Feasibility in reverse convex mixed-integer programming

European Journal of Operational Research 218 (2012) 58–67

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European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor

Discrete Optimization

Feasibility in reverse convex mixed-integer programming Wiesława T. Obuchowska Department of Mathematics, East Carolina University, Greenville, NC 27858, United States

a r t i c l e

i n f o

Article history: Received 29 March 2010 Accepted 13 October 2011 Available online 20 October 2011 Keywords: Integer programming Feasibility Concave integer minimization Reverse convex constraints Sensitivity analysis Irreducible infeasible sets

a b s t r a c t In this paper we address the problem of the infeasibility of systems defined by reverse convex inequality constraints, where some or all of the variables are integer. In particular, we provide a polynomial algorithm that identifies a set of all constraints critical to feasibility ðCF Þ, that is constraints that may affect a feasibility status of the system after some perturbation of the right-hand sides. Furthermore, we will investigate properties of the irreducible infeasible sets and infeasibility sets, showing in particular that every irreducible infeasible set as well as infeasibility sets in the considered system, are subsets of the set CF of constraints critical to feasibility.  2011 Elsevier B.V. All rights reserved.

1. Introduction In this paper we consider the problem of the feasibility in the following class of nonlinear integer optimization problems:

maximize f 0 ðxÞ; subject to x 2 G ¼ fx 2 Zn jfi ðxÞ P 0; i 2 J ¼ f1; 2; . . . ; mgg;

ð1Þ ð2Þ

where fi(x), i 2 J [ {0}, are either faithfully convex functions or a composition of convex increasing functions and convex polynomials. Some of the constraints in (2) may be in particular linear. The following continuous relaxation

maximize f 0 ðxÞ;

ð3Þ

subject to x 2 R ¼ fx 2 Rn jfi ðxÞ P 0; i 2 Jg;

ð4Þ

corresponding to the integer problem (1) and (2) is a multiextremal optimization problem, since it may have local maxima that are not global. It is well known [27] that the problem (3) and (4), in case the feasible region (4) is bounded, attains its global maximum at a vertex of the region R. The integer problem (1) and (2) and/or the corresponding relaxed problem were earlier studied in [5,17,20], where we have established sufficient and necessary conditions for existence of an upper bound in both problems, showing in particular [20] that if all entry data (coefficients) are rational, then boundedness of the integer problem (1) and (2) implies boundedness of the corresponding relaxed problem (3) and (4). The above problems belong to the class of one of the most important global optimization problems, which is due to their important

E-mail address: [email protected] 0377-2217/$ - see front matter  2011 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2011.10.011

applications in operations research, mathematical economics and various engineering problems [11–14,21,25]. In particular, integer programming problems with a concave objective (cost) function are often encountered in optimization models involving economies of scale, and reverse convex quadratic constraints occur for instance in packing problems and molecular conformation problems [1,2]. One of the most fundamental issues in constrained optimization is to determine whether or not a system of inequality constraints defining the feasible region is consistent. It is common to treat the latter problem as a separate minimization problem with an objective function that measures the degree of violation of the constraints at any given point. This approach requires the solution of a nonlinear problem which has still the same structure as the original problem, and contains one more variable. In particular, to determine feasibility of the region R defined in (4) we need to determine whether max{xn+1jfi(x) P xn+1, i 2 J} P 0. The latter problem can be equivalently restated as the such called piecewiseconvex maximization problem [26]. In linear programs a common approach to testing infeasibility relies either on the identification of an irreducible infeasible subset of constraints (IIS), that is, the set of constraints that is infeasible, but for which any proper subset of constraints is feasible or on identifying the infeasibility set (IN), i.e. a subset of constraints whose removal will transform the system into a feasible one. The importance and some aspects of detecting the irreducible infeasible sets and infeasibility sets has been discussed in [6–10]. In particular, given that all infeasible systems include at least one IIS, it follows that an inconsistent system of inequality constraints can be transformed into a feasible one by removing exactly one constraint from every IIS it contains. Another important observation in

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detecting infeasibility is that there are usually constraints that do not have any impact on the consistency of the system, regardless of the values of the right-hand sides. This aspect of the infeasibility analysis has been investigated in [15] in relation to the systems of quadratic convex inequality constraints and, in [16] the systems of faithfully convex constraints. In this paper we extend the results obtained earlier for convex systems in [16] to the more difficult nonconvex, and often disjoint regions defined by reverse convex constraints defined over the sets of integer. We show that all results proved for integer systems hold also for the corresponding relaxed systems. The paper is organized as follows. Section 2 contains some auxiliary results on unboundedness of the problems (1)–(4) from [17,20], necessary in the proofs of theorems presented in the remaining part of the paper. In Section 3 we present a method to identify a maximal subset of constraints that may have an impact on the feasibility status of the system after possible perturbation of the right-hand sides. Adjusting the terminology introduced in [15,16], we call the latter set ‘‘critical to feasibility constraints’’, and denote as CF . The method is given in a form of an algorithm, which at each iteration requires the identification of implicit equality constraints in a homogeneous linear system. Since the implicit equalities can be detected in a finite number of simplex steps and since each iteration of the algorithm reduces the number of constraints and the dimension of the associated convex polyhedral cone, the algorithm terminates in a finite number of steps. We show that the algorithm requires solving at most 2n(m  2) + m homogeneous linear programs. However, since the most computationally consuming part of the proposed method to determine the set CF involves finding an index set ID  J, which is produced by the algorithm to determine boundedness of the region R (and G), the set CF can be found (if ID is given) by solving only jID j homogeneous linear programs, all of which have a very similar structure. In Section 4 we investigate properties of the irreducible infeasible sets, which are expressed in terms of the implicit equalities in the corresponding linear system. We show in particular, that every irreducible infeasible set, as well as the infeasibility sets, in both the integer system and the relaxed system are subsets of the set of constraints critical to feasibility CF . Furthermore, we show that for any constraint with the index l 2 CF , it is possible to modify values of the right-hand sides of the constraints in CF , in such a way that there exists an irreducible infeasible system IISl containing l. The first of the above properties has a potential to reduce the number of constraints under consideration in the minimum-cardinality IIS set-covering problem, which is known to be NP-hard even for linear systems [6,9]. We observe that since all results proved in this paper hold for both the integer problem with reverse convex constraints (1) and (2) as well as for the corresponding relaxed problem (3) and (4), we can conclude that they also apply to the mixed-integer programming problem of the same structure, where only some of the variables are assumed to be integer. 2. Auxiliary results on unboundedness in reverse convex integer programming In this section we consider the following concave integer optimization problem

max ff0 ðxÞjx 2 G ¼ fx 2 Zn jfi ðxÞ P 0; i 2 Jgg;

ð5Þ

where fi(x), i 2 J [ {0}, are either faithfully convex functions or a composition of convex increasing functions and convex polynomials. The inequalities fi(x) P 0, i 2 J, are called reverse convex

constraints. The relaxed region R corresponding to the discrete region G will be defined as R ¼ fx 2 Rn jfi ðxÞ P 0; i 2 Jg. We will begin with recalling definitions of some of the terms used in this paper. The convex function f(x) is called faithfully convex [23] if it is constant along some segment only if it is constant along whole line containing this segment. As shown by Rockafellar in [23], every faithfully convex function fi can be represented in the form fi(x) = Fi(ci + Bix) + hbi, xi  di, where Fi are strictly convex functions defined on Rpi , (for some pi 6 n; pi 2 N), Bi 2 Rpi n , (where Bi are full-row rank matrices), ci 2 Rpi ; bi 2 Rn ; di 2 R; i 2 J [ f0g. A vector s – 0 is called a direction of recession of f(x) if for every x the function f(x + ts) is a nonincreasing function of t [22]. The symbol 0+f denotes the set of directions of recession of f(x) and D¼ f the constancy space of the function f. We assume that in case the functions fi, i 2 J [ {0} are faithfully convex, the functions are affine along any direction of recession, i.e. they satisfy the following condition:

s 2 0þ fi ) Bi s ¼ 0;

i 2 J [ f0g:

ð6Þ

Furthermore, we assume that when studying properties of the integer problem (1) and (2) and the function fi(x) is convex polynomial then its coefficients are integer (or rational), and in case fi(x) is a faithfully convex function then Bi, bi, ci are matrices (respectfully vectors) with integer (or rational) coefficients. It is well known that convex multivariate polynomials and convex functions of the form r f ðxÞ ¼ 1r ðhx; AxiÞ2 þ hb; xi  d, where r P 1, and A is positive semidefinite belong to the class of faithfully convex functions satisfying condition (6). It has been shown in [17] that Algorithm A below can be used to determine whether or not there exists a vector s 2 Rn satisfying conditions

s – 0;

ðBi s – 0 _ hbi ; si P 0Þ;

8 i 2 J:

ð7Þ

Algorithm A. [17] Step 1. Set I0 = {i 2 JjBi = 0}, and k = 0. If I0 = ;, then Stop since there is a solution to (7). Otherwise go to Step 2. Step 2. If there exists nonzero s 2 RIk ¼ fxjhbi ; xi P 0; 8 i 2 Ik g then go to Step 3. Otherwise go to Stop since there is no solution to (7). Step 3. Determine the set bI k ¼ fi 2 J n Ik jBi s ¼ 0; 8 s 2 RIk g. If bI k ¼ ; or Ik = J, then go to Stop since there is a solution to (7). Otherwise, set Ikþ1 ¼ Ik [ bI k ; k :¼ k þ 1, and go to Step 2. Stop: Set D :¼ k, and ID :¼ Ik . h As shown in [17], Algorithm A terminates after at most min {m  1, n  1} iterations indicating whether or not the system (7) has a real solution. Theorem 2.1 below states that the system (7) has an integer solution if and only if it has a real solution, which in particular implies that Algorithm A can also be used to determine whether or not the system (7) has an integer solution, and whether the nonempty region G is bounded. We will use the following theorem and lemmas proved in our earlier works. Theorem 2.1 [20]. Let us suppose that the functions fi(x), i 2 J, are faithfully convex, and satisfy condition (6) and that all coefficients in matrices Bi, i 2 J, and the vectors bi, i 2 J are integer (or rational). In particular, if fi(x) are convex polynomials, all coefficients in the functions are assumed to be integer. Then

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W.T. Obuchowska / European Journal of Operational Research 218 (2012) 58–67

Suppose that we consider the region

(i) If the system (7), i.e.

s – 0;

ðBi s – 0 _ hbi ; si P 0Þ;

F ¼ fx 2 Rn jfi ðxÞ P 0; i 2 Jg

8 i 2 J;

has a solution s 2 Rn , then the system has also an integer solution s 2 Zn , and if G – ;, then there exists n  P 0, such that  g  G, where x 2 G; n  2 N and where fxjxðnÞ ¼  x þ ns; n 2 N; n P n N denotes the set of natural numbers. (ii) The nonempty region G, defined over the set of integers, is unbounded if and only if the corresponding continuous region R is unbounded, i.e. if there exists a vector s satisfying the system (7). h Lemma 2.1 [17]. Let I  J, and RI = {xjhbi, xi P 0, " i 2 I}. Then the following statement:

8 i 2 J n I;

9si 2 RI ;

such that Bi si – 0 and Bj si ¼ 0;

8 j 2 I;

holds if and only if $s 2 RI, such that Bis – 0, " i 2 JnI, and Bjs = 0, " j 2 I. h

ð10Þ

and the corresponding integer region

G ¼ fx 2 Zn jfi ðxÞ P 0; i 2 Jg;

ð11Þ

where the functions fi, i 2 J are defined as fi(x) = hi(pi(x)), where the functions hi : R ! R are convex, strictly increasing and into R, and the functions pi : Rn ! R are convex polynomials with integer coefficients. In fact it is easy to see that the regions F and P ¼ 1 fx 2 Rn jpi ðxÞ P hi ð0Þ; i 2 Jg are identical, i.e. F ¼ P. For instance, 2 for pðxÞ ¼ x2 ; x 2 R; hðyÞ ¼ ey  1, we have fxjhðpðxÞÞ ¼ ex  1 2 P 0g ¼ fxjx P lnð1Þg. A similar statement is valid for the discrete region G, i.e. we have G ¼ P \ Zn [20]. A polynomial function p : Rn ! R with degree l is called a form of order l if pðtxÞ ¼ t l pðxÞ; 8 x 2 Rn ; 8 t 2 R. Each polynomial function p(x) of degree l P 1 can be represented as a sum of forms qj(x), j = 0, 1, . . . , l:

pðxÞ ¼ ql ðxÞ þ    þ q1 ðxÞ þ q0 ; Lemma 2.2 [20]. Let I  J, where RI be defined as in Lemma 2.1. Then if all coefficients in the matrices Bi, i 2 J and the vectors bi, i 2 I are integer, then the statement:

8 i 2 J n I;

9si 2 RI ;

such that Bi si – 0 and Bj si ¼ 0;

8 j 2 I;

n

holds if and only if 9s 2 RI \ Z , such that Bis – 0; 8 i 2 J n I, and Bjs ¼ 0; 8 j 2 I. h In order to prove the main result of this section stated in Theorem 2.2, we use the following two corollaries following from Theorem 2.1 and Lemma 2.2. It has been assumed there, as well as in the remaining part of the paper (whenever the integer problem (1) and (2) is considered), that coefficients of Bi, bi, and di are integer, unless stated otherwise.

where degree of qj(x) is j [3]. Indeed in case l > 1, the convex polynomial has an even degree. The following corollary will be used in the sequel of the paper. Corollary 2.3 [20]. If all coefficients of the polynomial p(x) are integer then linear subspace D¼ p may be described by a system of homogeneous linear equations with integer coefficients and the cone of recession 0+p can be represented by a system of homogeneous linear equations and one linear homogeneous inequality all with integer coefficients, i.e. there exist matrix G and vector g, both with integer entries only, such that

D¼p ¼ fx 2 Rn jGx ¼ 0; hg; xi ¼ 0g and

0þ p ¼ fRn jGx ¼ 0; hg; xi 6 0g: Corollary 2.1 [20]. If Bi – 0, " i 2 J and fi satisfy condition (6), then nonempty region G is unbounded. h Corollary 2.2 [20]. If the system

B0 s – 0 _ hb0 ; si > 0; Bi s – 0 _ hbi ; si P 0;

ð8Þ

8 i 2 J;

ð9Þ

has a real solution then there exists an integer solution satisfying the system. h The theorem below shows that the concave integer programming problem (5) is unbounded from above if and only if the corresponding relaxed concave programming problem (3) and (4) is unbounded.



As stated in [20], Corollary 2.3 follows from the following result proved in Bank and Mandel [3] (and the argument given in [18]): ‘‘If the quasi-convex form qm ðxÞ; x 2 Rn , has only integer coefficients, then the linear subspace L ¼ fx 2 Rn jqm ðxÞ ¼ 0g, may be described by a system of homogeneous linear equations with integer coefficients’’ (Lemma 3, p. 43, [3]). The proof of the lemma given in [3] is by construction, that is it provides a method to find a system of homogeneous linear equations with integer coefficients describing the linear subspace L. This method can be applied to find the matrix G and the vector g, which in Corollary 2.3 describe the linear subspace D¼ p and the cone of recession 0+p. Suppose that the polynomials pi(x), i 2 J [ {0}, of degree li P 1 are represented as the sums of the forms qij ðxÞ; j ¼ 0; 1; . . . ; li :

pi ðxÞ ¼ qili ðxÞ þ    þ qi1 ðxÞ þ qi0 ; Theorem 2.2 [20]. Let us suppose that the functions fi(x), i 2 J [ {0}, satisfy assumptions stated in Theorem 2.1. Then the function f0(x) is unbounded from above on G; ðG – ;Þ if and only if the function f0 is unbounded from above over the corresponding relaxed region R, i.e. if there exists a vector s satisfying conditions (8) and (9). h In the remaining part of this section the results stated in Theorems 2.1 and 2.2 for faithfully convex functions are extended to additional class of convex functions, namely functions which are composition of convex increasing functions and convex multivariate polynomials with integer (rational) coefficients.

where degree of qij ðxÞ is j. The following theorem provides necessary and sufficient conditions for boundedness of the discrete region G and its continuous relaxation F, defined in (11) and (10), respectively. Theorem 2.3 [20]. The nonempty regions F and G are unbounded if and only if there exists a nonzero vector s, such that (i) s 2

T

i2J ½ðR

or equivalently

n

n 0þ pi Þ [ D¼ pi ;

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W.T. Obuchowska / European Journal of Operational Research 218 (2012) 58–67

is infeasible and the system

(ii) the nonzero vector s satisfies the conditions 9si 2 fli ; li  2; . . . ; 2g; qisi ðsÞ – 0 _ qi1 ðsÞ P 0; i 2 J, equivalent to

s – 0;

ðGi s – 0 _ hg i ; si P 0Þ; pi n

where Gi 2 Z

which

are

8 i 2 J;

Bi s ¼ 0; ð13Þ



The following theorem provides necessary and sufficient conditions for unboundedness of the composite function f0(x) = h0(p0(x)) over both the discrete region G and the relaxed region F. Theorem 2.4 [20]. The function f0(x) = h0(p0(x)), where the function h0 : R ! R is convex and increasing and the function p0 : Rn ! R is convex polynomial with integer coefficients, is unbounded above over each of the nonempty regions F and G if and only if there exists a nonzero vector s, such that

\h

i h i ðRn n 0þ pi Þ [ D¼pi n 0þ p0 [ D¼g0 ;

q0s0 ðsÞ – 0 _ q01 ðsÞ > 0;

9si 2 fli ; li  2; . . . ; 2g;

i

qsi ðsÞ – 0 _ qi1 ðsÞ P 0;

i 2 J;

which are equivalent to

G0 s – 0 _ hg 0 ; si > 0;

8 i 2 J;

ð14Þ

where Gi 2 Zpi n ; g i 2 Zn ; i 2 J [ f0g and where Gi and gi satisfy the system of equations given in (13). h

3. Infeasibility analysis in reverse convex programs: constraints critical to feasibility In this section we will show how to identify the set of all inequalities in the integer system of reverse convex inequalities (where some or all of the variables may be integer), that may affect the feasibility status of the system after some perturbation of the right-hand sides.

i 2 J;

ð15Þ

belongs to the set CF of constraints critical to feasibility, (i.e. s 2 CF ), if there exist values a0s > 1, and ai > 1, i 2 J, such that the system

fi ðxÞ P ai ;

i 2 J;

ð16Þ

is infeasible and the system

fi ðxÞ P ai ;

i 2 J n fsg;

fs ðxÞ P a0s ;

ð17Þ

is feasible or the system (16) is feasible and (17) is inconsistent. Similarly we say that the sth inequality in the integer system

fi ðxÞ P 0;

i 2 J;

x 2 Zn

ð18Þ

belongs to the set CF of constraints critical to feasibility, if there 0s , and a i > 1; i 2 J, such that either the exist finite numbers a system

i ; fi ðxÞ P a

x 2 Zn ;

i 2 J;

8 i 2 J;

ð21Þ

is an implicit equality if it is satisfied as an equality, i.e. hbl, si = 0 for all s satisfying (21). We assume that the index set ID has been obtained as an outcome of applying Algorithm A to the system (15). Theorem 3.1. Let us suppose that the functions fi(x), i 2 J, are either faithfully convex, and satisfy condition (6), or they are a composition of convex increasing functions and convex polynomials, and that all coefficients in the polynomials and in the matrices Bi, i 2 J, as well as the vectors bi, i 2 J are integer. Then if either I0 = ; (where I0 = {i 2 JjBi = 0}), or the system

8 i 2 I0 ;

ð22Þ

Proof. Suppose that there are no implicit equalities in the system (22). Then there exists s, such that hbi, si > 0, " i 2 I0, which implies that the convex cone RI0 ¼ fx 2 Rn jhbi ; xi P 0; 8 i 2 I0 g, is fulldimensional. This property along with the definition of the index set bI 0 given in Algorithm A imply that bI 0 ¼ ;, and consequently Algorithm A terminates after the first iteration, with the message that there is a real solution to the system (7). Thus ID ¼ I0 , where ID is the index set obtained by applying Algorithm A to the system (15). We first note that 8 i 2 J n ID , there exists si, such that

hbj ; si i > 0;

8 j 2 ID ;

Bi si – 0:

ð23Þ

Then it follows from Lemma 2.1, that 9s 2 Posðsi ; i 2 J n ID Þ, (where Posðsi ; i 2 J n ID Þ denote the nonnegative hull of the vectors si ; i 2 J n ID ), such that Bi s – 0; 8 i 2 J n ID . Furthermore, since P si 2 int ðRI0 Þ, then for s ¼ i2JnID t i si ; t i P 0; i 2 J n ID ,

hbj ; si ¼

X

t i hbj ; si i > 0;

j 2 ID ;

i2JnID

that is s 2 int ðRID Þ, which means that s satisfies the system

Definition 3.1. We say that the sth inequality in the system

fi ðxÞ P 0;

hbi ; si P 0;

has no implicit equalities, then the regions R and G are nonempty and unbounded with int ðRÞ – ;, and the systems (15) and (18) have no constraints critical to feasibility.

or equivalently the vector s satisfies the following conditions

9s0 2 fl0 ; l0  2; . . . ; 2g;

8 i 2 J;

hbi ; si P 0;

i2J

Gi s – 0 _ hg i ; si P 0;

h

Definition 3.2. An inequality hbl, si P 0, in the system

fx 2 Rn jGi x ¼ 0; hg i ; xi 6 0g ¼ fx 2 Rn jqis ðxÞ ¼ 0; s

s2

ð20Þ

ð12Þ

; g i 2 Z , are such that

i 2 J:

x 2 Zn ;

is feasible or the system (19) is feasible and (20) has no solution.

n

¼ li ; li  2; . . . ; 2; qi1 ðxÞ 6 0g;

0s ; f s ðxÞ P a

i 2 J n fsg;

i ; fi ðxÞ P a

ð19Þ

hbj ; si > 0;

8 j 2 ID ;

Bi s – 0;

8 i 2 J n ID :

ð24Þ

Since the latter system has a real solution, Corollary 2.2 implies that the system (24) has also an integer solution, say s . Thus s R 0þ fi ; i 2 J, and therefore for the set of integer points on the half-line xðnÞ ¼ ns ; n 2 N, there exists n0, such that xðnÞ  G, for n P n0, (and xðtÞ  intðRÞ, for t P n0), or more specifically fi(x(n)) ? 1, as n ? 1, for all i 2 J. This indicates that the regions R and G are unbounded and nonempty for any values of the right-hand sides of the inequality constraints. Suppose now that I0 = ;. By Lemma 2.1, there exists s such that Bis – 0, " i 2 J, which by Theorem 2.1 (i) implies that the latter system has also an integer solution, say s. This, similarly as in the previous case (i.e. when I0 – ;and the system (22) has no implicit equalities), implies that fi ðnsÞ ! 1, as n ? 1, for all i 2 J. This completes the proof of the lemma. h Besides of an extreme case considered in Theorem 3.1, when there are no implicit equalities in the system of constraints with indices I0, another extreme possibility holds when all inequalities in the system

62

hbi ; si P 0;

W.T. Obuchowska / European Journal of Operational Research 218 (2012) 58–67

8 i 2 ID

ð25Þ

are implicit equalities (which happens for instance when the set R is bounded). It will follow from Theorem 3.3 presented in this section that in the latter case (i.e. when all inequalities in (25) are implicit equalities) it holds that CF ¼ ID . We will show that the set CF of constraints critical to feasibility in the relaxed system (15) is identical with the set of constraints critical to feasibility in the integer system (18), and that CF is a subset of ID , which in particular implies that the right-hand sides of all constraints with indices in J n ID do not have any impact on the feasibility status of neither the region R nor the region G. Theorem 3.2. Let us suppose that the functions fi(x), i 2 J, satisfy assumptions of Theorem 3.1. Then the constraint with the index s is critical to feasibility in the system (15) if and only if the sth constraint is critical to feasibility in the integer system (18).

is bounded from above. Let ms be the maximal value of the objec0 tive function of the problem (29). Since for any ds > ms , the system

hbi ; xi P di ;

i 2 ID n fsg;

0

hbs ; xi P ds ; 0

is infeasible, the constraint hbs ; xi P ds is critical to feasibility in the system

hbi ; xi P di ;

i 2 ID :

Now we will consider a case when at least some of the constraints with indices ID are nonlinear. In case the constraint with the index s 2 ID is nonlinear, consider k 6 D, such that s 2 bI k , where bI k is defined in the kth step of Algorithm A. Then for all s 2 RIk , we have Bss = 0. If sth constraint is linear, then s 2 I0 and Bs = 0, and thus Bss = 0 for any s. Furthermore, since sth constraint is an implicit equality in the system (28), then we have that for every vector s the following implication holds

s 2 RID ) hbs ; si ¼ 0: Proof. Suppose first that the constraint with the index s in the sys0 tem (15) is critical to feasibility and that R –;. Then 9ds > 0 such that the system

fi ðxÞ P 0;

i 2 J n fsg;

0

f s ðxÞ P ds ;

ð26Þ

is bounded. Let ai, i 2 Jn{s} be such that the system fi ðxÞ P ai ; i 2 J n fsg; x 2 Zn has a solution. Then Theorem 2.2 (and/or Theorem 2.4, depending on how fi(x), i 2 J are defined) allows us to conclude that the integer programming problem

max ffs ðxÞjfi ðxÞ P ai ; i 2 J n fsg; x 2 Zn g;

ð27Þ 00

is also bounded. The latter fact yields that 9ds > 0 such that the system 00

f s ðxÞ P ds ;

Bi s – 0 _ hbi ; si P 0;

Bi s ¼ 0 ^ hbi ; si P 0;

max ffs ðxÞjfi ðxÞ P 0; i 2 J n fsgg;

i 2 J n fsg;

It follows from the process of construction of ID , that a nonzero vector s is a solution to the system

8 i 2 ID ;

if and only if it satisfies the system

is infeasible. This implies that the problem

fi ðxÞ P ai ;

ð30Þ

8 i 2 ID :

Thus, if the vector s satisfies the latter system then s satisfies the set of inequalities hbi ; si P 0; 8 i 2 ID . Therefore, taking into account implication (30) we are able to conclude that the system

Bs s – 0 _ hbs ; si > 0;

Bi s – 0 _ hbi ; si P 0;

8 i 2 ID n fsg;

ð31Þ

has no solution. Assume that R – ;. Since the system (31) has no solution, Theorem 2.3 [17] implies that the problem

max ffs ðxÞjfi ðxÞ P 0; i 2 ID n fsgg; is bounded. Let ms ¼ sup ffs ðxÞjfi ðxÞ P 0; i 2 ID n fsgg. We have that 0 for any ds > ms , the system

x 2 Zn ;

has no solution, which immediately implies that the sth constraint in the system (18) is critical to infeasibility. Proof of the backward part of the theorem is analogous, given the fact that by Theorem 2.2 (and/or Theorem 2.4), the problem (26) is bounded if and only if the integer programming problem (27) is bounded, provided that the feasible region of the problem (27) is nonempty. h

fi ðxÞ P 0;

i 2 ID n fsg;

0

fs ðxÞ P ds ; is infeasible, which yields that the system

fi ðxÞ P 0;

i 2 J n fsg;

0

The following notation will be used in the proof of Theorem 3.3 presented below. The value of D and the sets RIk ; k ¼ 0; 1; . . . ; D, as well as the index sets Ik and bI k ; k ¼ 0; 1; . . . ; D are obtained by applying Algorithm A to the system (15). Theorem 3.3. Suppose that the functions fi, i 2 J satisfy assumptions of the Theorem 3.1. Then the index set of constraints critical to feasibility ðCF Þ in both the system (15) and the integer system (18) is identical with the index set of implicit equalities in the linear homogeneous system

hbi ; xi P 0;

i 2 ID :

ð28Þ

Proof. To prove the backward part of the theorem, suppose first for simplicity that all constraints fi ðxÞ; i 2 ID , are linear, and that sth constraint in the system (28) is an implicit equality. Assume that the region defined by the system hbi ; xi P di ; i 2 ID is consistent. (For simplicity, though without a loss of generality we assume that di are the right-hand sides in the latter system instead of some ai > 1; i 2 ID ). Therefore the problem

max fhbs ; xijhbi ; xi P di ; i 2 ID n fsgg;

ð29Þ

fs ðxÞ P ds ; is also inconsistent. This observation along with the assumption that R – ; implies that the sth constraint in the system (15) is critical to infeasibility, which ends the proof of the backward part of the theorem. To prove the forward part of the theorem, let us assume that the constraint with the index s in the system (15) is critical to 0 feasibility and R – ;. Thus 9ds > 0, such that the system

fi ðxÞ P 0;

i 2 J n fsg;

0

fs ðxÞ P ds ; is infeasible. This yields that the concave problem with reverse convex constraints

max ffs ðxÞjfi ðxÞ P 0; i 2 J n fsgg; is bounded, and consequently by Theorem 2.3 in [17] the system

Bs s – 0 _ hbs ; si > 0; Bi s – 0 _ hbi ; si P 0;

ð32Þ

8 i 2 J n fsg;

ð33Þ

does not have a nonzero solution s. We consider first the case when the region R is bounded. In this case Algorithm A, when applied to the system (15), terminates in Step 2, with the message that the

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set RID ¼ fxjhbi ; xi P 0; 8 i 2 ID g ¼ f0g. The latter fact implies that every inequality in the system (28) is an implicit equality, which completes the proof in case the region R is bounded. Suppose now that R is unbounded. This implies by Theorem 2.2 [17], that the system

Bi s – 0 _ hbi ; si P 0;

i 2 J;

ð34Þ

has a nonzero solution s, while the assumption that s  th constraint is critical to feasibility and Theorem 2.3 [17] imply that there is no solution to the system (32) and (33). This means that for every nonzero vector s satisfying the system (34) it holds that Bss = 0 ^ hbs, si = 0. Since every vector s satisfying (34) must be in RID , this immediately indicates that s 2 ID . (Note that it is not true in general that every vector in RID satisfies (34)). Now, in order to show that the inequality hbs, xi P 0 is an implicit equality in the system (28), it is sufficient only to show that there is no s 2 RID , for which hbs, si > 0. Suppose to the contrary that 9s 2 RID , such that hbs ; si > 0. Furthermore, it follows from the construction of the index set D that 8 j 2 J n ID ; 9sj 2 RID , such that Bjsj – 0, and by Lemma 2.1, there 9^s 2 RID , where ^s 2 Pos ðsj ; j 2 J n ID Þ, such that Bj^s – 0; 8 j 2 J n ID . Now we will demonstrate that 9t > 0, such that for ~s ¼ ^s þ ts; Bj~s – 0; 8 j 2 J n ID . To this end let us suppose for simplicity of notation that J n ID ¼ fj; j þ 1; . . . ; mg. The following procedure will be used to determine the value of the parameter t, and therefore to construct the vector ~s satisfying the above conditions. h Procedure A1 Step 1: l :¼ 0; ~s :¼ ^s; t max :¼ 0. Step 2: If Bjþls ¼ 0 then tl :¼ 1, and go to Step 3; otherwise (i.e. if Bjþls – 0) determine tl > tmax, such that B ~s t l >  Bjþl s, and go to the Step 3. jþl Step 3: Let tmax :¼ max {tmax, tl}. Set ~s :¼ ~s þ t ls. If l < m  j then l :¼ l + 1 and go to the Step 2, otherwise go to P j Stop with t ¼ m l¼0 t l . Stop. P j Indeed we have that ~s ¼ ^s þ ts for t ¼ m l¼0 t l , (where tl are determined by applying Procedure A1), satisfies Bj~s – 0; 8 j 2 J n ID . Since t > 0, and ^s; s 2 RID , as well as hbs ; si > 0, then ~s ¼ ^s þ ts 2 RID , with hbs ; ^s þ tsi > 0. Therefore we arrive at a contradiction with an earlier conclusion that there is no s satisfying the system (32) and (33). This completes the proof of the theorem for the region R. Since by Theorem 3.2 the constraint in the system (15) is critical to feasibility if and only if it is critical to feasibility in the corresponding integer system (18), the result remains true for both systems. On the other hand the proof given above for the relaxed system of reverse convex constraints (15) can be applied with appropriate modifications to the integer system (18). The mentioned modifications would involve use of Theorems 2.1 and 2.2, and Lemma 2.2 as well as selecting parameters tl with positive integer values when applying Procedure A1 to find value of t. h The following corollary follows directly from Theorem 3.2.

  0 Gd0 ¼ x 2 Zn jfi ðxÞ P di ; i 2 J ;

In the corollary below we show some relationship between cardinality of CF , and boundedness of the set R. Corollary 3.2. If jCF j < jID j, then the sets R and G are either empty or 0 unbounded. Furthermore, for any numbers di 2 R; i 2 J, each of the sets

ð35Þ

ð36Þ

is either empty or unbounded. Proof. Inequality jCF j < jID j and Theorem 3.3 indicate that there are jID j  jCF j inequalities in the system

hbi ; si P 0; 8 i 2 ID ;

ð37Þ

which are not implicit equalities. This implies that the system (37) has a nonzero solution s. Now it follows from Lemma 2.1 that there exists real vector s⁄ satisfying the system (7). In case all entries of Bi and bi are integer, Lemma 2.2 assures the existence of the integer vector s , for which conditions (7) hold. To be more specific, both vectors s⁄ and s satisfy (37) and Bi s – 0; 8 i 2 J n ID . Thus in case the set G is nonempty, the half-line xðtÞ ¼ x0 þ ts 2 R; 8t P t 0 , for some t0 P 0, and x0 2 R and xðnÞ ¼ x þ ns 2 G; 8 n 2 N; n P n0 for some n0 2 N, and x 2 G, implying unboundedness of R and G. The proof of the second statement in the corollary follows from the fact that by Corollary 3.1 values of the right-hand sides of the 0 constraints fi ðxÞ P di ; i 2 J n CF do not have any impact on the feasibility (and boundedness) status of the systems in (35) and (36), and the right-hand sides of the constraints of the subsystem 0 fi ðxÞ P di ; i 2 CF do not have an impact on boundedness of the nonempty regions Rd0 and Gd0 . h 4. Irreducible infeasible sets It is well known (Helly’s theorem), that in case of relaxed systems of convex inequality constraints, the system is infeasible only if there exists an infeasible subset of constraints of cardinality not greater than n + 1. Similar problem of finding an upper bound for the minimal cardinality of an infeasible subset of inconsistent integer linear system has been studied in [4] by Bell and in [24] by Scarf, who proved that every inconsistent system of linear inequality constraints defined over the set of integers contains an infeasible subset of constraints of cardinality not greater than 2n. In [19], we generalized the latter result by showing that any system of convex inequality constraints (satisfying some mild assumptions), defined over the set of integers, contains an irreducible infeasible subset of cardinality not greater than 2n. However, it is easy to see that unfortunately there is no upper bound for the number of constraints in the irreducible infeasible set of constraints in the integer system

fi ðxÞ P 0;

i 2 J;

x 2 Zn

ð38Þ

or in the corresponding relaxed system

fi ðxÞ P 0;

i 2 J:

ð39Þ

In order to demonstrate this fact we consider the following inconsistent system in Z2 :

ðx1  iÞ2 þ ðx2  jÞ2 P

Corollary 3.1. Constraints fi ðxÞ P ai ; i 2 J n CF do not have any impact on the feasibility status of the systems (15) and (18) regardless of the values of the right-hand-sides ai ; i 2 J n CF . h

  0 Rd0 ¼ x 2 Rn jfi ðxÞ P di ; i 2 J ;

and

1 1 6 x1 6 m þ ; 2 2

1 ; 3

i ¼ 1; 2; . . . ; m;

1 1 6 x2 6 m þ ; 2 2

x1 ;

j ¼ 1; 2; . . . ; m; x2 2 Z:

For any natural number m the system has m2 + 4 constraints, all of which are necessary, since removing any of the constraints make the system feasible. Similar example, though with x1 ; x2 2 R, and pffiffi 2 with the right-hand sides 13 being replaced with 22 þ  , (where pffiffi 0 <  < 1  22), can be used to show that there is no upper bound for the number of constraints in the irreducible infeasible set of constraints defined in (39).

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We will use the symbol IIS (irreducible infeasible subset) with respect to the system of reverse convex inequality constraints with a similar meaning that has been used by other authors in [6–10] with respect to the system of linear inequality constraints, and in [15] with respect to convex quadratic constraints as well as in [16] with respect to faithfully convex constraints. More precisely, we define:

are expressed in terms of the implicit equalities in the corresponding homogeneous linear system. We note that while the system (38) and its continuous relaxation (39) have identical set of constraints critical to feasibility CF , the index sets of irreducibly infeasible sets IIS and IIS corresponding to these systems are different in general, and they depend on values of the right-hand sides.

Definition 4.1. The subset of constraints in (39) is called an irreducible infeasible subset of the system (39) (denoted as IIS), if it is infeasible, but for which any proper subset of constraints is consistent. The subset of constraints in (38) is called an irreducible integer infeasible subset of the system (38) (and it will be denoted as IIS), if it is infeasible in the set of integers, but for which any proper subset of constraints has a feasible integer solution. h

Theorem 4.1. Let IIS denote the irreducibly infeasible subset of the system (39) and IIS the irreducible integer infeasible subset of the system (38). Then

Algorithm B below which is a straightforward extension of the deletion filtering algorithm for linear programs [7], identifies an irreducible infeasible subset of the system (38) or (39). If m is significantly larger than jCF j, then constraints in the set CF  ID , where ID is obtained by the application of Algorithm A can be considered as an input set of Algorithm B identifying an IIS. Let us assume, without loss of generality that CF ¼ f1; 2; . . . ; pg, and that the system of constraints with indices in CF is infeasible.

(i) Every inequality in each of the systems

Bi s ¼ 0;

hbi ; si P 0;

8 i 2 IIS

ð41Þ

Bi s ¼ 0;

hbi ; si P 0;

8 i 2 IIS;

ð42Þ

and

is an implicit equality. (ii) Any IIS and any IIS belong to CF , that is

[

IIS a # CF

and

a

[

IISa # CF ;

a

where IIS a and IISa are members of the indexed family of the irreducible infeasible subsets of the system (39) and (38) respectively.

Algorithm B. To identify the irreducible infeasible set 1. Let IIS ¼ CF ; s ¼ 1. 2. Determine whether or not the system

fi ðxÞ P 0;

i 2 IIS n fsg;

x 2 Zn ;

ð40Þ

is feasible. 3. If NO then IIS :¼ IIS n fsg. If s < p go to Step 2 with s = s + 1, otherwise Stop. If YES and s < p then s :¼ s + 1 and go to Step 2, otherwise Stop. h

In case the condition x 2 Zn is dropped from the system (40), Algorithm B identifies an irreducible infeasible set IIS in the system (39). The proof of Algorithm B is contained in the lemma below. Lemma 4.1. If the system (38) (or (39)) is infeasible then the output of Algorithm B will contain exactly one irreducible integer infeasible set. Furthermore, there are at most jCF j! irreducible infeasible subsets in (38) and (39).

Proof. (i) The proof will be given for the irreducible infeasible subset IIS of the system (38), although it remains valid for the relaxed system (39) after removing restrictions that the variable x is integer. Suppose that the system (42) contains a constraint (say with the index k), which is not an implicit equality. Therefore there exists ^s 2 Rn , such that

Bi^s ¼ 0;

8 i 2 IIS;

8 i 2 IIS n fkg;

hbi ; ^si P 0;

hbk ; ^si > 0:

Since the coefficients of the latter system are integer, then Corollary 4.1 [18] shows the existence of the integer solution to the above system, which without a loss of generality can be denoted using the same symbol ^s. Now it follows from the definition of the index set IIS that the system

fi ðxÞ P 0;

i 2 IIS n fkg;

x 2 Zn ;

is feasible. Thus for ^x satisfying the latter system, it holds that

fi ð^x þ n^sÞ ¼ F i ðBi ð^x þ n^sÞ þ ci Þ þ hbi ; ^x þ n^si  di ¼ fi ð^xÞ þ hbi ; ^sin P hbi ; ^sin P 0;

8 n 2 N; 8i 2 IIS n fkg, and Proof. The proof of the first part of the lemma follows from the fact that the set IIS produced by Algorithm B is infeasible in the set of integers, while removal of any constraints in IIS will transform the system into a system feasible over Zn . Since each irreducible infeasible set can be obtained as an output of Algorithm B for some permutation of the set CF , the second part of the lemma follows. h It is easy to see that various IIS (or IISÞ may have different number of elements as the following simple example of the system of 5 linear inequalities demonstrates:

x1  x2 P 1;

x1 þ x2 P 0;

x1 P 0;

x1 P 1;

x1 P 2;

where the first three inequalities form IIS as well as the last two. Theorems 4.1 and 4.2 below provide several properties of the irreducible infeasible subsets of constraints (IIS and IISÞ. They

fk ð^x þ n^sÞ ¼ fk ð^xÞ þ hbk ; ^sin > 0;

8n >

fk ð^xÞ ; hbk ; ^si

8 n 2 N;

which contradicts the fact that the system of constraints with indices in the set IIS has no integer solution.(ii) Let IISa0 be an arbitrary irreducible integer infeasible set. By part (i) of this theorem, for every j 2 IISa0 , jth inequality in the system

Bi s ¼ 0;

hbi ; si P 0;

i 2 IISa0 ;

is an implicit equality. Suppose that IISa0  CF . Consider first the case when 9i 2 ðIISa0 n CF Þ \ ID and IISa0 \ ðJ n ID Þ ¼ ;. Since in this case we have that IISa0 # ID , then by Theorem 3.3 this would imply that the inequalities hbi ; si P 0; 8 i 2 IISa0 n CF are not implicit equalities in the system

Bi s ¼ 0;

hbi ; si P 0;

8 i 2 ID :

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W.T. Obuchowska / European Journal of Operational Research 218 (2012) 58–67

Furthermore, since IISa0 # ID , then the latter fact would allow us to conclude that 8 i 2 IISa0 n CF , the inequality hbi, si P 0, is not an implicit equality in the system

Bi s ¼ 0;

Theorem 4.2. Let IIS denote the irreducible integer infeasible subset of the system (38) and IIS the irreducible subset of the system (39). Then for any l 2 CF , there exist di ; i 2 CF , such that the system

8 i 2 IISa0

hbi ; si P 0;

and therefore there exists s0, such that

Bi s0 ¼ 0;

hbi ; s0 i P 0;

8 i 2 IISa0 \ CF ;

Bi s0 ¼ 0;

hbi ; s0 i > 0;

8 i 2 IISa0 n CF :

fi ðxÞ P di ;

8 i 2 IIS a0 n ID :

ð44Þ

b I , we have Bi s ¼ 0; 8 i 2 ID , then it follows that Since for every s 2 R D ^s also satisfies

Bi^s ¼ 0;

hbi ; ^si P 0;

8 i 2 IIS a0 \ ID :

ð45Þ

Since the system fi ðxÞ P 0; i 2 IIS a0 is irreducible, then the system

fi ðxÞ P 0;

i 2 IIS a0 \ ID ;

ð46Þ

is feasible. Suppose that ^x is a solution to the system (46). Then for the vector ^s we have that 9^t > 0, such that xðtÞ ¼ ^x þ t^s 2 R1 ¼ fx 2 Rn jfi ðxÞ P 0; i 2 IIS a \ ID g; 8 t P ^t. Since the vector ^s satisfies the systems (44) and (45), then 9t P 0, such that xðtÞ 2 fx 2 Rn jfi ðxÞ P 0; i 2 IIS a0 g; 8 t P max ft; ^tg. This contradicts the fact that the system of constraints with indices IIS a0 is an infeasible set. Now assume that IISa0 \ ðJ n ID Þ – ;. Then by Lemma 2.2 b I ¼ fxjhbi ; xi P 0; i 2 ID g \ Zn , for which the we get that 9~s 2 R D conditions

Bi~s – 0;

8 i 2 IISa0 n ID

ð47Þ

b I , it holds that Bi s ¼ 0; 8 i 2 ID , are satisfied. Since for every s 2 R D then we get that the integer vector ~s also satisfies the system

Bi~s ¼ 0; hbi ; ~si P 0;

8 i 2 IISa0 \ ID :

ð48Þ

Furthermore, since the system fi ðxÞ P 0; x 2 Zn ; ı 2 IISa0 is irreducible, and IISa0 \ ID – IISa0 , then the system of constraints

fi ðxÞ P 0;

x 2 Zn ;

i 2 IISa0 \ ID ;

i 2 CF ;

x 2 Zn ;

ð50Þ

ð43Þ

Since ðIISa0 \ CF Þ [ ðIISa0 n CF Þ ¼ IISa0 , the existence of a nonzero solution to the system (43) would contradict the property of the set IISa0 proved in part (i) of this theorem that all inequalities in the system (41) are implicit equalities. This completes the proof of the part (ii) of the theorem in case IISa0 # ID . The proof in case IIS a0 # ID is analogous. Now suppose that IIS a0 \ ðJ n ID Þ – ;. Then by Lemma 2.1 we b I ¼ fxjhbi ; xi P 0; i 2 ID g such that get that 9^s 2 R D

Bi^s – 0;

fi ðxÞ P 0; i 2 CF , so that the system contains an irreducible subset containing constraint with the index l.

ð49Þ

is feasible. Suppose that ~x is an integer solution to the latter system. Taking into account that the matrices Bi as well as the vectors bi have all coefficients integer, by Theorem 2.1 (considering that any vector s satisfying (48) satisfies Bi s – 0 _ hbi ; si P 0; 8 i 2 IISa0 \ID ), the system consisting of (47) and (48) has an integer solution s⁄. Thus for an integer vector x⁄ satisfying the system (49) we have ^ 1 2 N, such that xðnÞ ¼ x þ ns 2 R1 \ Zn ; 8 n P n ^ 1 . Since that 9n the integer vector s⁄ satisfies the systems (47) and (48), then ^ 2 N, such that xðnÞ 2 fx 2 Zn jfi ðxÞ P 0; i 2 IISa0 g; 8 n P n ^ . This 9n contradicts the fact that the system of inequalities with indices IISa0 is an irreducible integer infeasible set, which completes the proof of the part (ii) of the theorem. h Theorem 4.2 below shows that Theorem 4.1 (ii) cannot be stated in a stronger form, in the sense that for any l 2 CF it is always possible to modify values of the right-hand sides in the system

contains irreducible integer infeasible subsystem IIS, with l 2 IIS. Sim0 ilarly, for any l 2 CF , there exist di ; i 2 CF , such that the system 0

fi ðxÞ P di ;

i 2 CF ;

contains irreducible infeasible subsystem IIS, containing constraint with the index l. Proof. Let us consider the following procedure to determine updated vector of the right-hand sides of the system (50), i.e. di ; i 2 CF so that the conclusion of the theorem holds. Procedure A2 Step 1. Let l 2 CF and IISl :¼ CF . Step 2. Find the constant d such that the system

fi ðxÞ P d;

i 2 IISl ;

x 2 Zn ;

is feasible. Step 3. Find the maximum ~f l of the problem.

~f ¼ max f ðxÞ; l l subject to f i ðxÞ P d;

i 2 IISl n flg;

x 2 Zn :

Set dl ¼ ~f l þ , for some  > 0. Step 4. For s 2 IISl n flg determine whether or not the system

fi ðxÞ P d; i 2 IISl n flg n fsg; fl ðxÞ P dl ; x 2 Zn ; is feasible. Step 5. If NO then IISl :¼ IISl n fsg and go to Step 4 with the next s, (s – l). Step 6. If YES then go to Step 4 with the next s 2 IISl ðs – lÞ. If all s 2 CF n flg were examined, then go to Stop. Stop.

We will show that for any index l 2 CF the Procedure A2 generates a subset IISl of CF , containing l, and the updated vector of the right-hand sides, such that the system of constraints with indices in IISl is an irreducible infeasible system. To this end we observe that Steps 2 and 3 assure that the system of constraints

fi ðxÞ P d;

i 2 IISl n flg;

f l ðxÞ P dl ;

x 2 Zn ;

is infeasible, but removal of the constraint with the index l will change the status of the system into a feasible one. Similarly, Steps 4–6 assure that removal of any constraint with the index in IISl n flg will transform the system into a feasible one. Therefore, every proper subset of IISl is feasible, which implies that IISl is an irreducible integer infeasible set and by the construction process, l 2 IISl , which completes proof of the first part of the theorem. The proof of the second part of the theorem for the relaxed system (39) is very similar, and it can be obtained by applying Procedure A2 with removed integral condition x 2 Zn . h

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W.T. Obuchowska / European Journal of Operational Research 218 (2012) 58–67

Corollary 4.1. Let us suppose that the functions fi, i 2 J defining the system (39) and (38) satisfy assumptions of Theorem 3.1, i.e. fi(x) are either faithfully convex, and satisfy condition (6), or they are a composition of convex increasing functions and convex polynomials, and that all coefficients in the polynomials and in the matrices Bi, i 2 J, as well as the vectors bi, i 2 J are integer. Then any infeasible set of reverse convex constraints and in particular any irreducible infeasible set IIS (and IIS) contains at least two linear inequality constraints. Proof. It follows from Theorem 3.1, that if R (respectively G) is infeasible, then there exists at least one linear constraint. If there is only one linear inequality constraint in the system, then the system hbi, si P 0, i 2 I0, has no implicit equality, and by Theorem 3.1 the region R (respectively G) is nonempty, contradicting the assumption. h Infeasibility set (denoted as IN ) is a subset of constraints, whose removal will transform the system into a feasible one, provided that no subset of IN has the latter property. The following corollary is a trivial consequence of the definition of the infeasibility set and of the constraints critical to feasibility. It remains valid for both integer as well relaxed systems, even if the constraint functions fi are not necessarily convex or continuous. Corollary 4.2. Let the system (38) be infeasible. Let fIN a ; a 2 Xg, be an indexed family of all infeasibility sets. Then

[

IN a # CF :

a2X

S Proof. Let a2D IISa be a union of the family of all irreducible integer infeasible sets of J. Since a subset IN a of J is an infeasibility set of the system (38) if and only if IN a contains an element of each irreducible infeasible set of J, then

[

IN a #

a2X

[

IISa :

a2D

By part (ii) of Theorem 4.1 we have

[

IISa # CF ;

a2D

which completes the proof of the theorem.

h

Example 4.1. We apply Algorithm A to the system

fi ðxÞ ¼ hbi ; xi  di P 0;

i ¼ 1; 2; 3;

f4 ðxÞ ¼ hx; B4 xi4 þ hb4 ; xi  d4 P 0; 5

f5 ðxÞ ¼ hx; B5 xi2 þ hb5 ; xi  d5 P 0 with m = 5, n = 4, B1 = B2 = B3 = 0,

2

1 1 1 1

3

2

7 6 61 1 1 17 7 B4 ¼ 6 6 1 1 1 1 7; 5 4

T

T b4

¼ ð1; 1; 1; 2Þ;

3

7 6 61 1 1 17 7 B5 ¼ 6 6 1 1 1 1 7; 5 4

1 1 1 3 b1 ¼ ð1; 1; 1; 1Þ;

1 1 1 1

1 1 1 4 T

b2 ¼ ð1; 1; 1; 1Þ; T b5

d3 ¼ d4 ¼ d5 ¼ 100:

¼ ð1; 1; 1; 4Þ;

T

b3 ¼ ð1; 1; 1; 1Þ;

d1 ¼ 25;

d2 ¼ 15;

We start the algorithm with k = 0. Since I0 = {1, 2, 3} – ;, then we go to Step 2 and since RI0 ¼ fxjhbi ; xi P 0; i ¼ 1; 2; 3g – f0g we proceed to Step 3. We determine that bI 0 ¼ ;, since

ð1; 1; 1; 1ÞT x ¼ 0 ^ ð1; 1; 1; 1ÞT x P 0 ; ð1; 1; 1; 1ÞT x ¼ 0 ^ ð1; 1; 1; 3ÞT x ¼ 0 and

ð1; 1; 1; 1ÞT x ¼ 0 ^ ð1; 1; 1; 1ÞT P 0 ; ð1; 1; 1; 1ÞT x ¼ 0 ^ ð1; 1; 1; 4ÞT x ¼ 0: Theorem 3.3 implies that the index set CF of constraints critical to feasibility in both the integer system and the corresponding relaxed system is identical with the index set of implicit equalities in the linear homogeneous system hbi, xi P 0, i = 1, 2, 3. The inequality T b3 s P 0 is not an implicit equality, which can be verified using the vector sT1 ¼ ð1; 1; 1; 1ÞT . Therefore CF ¼ f1; 2g, which implies that the feasibility of the original system is equivalent to the feasibility of the following reduced system

 x1 þ x2  x3 þ x4 P 25; x1  x2 þ x3  x4 P 15: Note that if either the value d2 is changed from d2 = 15 to d2 > 25 or the value d1 is increased from d1 = 25 to d1 > 15, then the original system becomes infeasible. We note that although the system considered in this example is feasible, any infeasible system obtained by the perturbation of the right-hand sides, has easily identified irreducible infeasible set and infeasibility sets. They are respectively equal: IIS ¼ CF ¼ f1; 2g; IN 1 ¼ f1g; IN2 ¼ f2g; all being subsets of CF . h We investigated the performance of Algorithm A on several other systems involving reverse convex multivariate polynomial inequalities. The results obtained for these examples suggest that the proposed method to determine the set of constraints critical to feasibility CF may be particularly beneficial for large sparse systems which contain subsystem of linear inequalities forming unbounded region. Indeed, if either the system has no more than one linear inequality or the linear homogeneous system corresponding to the linear subsystem forms a full dimensional polyhedral cone, then none of the constraints belongs to the set CF as shown in Theorem 3.1, i.e. CF ¼ ;. On the other hand, if the feasible region R is known in advance to be bounded, then CF ¼ J. 5. Conclusions In this paper we considered the problem of infeasibility of systems defined by reverse convex inequality constraints, where some or all of the variables are integer. We first presented a method to detect all constraints which do not have an impact on the feasibility status of the system regardless of the values of the right-hand sides. Application of the method allows us to obtain a new system with the reduced number of constraints, which is feasible if and only if the original system is feasible. Furthermore, we develop several results on the irreducible infeasible sets, showing in particular that every irreducible infeasible set in the considered system, is a subset of the set CF of constraints critical to feasibility. All conditions for boundedness and feasibility provided in this paper can be verified by using the algorithm, which requires solving a finite number of homogeneous linear programming problems (HLP). More specifically, we observe that if jI0j 6 1 then Algorithm A terminates after the first iteration with ID containing no more than one index, which by Theorem 3.3 implies that CF ¼ ;. Therefore in order to determine whether i 2 bI k ; k ¼ 0; 1; . . . ; D, we need

W.T. Obuchowska / European Journal of Operational Research 218 (2012) 58–67

to solve 2n HLPs with at most m  1 constraints. The latter observations along with the fact that implementation of the Step 2 requires at most m HLPs, allows us to conclude that the existence of the solution to the system (7) can be checked by solving at most 2n(m  2) + m linear programs, each with at most m  1 linear constraints with zero right-hand sides. In particular, the index set ID and the set CF of all constraints critical to feasibility in (38) and (39) can also be found by solving at most 2n(m  2) + m HLPs. Furthermore, since the index set ID is generated by the algorithm to determine boundedness of the regions G, and R at no additional cost, the set CF can be found (if ID is known) by solving only jID j linear programs with zero right-hand sides, all of which have a very similar structure. In fact, it is beneficial to determine the set CF only when Algorithm A concludes that the nonempty region R is unbounded, since otherwise ID ¼ J. While, it was important for us to note that the method to determine the set of constraints critical to feasibility is polynomial, the Procedures A1 and A2 are used only to prove some of the theoretical results proposed in this paper. Concluding, since all results developed in this paper are valid for both the integer problem (1), (2) as well as for the corresponding relaxed problem (3), (4), we are able to infer that these results also apply to the mixed-integer programming problem of the same structure, where only some of the variables are assumed to be integer. Acknowledgments The author thanks two anonymous referees for their helpful comments and suggestions. References [1] B. Addis, M. Locatelli, F. Schoen, Efficiently packing unequal disks in a circle, Operations Research Letters 36 (2008) 342–343. [2] B. Addis, M. Locatelli, F. Schoen, Disk packing in a square: A new global optimization approach, INFORMS Journal on Computing 20 (4) (2008) 516– 524. [3] B. Bank, R. Mandel, Parametric Integer Optimization, Mathematical Research, vol. 39, Academie-Verlag, Berlin, 1988. [4] D.E. Bell, A theorem concerning the integer lattice, Studies in Applied Mathematics 56 (1977) 187–188.

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[5] R.J. Caron, W.T. Obuchowska, Unboundedness of a convex quadratic function subject to concave and convex quadratic constraints, European Journal of Operational Research 63 (1992) 114–123. [6] N. Chakravarti, Some results concerning post-infeasibility analysis, European Journal of Operational Research 73 (1994) 139–143. [7] J.W. Chinneck, E.W. Dravnieks, Locating minimal infeasible constraint sets in linear programs, ORSA Journal on Computing 3 (1991) 157–168. [8] J.W. Chinneck, Analyzing infeasible nonlinear programs, Computational Optimization and Applications 4 (1995) 167–179. [9] J.W. Chinneck, Feasibility and Infeasibility in Optimization, Algorithms and Computational Methods, Springer, 2008. [10] H.J. Greenberg, Consistency, redundancy, and implied equalities in linear systems, Annals of Mathematics and Artificial Intelligence 17 (1996) 37–83. [11] R.J. Hillestad, S.E. Jacobsen, Reverse Convex Programming, Applied Mathematics and Optimization 6 (1980) 63–78. [12] R. Horst, H. Tuy, Global Optimization, Springer-Verlag, 1993. [13] D. Li, X. Sun, Nonlinear Integer Programming, Springer, 2006. [14] G.L. Nemhauser, L.A. Wolsey, Integer and Combinatorial Optimization, Interscience Series in Discrete Mathematics and Optimization, Wiley, London, 1988. [15] W.T. Obuchowska, Infeasibility analysis for systems of quadratic convex inequalities, European Journal of Operational Research 107 (1998) 633–643. [16] W.T. Obuchowska, On infeasibility of systems of convex analytic inequalities, Journal of Mathematical Analysis and Applications 234 (1999) 223–245. [17] W.T. Obuchowska, Conditions for boundedness in concave programming under reverse convex and convex constraints, Mathematical Methods of Operations Research 65 (2) (2007) 261–279. [18] W.T. Obuchowska, On boundedness of (quasi-)convex integer optimization problems, Mathematical Methods of Operations Research 68 (3) (2008) 445– 467. [19] W.T. Obuchowska, Minimal infeasible constraint sets in convex integer programs, Journal of Global Optimization 46 (3) (2010) 423–433. [20] W.T. Obuchowska, Unboundedness in reverse convex and concave integer programming, Mathematical Methods of Operations Research 72 (2010) 187– 204. [21] P.M. Pardalos, H.E. Romeijn (Eds.), Nonconvex Optimization and its Applications, vol. 62, Kluwer Academic Publishers, Dordrecht, 2004. [22] R.T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, NJ, 1979. [23] R.T. Rockafellar, Some convex programs whose duals are linearly constrained, in: J.B. Rosen, O.L. Mangasarian, K. Ritter (Eds.), Nonlinear Programming, Academic Press, New York, 1970. [24] H.E. Scarf, An observation on the structure of production sets with indivisibilities, Proceedings of the National Academy of Sciences of the United States of America 74 (9) (1977) 3637–3641. [25] A. Schrijver, Theory of Linear and Integer Programming, Wiley, 1986. [26] I. Tsevendorj, Piecewise-convex maximization problems; global optimization condition, Journal of Global Optimization 21 (2001) 1–14. [27] U. Ueing, A combinatorial method to compute a global solution of certain nonconvex optimization problems, in: F.L. Lootsma (Ed.), Numerical Methods for Non-linear optimization, Academic Press, New York, 1972.