤ 0pS − pT 1 − 1 since f y − f x ∈ Z. Hence, we have the claim. Let x∗ ∈ domZ f satisfy f "−p#x∗  ≤ inf f "−p# + ε0 /2, and y ∈ domZ f be any vector with y − x∗ ∞ ≤ 1. If f "−p#y < f "−p#x∗ , then −ε0 ≥ f "−p#y − f "−p#x∗  ≥ inf f "−p# − inf f "−p# + ε0 /2 ≥ −ε0 /2 a contradiction. Hence, we have f "−p#y ≥ f "−p#x∗  for any y ∈ domZ f with y − x∗ ∞ ≤ 1. By Theorem 6.4 we have x∗ ∈ arg min f "−p#. For any f R V → R ∪ ±∞, we define fZ ZV → R ∪ ±∞ by
f x x ∈ ZV , fZ x = +∞ x ∈ ZV . Theorem 6.14.

(67)

(i) For f ∈ int , it holds that f • ∈ int and

f • p = sup p x − f x x∈Z

V

p ∈ R V 

(68)

In particular, for any p ∈ dom f • there exists some x ∈ dom f ∩ ZV such that f • p = p x − f x. (ii)

For g ∈ int , it holds that g• ∈ int and g• x = sup p x − gp p∈Z

V

x ∈ R V 

In particular, for any x ∈ dom g• there exists some p ∈ dom g ∩ ZV such that g• x = p x − gp. (iii) The mappings f (→ f • f ∈ int  and g (→ g• g ∈ int  provide a one-to-one correspondence between int and int and are the inverse of each other.

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Proof. We prove claim (i) only, since (ii) can be shown similarly to (i), and (iii) is an immediate corollary to (i), (ii), and Theorem 4.1. Equation (68) follows from the integral convexity of fZ shown in Theorem 6.5. For any p ∈ dom f • Lemma 6.13 assures the existence of x ∈ dom f ∩ ZV with f • p = p x − f x. We abbreviate f • Z to fZ• . Theorem 6.12 and Eq. (68) yield fZ• ∈ "Z#. Therefore, it suffices to show f • p = fZ• p p ∈ R V . We have f • p = fZ• p for integral vectors p ∈ ZV and f • p ≤ fZ• p p ∈ R V . Hence, we may assume f • p < +∞ since otherwise f • p = fZ• p = +∞ holds. Then, there exists some x0 ∈ domZ fZ such that f • p = p x0  − fZ x0 . Put D = q ∈ R V  f • q = q x0  − fZ x0  Then, we have D = ∂fZ x0  = ∂fZ x0  ∩ ZV 

∂fZ x0  ∩ ZV ∈ 0 "Z#

(see, e.g., [32, Theorem 4.7; 35, Theorem 2.41]), i.e., D is an integral polyhedron. Thus, there exist some integral vectors qi ki=1 k ≥ 1 in ∂fZ x0  ∩ ZV such that p can be represented by a convex combination of qi ki=1 . Since f • is linear over D, f • qi  = fZ• qi  for i = 1  k, and f • p ≤ fZ• p, convexity of f • implies f • p = fZ• p. 7. DUALITY 7.1. Duality Theorems In this section, we discuss duality theorems for polyhedral M/L-convex functions. We first state the Fenchel duality for convex/concave functions. For a convex function f R V → R ∪ +∞ and a concave function g R V → R ∪ −∞ with dom f = , dom g = , we have inf f x − gx ≥ inf f x − gx

x∈ZV

x∈R V

≥ sup g◦ p − f • p p∈R V

≥ sup g◦ p − f • p p∈ZV

where the concave conjugate g◦ R V → R ∪ ±∞ is defined by g◦ p = inf p x − gx x∈R V

p ∈ R V 

(69)

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The Fenchel duality (70) below shows that the second inequality in (69) is satisfied with equality under a certain assumption. This is just an application of the existing result in convex analysis to polyhedral convex/concave functions, and it stands independently of M-convexity/M-concavity or L-convexity/L-concavity of functions. On the other hand, the claim (ii) of Theorem 7.1 below shows that all inequalities in (69) are satisfied with equality if f and g are integral M-convex/M-concave. This integrality result is not obtained from the convexity/concavity alone, but from the combination of the convexity/concavity and the combinatorial properties of f and g. Theorem 7.1. (i) Let f R V → R ∪ +∞ and g R V → R ∪ −∞ be polyhedral convex/concave functions with dom f = , dom g = . If either of (a) dom f ∩ dom g =  and (b) dom f • ∩ dom g◦ =  holds, then we have inf f x − gx = sup g◦ p − f • p

x∈R V

p∈R V

(70)

The supremum is attained at some p∗ ∈ R V if (a) is satisfied, and the infimum is attained at some x∗ ∈ R V if (b) is satisfied. (ii) If f −g ∈ int and either (a) or (b) is fulfilled, then we have inf f x − gx = inf f x − gx

x∈ZV

x∈R V

= sup g◦ p − f • p = sup g◦ p − f • p p∈R V

p∈ZV

The supremum is attained at some integral p∗ ∈ ZV if (a) is satisfied, and the infimum is attained at some integral x∗ ∈ ZV if (b) is satisfied. Proof. The claim (i) is a standard result; see [41, 44]. We derive (ii) from the corresponding result for discrete M-convex/concave functions established in [32]. We define the function fZ ZV → Z ∪ +∞ by (67), and gZ ZV → Z ∪ −∞ by
gx x ∈ ZV , gZ x = −∞ x ∈ ZV . Then we have fZ  −gZ ∈ "Z#. We also define fZ• = f • Z ZV → Z ∪ +∞ and gZ◦ = g◦ Z ZV → Z ∪ −∞ in the similar way, where it is noted that (cf. (68)) fZ• p = sup p x − fZ x

p ∈ ZV 

gZ◦ p

p ∈ ZV 

x∈Z

V

= inf p x − gZ x x∈ZV

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We have dom f = domZ fZ and dom g = domZ gZ , and from Theorem 6.14 follows dom f • = domZ fZ• and dom g◦ = domZ gZ◦ . These properties imply that the condition (a) is equivalent to domZ fZ ∩ domZ gZ = , and (b) is equivalent to domZ fZ• ∩ domZ gZ◦ = . Thus, the first and the last term in (69) are equal by the Fenchel-type duality for discrete M-convex/Mconcave functions [32, Theorem 5.2], and each inequality in (69) holds with equality. When f and g are polyhedral M-convex and M-concave, respectively, in (70), then f • and g◦ are polyhedral L-convex and L-concave, respectively, by Theorem 5.1. Hence, Fenchel duality also reads that the minimization of the sum of polyhedral M-convex functions is equivalent to the minimization of the sum of their conjugate polyhedral L-convex functions. Note that the sum of polyhedral L-convex functions is also polyhedral Lconvex, whereas the sum of polyhedral M-convex functions is not necessarily polyhedral M-convex (cf. Remark 3.5). As a corollary of Theorem 7.1, we obtain an optimality criterion of the minimization of the sum of two polyhedral convex functions. Corollary 7.2. (i) Let fi R V → R ∪ +∞ i = 1 2 be a polyhedral convex function with dom f1 ∩ dom f2 = . Then there exists p∗ ∈ R V such that inf f1 x + f2 x = inf f1 "−p∗ #x + inf f2 "p∗ #x

x∈R V

x∈R V

x∈R V

(71)

(ii) If f1  f2 ∈ int and dom f1 ∩ dom f2 = , then there exists an integral p∗ ∈ ZV such that inf f1 x + f2 x = inf f1 x + f2 x

x∈ZV

x∈R V

= inf f1 "−p∗ #x + inf f2 "p∗ #x x∈R V

x∈R V

= inf f1 "−p∗ #x + inf f2 "p∗ #x x∈ZV

x∈ZV

Remark 7.3. The conditions (a) and (b) in Theorem 7.1 cannot be removed, as shown in the following example. Let f R 2 → R ∪ +∞ and g R 2 → R ∪ −∞ be defined by
x1 + x2 = 1, x1 f x1  x2  = +∞ x1 + x2 = 1,
x1 + x2 = −1, −x1 gx1  x2  = −∞ x1 + x2 = −1. Then we have f −g ∈  ∩ int , dom f ∩ dom g = , and inf

x1  x2 ∈R 2

f x1  x2  − gx1  x2  = +∞

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On the other hand, it holds that
p2 f • p1  p2  = +∞
−p2 g◦ p1  p2  = −∞

p1 − p2 = 1, p1 − p2 = 1 p1 − p2 = −1, p1 − p2 = −1

We have f •  −g• ∈  ∩ int , dom f • ∩ dom g◦ = , and sup p1  p2 ∈R

2

g◦ p1  p2  − f • p1  p2  = −∞

Neither (a) nor (b) is fulfilled, and Eq. (70) does not hold. We state separation theorems for polyhedral convex/concave functions. Whereas the claim (i) of Theorems 7.4 below is just the standard separation theorem in convex analysis, (ii) and (iii) are based on the combinatorial properties of M/L-convexity and their essence lies in the discrete separation theorems shown in [29, 30, 32]. See also [18] for the proof of (ii) and (iii). Theorem 7.4. Let f R V → R ∪ +∞ and g R V → R ∪ −∞ be functions with dom f ∩ dom g = . Suppose f x ≥ gx ∀x ∈ R V  holds. (i) If f and g are polyhedral convex and concave, respectively, then there exist p∗ ∈ R V and α∗ ∈ R such that f x ≥ p∗  x + α∗ ≥ gx

x ∈ R V  V

(72)

(ii) (72).

If f −g ∈ int , then we can take integral p∗ ∈ Z

(iii) (72).

If f −g ∈ int , then we can take integral p∗ ∈ ZV and α∗ ∈ Z in

and α∗ ∈ Z in

7.2. Submodular Flow Problem with M-Convex Cost Function We consider a generalization of the ordinary submodular flow problem and state a duality theorem for the problem. We also provide two optimality criteria, one of which is by negative cycles and the other by potentials. The generalization here is a polyhedral extension of the framework proposed in[29]. See, e.g., [3, 10, 12, 17, 40] for the ordinary submodular flow problem. Let G = V A be a directed graph. A flow is a vector ξ ∈ R A . Suppose that we are given a flow cost function fa R → R ∪ +∞ for each a ∈ A and a boundary cost function f R V → R ∪ +∞. Then, the minimum cost flow problem with a boundary cost function is formulated as  (MFBC) Minimize Fξ = f ∂ξ + fa ξa subject to ξ ∈ R A  a∈A

polyhedral m/l-convex functions

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where ∂ξ ∈ R V is the boundary of ξ defined by (1). The problem (MFBC) coincides with the ordinary (linear) submodular flow problem if each fa is a linear function in a nonempty closed interval and f is the indicator function of some M-convex polyhedron; i.e., f is a polyhedral M-convex function with f R V → 0 +∞. We call a flow ξ feasible if Fξ < +∞. We then consider the dual of the problem (MFBC). A potential is a vector p ∈ R V . Suppose that we are given a tension cost function ga R → R ∪ +∞ for each a ∈ A and a potential cost function g R V → R ∪ +∞. Then the minimum cost tension problem with a potential cost function is formulated as  (MTPC) Minimize Gp = gp + ga −δpa subject to p ∈ R V  a∈A

where δp ∈ R A is the coboundary of p defined by (3). The problem (MTPC) coincides with the ordinary minimum (convex-)cost tension problem if gp = 0 for any p ∈ R V . We call a potential p feasible if Gp < +∞. Theorem 7.1 (i) yields the duality between (MFBC) and (MTPC). Note that this is independent of M/L-convexity. Theorem 7.5. Let f and g be polyhedral convex functions such that dom f =  dom g = , and g = f • , and for each a ∈ A let fa  ga ∈
1 satisfy ga = fa• . Suppose that at least one of two problems (MFBC) and (MTPC) has a feasible solution. Then inf Fξ = − inf Gp

ξ∈R A

p∈R V

If (MFBC) is feasible, then Gp∗  = inf Gp for some p∗ ∈ R V , and if (MTPC) is feasible, then Fξ∗  = inf Fξ for some ξ∗ ∈ R A . Define functions f1  f2 R V → R ∪ ±∞ by   fa ξa  ∂ξ = −x  f2 x = f −x x ∈ R V  f1 x = inf

Proof.

ξ∈R A

a∈A

Similarly, we define functions g1  g2 R V → R ∪ ±∞ by  g1 p = ga −δpa g2 p = gp p ∈ R V 

(73)

a∈A

Note that f1 and g1 are special cases of functions f and g in Example 2.4, where T = V . Moreover, if (MFBC) is feasible, then f1 x0  is finite for some x0 ∈ R V , and if (MTPC) is feasible, then g1 p0  is finite for some p0 ∈ R V . Therefore, f1 and g1 are polyhedral convex functions such that f1 > −∞, g1 > −∞, dom f1 = , dom g1 = , and g1 = f1• . Moreover,

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we have −f2 ◦ p = −g2 p ∀p ∈ R V . Thus, Theorem 7.1(i) yields the equation inf Fξ = inf f1 x + f2 x

ξ∈R A

x∈R V

= − inf g2 p + g1 p = − inf Gp p∈R V

p∈R V

To state the first optimality criterion, we define the auxiliary network Gξ = V Aξ  wξ  associated with a feasible flow ξ ∈ R A . The underlying graph Gξ has the vertex set V and the arc set Aξ consisting of three disjoint parts, Aξ = Fξ ∪ Bξ ∪ Jξ , where Fξ = a  a ∈ A fa ξa+ +1 < +∞ Bξ = a¯  a ∈ A fa ξa+ −1 < +∞

a¯ reorientation of a

Jξ = u v  u v ∈ V u = v f  ∂ξ+ v u < +∞ The weight function wξ Aξ → R is defined by   a ∈ Fξ ,  fa ξa+ +1 ¯ −1 a ∈ Bξ , wξ a = fa¯ ξa+  f  ∂ξ+ v u a ∈ Jξ  For

any cycle C ⊆ Aξ  in Gξ , we define the weight of C by wξ C = wξ a  a ∈ C, and call C a negative cycle if wξ C < 0. Theorem 7.6. Let f be a polyhedral convex function with dom f = , and ξ ∈ R A be a feasible flow. (i) There is no negative cycle in Gξ  wξ  if ξ is optimal for (MFBC). (ii) The converse is also true if f is M-convex. That is, for f ∈ : ξ is optimal for (MFBC) ⇐⇒ there is no negative cycle in Gξ  wξ . Proof. (i) Suppose that there exists a negative cycle C ⊆ Aξ  in Gξ . We show that Fξ∗  < Fξ holds for some flow ξ∗ ∈ R A . Put  CF = C ∩ Fξ  CB = C ∩ Bξ  CJ = C ∩ Jξ  y= χv − χu  u v∈CJ

Then, there exists a sufficiently small α > 0 such that  f ∂ξ + αy − f ∂ξ = αf  ∂ξ+ y ≤ α u v∈CJ

wξ u v

(74)

fa ξa + α − fa ξa = αwξ a

a ∈ CF 

(75)

fa¯ ξa ¯ − α − fa¯ ξa ¯ = αwξ a ¯

a ∈ CB 

(76)

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419

where the inequality in (74) is from Theorems 4.2 and 4.4. We define a flow ξ∗ ∈ R A by  a ∈ CF   ξa + α ¯ −α a ∈ CB  ξ∗ a = ξa  ξa otherwise Combining the inequalities (74), (75), and (76), we obtain  Fξ∗  − Fξ = f ∂ξ + αy − f ∂ξ + fa ξa + α − fa ξa +

 a∈CB

a∈CF

fa¯ ξa ¯ − α − fa¯ ξa ¯

≤ αwξ C < 0 (ii) Suppose that there exists a flow ξ∗ ∈ R V with Fξ∗  < Fξ. To prove the existence of a negative cycle in Gξ  wξ , it suffices to show the existence of a “circulation” in Gξ  wξ  with

negative cost w.r.t. wξ , i.e., the existence of ζ Aξ → R satisfying ζ ≥ 0, a∈Aξ wξ aζa < 0 and 

ζa  a∈ Aξ  leaves v =

We have

 ζa  a∈ Aξ  enters v v ∈ V  (77)

fa ξ∗ a − fa ξa ≥ ξ∗ a − ξafa ξa+ +1 ∀a ∈ supp+ ξ∗ − ξ

(78)

fa ξ∗ a − fa ξa ≥ ξa − ξ∗ afa ξa+ −1 ∀a ∈ supp− ξ∗ − ξ f ∂ξ∗  − f ∂ξ ≥ f  ∂ξ+ ∂ξ∗ − ∂ξ

(79) (80)

By Theorem 4.15 (ii), there exists some λ = λuv  u v ∈ Jξ  such that  λuv ≥ 0 u v ∈ Jξ  λuv χv − χu  = ∂ξ∗ − ∂ξ u v∈Jξ



u v∈Jξ

λuv f  ∂ξ+ v u = f  ∂ξ+ ∂ξ∗ − ∂ξ

We now define a function ζ Aξ → R as   maxξ∗ a − ξa 0 ¯ − ξ∗ a ¯ 0 ζa = maxξa λ uv

a ∈ Fξ  a ∈ Bξ  a = u v ∈ Jξ .

(81)

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It is easy to see that ζ ≥ 0 and ζ satisfies (77). Moreover, (78), (79), and (81) imply   wξ aζa = ξ∗ a − ξafa ξa+ +1 a∈supp+ ξ∗ −ξ

a∈Aξ



+

−

ξa − ξ∗ afa ξa+ −1

a∈supp ξ∗ −ξ

+



u v∈Jξ

λuv f  ∂ξ+ v u

≤ Fξ∗  − Fξ < 0 Hence, ζ is a circulation in Gξ  wξ  with negative cost. Remark 7.7. The claim (ii) of Theorem 7.6 does not hold in general when f ∈ . Let G = V A be a directed graph with V = u v w and A = u w v w, fa R → R ∪ +∞ a ∈ A a family of flow cost functions such that dom fa = "0 1#

fa α = 0 α ∈ dom fa 

and f R V → R ∪ +∞ a boundary cost function such that
− minxu xv xu ≥ 0 xv ≥ 0 xV  = 0 f x = +∞ (otherwise). It is easy to see that f is polyhedral convex, but not polyhedral M-convex. For a feasible flow ξ = 0 0 ∈ R A , the auxiliary network Gξ = V Aξ  wξ  is such that Aξ = u w v w w u w v

wξ a = 0 a ∈ Aξ 

There is no negative cycle in Gξ  wξ , but ξ is not optimal; a feasible flow ξ∗ ∈ R A is optimal if and only if either ξ∗ u w = 1 or ξ∗ v w = 1 holds. We next state an optimality criterion by potentials. Recall the definition of the coboundary δp of a potential p ∈ R V in (3). Theorem 7.8. Let ξ∗ ∈ R A be a feasible flow. (i)

If the condition

∂ξ∗ ∈ arg min f "−p∗ #

ξ∗ a ∈ arg min fa "−ηa#

∀a ∈ A

(82)

holds for some p∗ ∈ R V and η = −δp∗ , then ξ∗ is optimal for (MFBC). (ii) Suppose that f is polyhedral convex with dom f = , and fa ∈
1 a ∈ A. Then, ξ∗ is optimal for (MFBC) if and only if (82) holds for some p∗ ∈ R V and η = −δp∗ .

polyhedral m/l-convex functions (iii)

421

If p∗ ∈ R V and η = −δp∗ satisfy (82), then we have p∗ v − p∗ u ≤ wξ u v

∀u v ∈ Aξ∗ 

(83)

(iv) Suppose f ∈  and fa ∈
1 a ∈ A. Then ξ∗ is optimal for (MFBC) if and only if (83) holds for some p∗ ∈ R V . Proof.

Note that Fξ = f "−p∗ #∂ξ +

 a∈A

fa "−ηa#ξa

for any p∗ ∈ R V and η = −δp∗ . Hence, the claim (i) follows immediately. The claim (iii) is also immediate from the definition of wξ . (ii) Suppose ξ∗ is optimal for (MFBC). Define fi R V → R ∪ ±∞ i = 1 2 by   f1 x = f x f2 x = inf fa ξa  ∂ξ = x x ∈ R V  ξ∈R A

a∈A

Then Corollary 7.2 implies the existence of p∗ ∈ R V with (71). Note that    inf f2 "p∗ #x = inf p∗  ∂ξV + fa ξa x∈R V

ξ∈R A

=



a∈A

a∈A

inf δp∗ aα + fa α

α∈R

where the second equality is by p∗  ∂ξV = δp∗  ξA . Hence, we have (82). (iv) By Theorem 4.12 for f ∈  and the convexity of fa a ∈ A, the condition (82) is equivalent to (83). Hence, the claim follows from (ii). Remark 7.9. The following claim is easy to show: for a feasible flow ξ ∈ R A , if there exists p∗ ∈ R V with (83), then there is no negative cycle in the auxiliary network Gξ  wξ . Indeed, for any cycle C in Gξ  wξ    wξ C = wξ u v ≥ p∗ v − p∗ u = 0 u v∈C

u v∈C

follows from (83). The converse is also a well-known fact in graph theory. Summarizing the above results for f ∈ , we obtain the following statement. Corollary 7.10. Suppose f ∈ . For any feasible flow ξ ∈ R A , the following three conditions are equivalent: (OPT) ξ is optimal for (MFBC). (NNC) There is no negative cycle in the auxiliary network Gξ  wξ . (POT) There exists a potential p ∈ R V with (83).

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murota and shioura 7.3. Network Induction

As shown in Section 4 (Theorems 4.18 and 4.35 in particular), there are various operations for polyhedral M/L-convex functions. In this section, we explain a more powerful operation called network induction. Network induction is a transformation by using networks and includes many operations such as translation, restriction, projection, etc., as its special cases. We show that polyhedral M/L-convex functions can be transformed into polyhedral M/L-convex functions by network induction. Let G = V A+ S T  be a directed graph with vertex set V , arc set A, and two disjoint vertex subsets S and T called entrance set and exit set, respectively. For f R S → R ∪ +∞ and fa R → R ∪ +∞ a ∈ A, define f˜ R T → R ∪ ±∞ by   fa ξa  ∂ξ = x −y 0 ξ ∈ R A  f˜y = inf f x + ξx

a∈A

x −y 0 ∈ R S × R T × R V \S∪T 



y ∈ R T 

(84)

For g R S → R ∪ +∞ and ga R → R ∪ +∞ a ∈ A, define g˜ R T → R ∪ ±∞ by   ga ηa  η = −δp q r ∈ R A  gq ˜ = inf gp + ηpr

a∈A

S

p q r ∈ R × R T × R V \S∪T 



q ∈ R T 

Theorem 7.11. Let f , g be polyhedral convex functions such that dom f = , dom g = , and g = f • , and fa  ga ∈
1 with ga = fa• a ∈ A. Suppose that at least one of the following three conditions holds: (a)

f˜y0  is finite for some y0 ∈ R T ,

gq ˜ 0  is finite for some q0 ∈ R T , ˜ 0  < +∞ for some y0 ∈ R T and q0 ∈ R T . (c) f˜y0  < +∞ and gq

(b)

Then, the following statements hold: (i) both f˜ and g˜ are polyhedral convex functions such that dom f˜ = , dom g˜ = , f˜ > −∞, g˜ > −∞, and g˜ = f˜• , (ii) if f is polyhedral M-convex, then f˜ is also polyhedral M-convex, (iii) Proof.

if g is polyhedral L-convex, then g˜ is also polyhedral L-convex. (i)

We first prove two claims.

Claim 1. For y ∈ R T , we have f˜y = g ˜ • y if either f˜y < +∞ or gq ˜   < +∞ for some q ∈ R T .

polyhedral m/l-convex functions

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Define a function fˆ R S × R T × R V \S∪T  → R ∪ +∞ by
y  = −y z  = 0 f x  fˆx  y   z   = +∞ (otherwise). Then its conjugate fˆ• R S × R T × R V \S∪T  → R ∪ +∞ is given by fˆ• p  q  r   = gp  − q  yT

p  q  r   ∈ R S × R T × R V \S∪T  

By Theorem 7.5, we have 
 V ˜ ˆ fa ξa  ξ ∈ R f y = inf f ∂ξ + a∈A




= − inf fˆ• p  q  r   +

 a∈A

ga ηa  η = −δp  q  r   ∈ R A 

p  q  r   ∈ R S × R T × R V \S∪T 



  = − inf −q  yT + gq ˜    q ∈ R T = g ˜ • y This ends the proof for Claim 1. ˜ = f˜• q if either gq ˜ < +∞ or Claim 2. For q ∈ R T , we have gq   T ˜ f y  < +∞ for some y ∈ R . Define a function gˆ R S × R T × R V \S∪T  → R ∪ +∞ by
q = q, gp     gp ˆ q r  = +∞ (otherwise). Then its conjugate g ˆ • R S × R T × R V \S∪T  → R ∪ +∞ is given by
z  = 0, f x  + q y   g ˆ • x  y   z   = +∞ (otherwise). By Theorem 7.5, we have
 ga ηa  η = −δp  q  r   ∈ R A  gq ˜ = inf gp ˆ   q  r   + a∈A

p  q  r   ∈ R S × R T × R V \S∪T 


= − inf g ˆ • x  y   z   +

 a∈A







S



fa ξa  ∂ξ = x  y   z   ξ ∈ R A  T

x  y  z  ∈ R × R × R

V \S∪T 



  = − inf q y  T + f˜−y    y  ∈ R T = f˜• q This ends the proof for Claim 2.

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We see from Claim 1 (resp. Claim 2) that the conditions (a) and (c) (resp. (b) and (c)) are equivalent. Hence, the statement (i) follows. (ii) Suppose that g is polyhedral L-convex. Define r0 = gp + λ1 − gp/λ, which is independent of p ∈ R S and λ ∈ R . Since δp + λ1 q + λ1 r + λ1 = δp q r we have




gq ˜ + λ1 = inf

ηp r 

gp  +




 a∈A

= inf gp + λ1 + ηpr

∀λ ∈ R  

ga ηa  η = −δp  q + λ1 r    a∈A

 ga ηa  η = −δp q r

= gq ˜ + λr0 For q1  q2 ∈ dom g, ˜ we show gq ˜ 1  + gq ˜ 2  ≥ gq ˜ 1 ∧ q ˜ 1 ∨ q2 . 2  + gq For i = 1 2, let pi  qi  ri  ∈ R V satisfy gq ˜ i  = gpi  + a∈A ga ηi a, where ηi = −δpi  qi  ri . Putting η∧ = −δp1 ∧ p2  q1 ∧ q2  r1 ∧ r2 

η∨ = −δp1 ∨ p2  q1 ∨ q2  r1 ∨ r2 

we obtain ga η1 a + ga η2 a ≥ ga η∧ a + ga η∨ a

a ∈ A

by the convexity of ga , while gp1  + gp2  ≥ gp1 ∧ p2  + gp1 ∨ p2  holds by the submodularity of g. Hence it follows  gq ˜ 1  + gq ˜ 2  ≥ gp1 ∧ p2  + gp1 ∨ p2  + ga η∧ a + ga η∨ a a∈A

˜ 1 ∨ q2  ≥ gq ˜ 1 ∧ q2  + gq (iii)

This is immediate from (i), (iii), and Theorem 5.1.

Proof of Theorem 4 18 7. Consider the bipartite graph G = V ∪ v0  ∪ U   A+ S T  with S = V , T = U  ∪ v0 , and A = v v0   v ∈ V \ U ∪ u u   u ∈ U, where u ∈ U  is the copy of u ∈ U. Then we have fˆy0  y = f˜−y0  −y for y0  y ∈ R × R U , where fˆ is defined by (38) and f˜ is the induced function defined by (84). Thus, fˆ is polyhedral M-convex by Theorem 7.11(ii).

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Remark 7.12. The convolution of two functions can be represented as a special case of network induction, and vice versa. We first show how to represent the convolution of two functions as a network induction. Let fi R V → R ∪ +∞ i = 1 2. Let V1 and V2 be disjoint copies of V and consider a bipartite graph G = V1 ∪ V2 ∪ V A+ S T  with S = V1 ∪ V2 , T = V , and A = v1  v  v ∈ V  ∪ v2  v  v ∈ V , where vi ∈ Vi is the copy of v ∈ V i = 1 2. Define f R V1 ∪V2 → R ∪ +∞ and fa R → R ∪ +∞ a ∈ A by f x1  x2  = f1 x1  + f2 x2  xi ∈ R Vi 

fa α = 0 α ∈ R 

Then f1 f2 x = f˜−x x ∈ R V , where f˜ is the induced function defined by (84). We next show that the function f˜ obtained by the network induction (84) can be represented as the convolution of two functions. Suppose we are given a directed graph G = V A+ S T  with an entrance set S and an exit set T , and functions f R S → R ∪ +∞, fa R → R ∪ +∞ a ∈ A. We define fˆi R S∪T → R ∪ ±∞ i = 1 2 as


f x y = 0T , x ∈ R S  y ∈ R T  +∞ (otherwise)  
  ξ ∈ RA f ξa  ˆ a f2 x y = inf x ∈ R S  y ∈ R T   ∂ξ = −x −y 0 a∈A fˆ1 x y =

Then the convolution fˆ1

fˆ2 R S∪T → R ∪ ±∞ is written as

   x1 + x2 = x xi ∈ R S    y1 + y2 = y yi ∈ R T   = inf fˆ1 x1  0T  + fˆ2 x2  y  x1 + x2 = x xi ∈ R S   = inf f x  + fˆ2 x − x  y  x ∈ R S 
  ξ ∈ R A  x ∈ R S     = inf f x  + fa ξa  ∂ξ = x − x −y 0 a∈A


fˆ1 fˆ2 x y = inf fˆ1 x1  y1  + fˆ2 x2  y2 

for x ∈ R S and y ∈ R T . Thus, fˆ1

fˆ2 0S  y = f˜y holds for y ∈ R T .

ACKNOWLEDGMENT The authors thank Satoru Fujishige for his helpful comments.

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murota and shioura REFERENCES

1. R. K. Ahuja, T. L. Magnanti, and J. B. Orlin, “Network Flows—Theory, Algorithms, and Applications,” Prentice Hall, Englewood Cliffs, NJ, 1993. 2. G. Birkhoff, Rings of sets, Duke Math. J. 3 (1937), 443–454. 3. R. E. Bixby and W. H. Cunningham, Matroid optimization and algorithms, in “Handbook of Combinatorics” (R. L. Graham, M. Gr¨ otschel, and L. Lov´asz, Eds.), Vol. I, Chap. 11, pp. 551–609, Elsevier, Amsterdam, 1995. 4. A. Bouchet and W. H. Cunningham, Delta-matroids, jump systems, and bisubmodular polyhedra, SIAM J. Discrete Math. 8 (1995), 17–32. 5. W. J. Cook, W. H. Cunningham, W. R. Pulleyblank, and A. Schrijver, “Combinatorial Optimization,” Wiley, New York, 1998. 6. V. Danilov, G. Koshevoy, and K. Murota, Equilibria in economies with indivisible goods and money, RIMS preprint, No. 1204, Kyoto University, 1998. 7. A. W. M. Dress and W. Wenzel, Valuated matroid: A new look at the greedy algorithm, Appl. Math. Lett. 3 (1990), 33–35. 8. A. W. M. Dress and W. Wenzel, Valuated matroids, Adv. Math. 93 (1992), 214–250. 9. J. Edmonds, Submodular functions, matroids and certain polyhedra, in “Combinatorial Structures and Their Applications” (R. Guy, H. Hanani, N. Sauer, and J. Sch¨ onheim, Eds.), pp. 69–87, Gordon & Breach, New York, 1970. 10. U. Faigle, Matroids in combinatorial optimization, in “Combinatorial Geometries” (N. White, Ed.), pp. 161–210, Cambridge Univ. Press, London, 1987. 11. P. Favati and F. Tardella, Convexity in nonlinear integer programming, Ricerca Operativa 53 (1990), 3–44. 12. A. Frank, An algorithm for submodular functions on graphs, Ann. Discrete Math. 16 (1982), 97–120. 13. A. Frank, Generalized polymatroids, in “Finite and Infinite Sets” (A. Hajnal, L. Lov´asz, and V. T. S´ os, Eds.), pp. 285–294, North-Holland, Amsterdam, 1984. ´ Tardos, Generalized polymatroids and submodular flows, Math. Program14. A. Frank and E. ming 42 (1988), 489–563. 15. S. Fujishige, A note on Frank’s generalized polymatroids, Discrete Appl. Math. 7 (1984), 105–109. 16. S. Fujishige, Submodular systems and related topics, Math. Programming Study 22 (1984), 113–131. 17. S. Fujishige, “Submodular Functions and Optimization,” Annals of Discrete Mathematics, Vol. 47, North-Holland, Amsterdam, 1991. 18. S. Fujishige and K. Murota, Notes on L-/M-convex functions and the separation theorems, Math. Programming 88 (2000), 129–146. 19. T. Ibaraki and N. Katoh, “Resource Allocation Problems: Algorithmic Approaches,” MIT Press, Cambridge, MA, 1988. 20. M. Iri, “Network Flow, Transportation and Scheduling—Theory and Algorithms,” Academic Press, New York, 1969. 21. S. Iwata, A capacity scaling algorithm for convex cost submodular flows, Math. Programming 76 (1997), 299–308. 22. N. Katoh and T. Ibaraki, Resource allocation problems, in “Handbook of Combinatorial Optimization II” (D.-Z. Du and P. M. Pardalos, Eds.), pp. 159–260, Kluwer Academic, Boston, MA, 1998. 23. E. L. Lawler, “Combinatorial Optimization: Networks and Matroids,” Holt, Rinehart & Winston, New York, 1976.

polyhedral m/l-convex functions

427

24. L. Lov´asz, Submodular functions and convexity, in “Mathematical Programming—The State of the Art” (A. Bachem, M. Gr¨ otschel, and B. Korte, Eds.), pp. 235–257, Springer-Verlag, Berlin, 1983. 25. K. Murota, Finding optimal minors of valuated bimatroids, Appl. Math. Lett. 8 (1995), 37–42. 26. K. Murota, Valuated matroid intersection. I. Optimality criteria, SIAM J. Discrete Math. 9 (1996), 545–561. 27. K. Murota, Valuated matroid intersection. II. Algorithms, SIAM J. Discrete Math. 9 (1996), 562–576. 28. K. Murota, Matroid valuation on independent sets, J. Combin. Theory Ser. B 69 (1997), 59–78. 29. K. Murota, Submodular flow problem with a nonseparable cost function, Combinatorica 19 (1999), 87–109. 30. K. Murota, Convexity and Steinitz’s exchange property, Adv. Math. 124 (1996), 272–311. 31. K. Murota, Structural approach in systems analysis by mixed matrices—An exposition for index of DAE, in “ICIAM 95” (K. Kirchg¨assner, O. Mahrenholtz, and R. Mennicken, Eds.), pp. 257–279, Mathematical Research, Vol. 87, Akademie Verlag, Berlin, 1996. 32. K. Murota, Discrete convex analysis, Math. Programming 83 (1998), 313–371. 33. K. Murota, On the degree of mixed polynomial matrices, SIAM J. Matrix Anal. Appl. 20 (1999), 196–227. 34. K. Murota, Discrete convex analysis—Exposition on conjugacy and duality, in “Graph Theory and Combinatorial Biology” (L. Lov´asz et al., Eds.), pp. 253–278, The J´anos Bolyai Mathematical Society, 1999. 35. K. Murota, Discrete convex analysis, in “Discrete Structures and Algorithms V” (S. Fujishige, Ed.), pp. 51–100, Kindai-Kagakusha, Tokyo, 1998. [In Japanese] 36. K. Murota, “Matrices and Matroids for Systems Analysis,” Springer-Verlag, Berlin, 2000. 37. K. Murota and A. Shioura, M-convex function on generalized polymatroid, Math. Oper. Res. 24 (1999), 95–105. 38. K. Murota and A. Shioura, Relationship of M-/L-convex functions with discrete convex functions by Miller and by Favati–Tardella, Discrete Appl. Math., in press. 39. C. H. Papadimitriou and K. Steiglitz, “Combinatorial Optimization: Algorithms and Complexity,” Prentice-Hall, Englewood Cliffs, NJ, 1982. 40. W. R. Pulleyblank, Polyhedral combinatorics, in “Optimization” (G. L. Nemhauser, A. H. G. Rinnooy Kan, and M. J. Todd, Eds.), Handbooks in Operations Research and Management Science, Vol. 1, Chap. V, pp. 371–446, Elsevier, Amsterdam, 1989. 41. R. T. Rockafellar, “Convex Analysis,” Princeton Univ. Press, Princeton, 1970. 42. R. T. Rockafellar, “Network Flows and Monotropic Optimization,” Wiley, New York, 1984. 43. A. Shioura, Minimization of an M-convex function, Discrete Appl. Math. 84 (1998), 215–220. 44. J. Stoer and C. Witzgall, “Convexity and Optimization in Finite Dimension, I,” SpringerVerlag, Berlin, 1970. ´ Tardos, Generalized matroids and supermodular colourings, in “Matroid Theory” 45. E. (A. Recski and L. Lov´asz, Eds.), Colloquia Mathematica Societatis J´anos Bolyai, Vol. 40, pp. 359–382, North-Holland, Amsterdam, 1985. 46. D. Welsh, “Matroid Theory,” Academic Press, New York, 1976. 47. H. Whitney, On the abstract properties of linear dependence, Amer. J. Math. 57 (1935), 509–533. 48. N. White (Ed.), “Theory of Matroids,” Cambridge Univ. Press, London, 1986. 49. N. White (Ed.), “Combinatorial Geometries,” Cambridge Univ. Press, London, 1987. 50. N. White (Ed.), “Matroid Applications,” Cambridge Univ. Press, London, 1992.