Lexicographical order and duality in multiobjective programming

342

European Journal of Operational Research 33 (1988) 342-348 North-Holland

Lexicographical order and duality in multiobjective programming Juan Enrique MARTINEZ-LEGAZ Departamento de Ecuaciones Funeionales, Facultad de Matemdticas, Universitat de Barcelona, Spain

Abstract: This paper studies the applications of lexicographical order relation for vectors in the mathematical theory of multiobjective programming. We show that any Pareto minimum of an unconstrained convex problem is the lexicographical minimum for the problem associated to a matrix multiplier having lexicographical positive columns. A similar result is also obtained for inequality constrained problems. Our approach to the theory of duality follows the pattern of Jahn [3], but we substitute vectors by matrices in the formulation of the dual problem and the usual scalar order relation by the lexicographical order relation. This allows us to state the Strong Duality Theorem in terms of Pareto minima and to eliminate some regularity assumptions.

Keywords: Multiple criteria, convex programming, Lagrange multipliers

O. Introduction Separation theorem of convex sets constitute an essential tool in proving existence theorems in optimization theory. In [7] we proved one of them which is valid for any (not necessarily closed) convex set, and used it in quasiconvex conjugation theory. Some other applications to linear and convex inequality systems and optimization, together with a study of the lexicographical order relation, are given in [8]. Its consequences in the field of multiobjective programming are examined in this paper, mainly in the area of duality. The motivation of this approach is in extending some known results to problems which do not meet some regularity assumptions. In the case of multiobjective linear programming Isermann [5] established a duality theory. In the nonlinear case there exist several different approaches (see, for example, [2-6,9-13]). Our development is based on [6]. Let R = R t~ { + oe }. We shall consider < / z , the lexicographical ordering on R k, that is, given t, t ' ~ R k such that t a t ' we have t < L t ' if

Received November 1983

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t, < t', where t = (q . . . . . tk) T, t ' = (q . . . . . t() T and s is the first index for which t, 4: t~ (all vectors are considered column vectors and the superscript T indicates transposition). The relations ~L and >~L will have the corresponding obvious meanings. A matrix P is lexicographically nonnegative, P >~L0, if all of its columns are lexicographically nonnegative. The symbols >/ and > are reserved for the termwise order relations, while P >_ 0 means that every column of P is nonnegative and different from zero. A lower-triangular matrix L will be called unitary if its diagonal elements are one. Given a ~ R we shall denote by ( a ) k the vector (0 . . . . . 0, a ) T ~ R k. An inequality system will be called consistent if it has some solution. The following results will be used (see Corollaries 2.1 and 2.2a, Proposition 3 and Corollary 5.2 in [8], where a more detailed treatment of the lexicographical order relation and its applications to the theory of inequality systems is given). We shall give the proofs in order to make this paper self-contained.

Lemma 0.1. A matrix A & lexicographically nonnegative if and only if A = L P for some unitary lower-triangular matrix L and some P >10.

0377-2217/88/$3.50 © 1988, ElsevierSciencePublishers B.V. (North-Holland)

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J.E. Martinez-Legaz / L e x i c o g r a p h i c a l order and duality

Proof. Let us s u p p o s e that A = L P , L = (lij), P = (Pij). W e have

A = (a,j),

i-1

i--1

a , j = ~_, likPkj+Piy.

E ti,%(x) + f , ( x ) + e > 0

k=l

If p , j = 0 for i = 1 . . . . . h, clearly a ~ j = 0 for the s a m e values of i, a n d if m o r e o v e r P h + l , j > 0, then ah+l.j =Ph+l,j > 0. This shows that every c o l u m n of A is l e x i c o g r a p h i c a l l y nonnegative. T h e converse will be p r o v e d b y showing that there is an u n i t a r y l o w e r - t r i a n g u l a r m a t r i x L such that T~4 >~ 0. It will suffice, since A = L - ~ L 4 a n d Z -~ is again u n i t a r y a n d lower-triangular. W e shall a p p l y i n d u c t i o n on the n u m b e r of rows of A. If it is one, trivially L = (1). Let n o w A b e an m × n matrix• By the i n d u c t i o n h y p o t h e s i s , ]_,m_lAm_l >10 for some u n i t a r y a n d l o w e r - t r i a n g u l a r L,~ ~, where Am_ 1 is the m a t r i x o b t a i n e d b y d r o p p i n g the last row in A. Let us d e n o t e b y r~ . . . . . rm v the rows of A = (aij), a n d J = { j i m 1 = m i n { i I aij 4= 0 } } . W e put a m _ 1 >/ max{ - am J a m _ 1j ] J ~ d } if d =~fl a n d a r b i t r a r y t i m - i if d = f~. I n this w a y t h e m a t r i x ( r l . . . . . rm-2, tim ~rm-1 + rm) T is l e x i c o g r a p h i c a l l y nonnegative. By r e p e a t i n g this process we can get scalars ti1 . . . . . am-2 such that ti1 rw + . . . +tim_~rT_~ + rT >/0. T h e d e s i r e d m a t r i x is the one o b t a i n e d b y a d d i n g {ti1 . . . . . t i m - > 1) as a last row to I-m-~ ( a n d o b v i o u s l y zeros as the m - 1 first terms of the m-th column). L e m m a 0.2. Let C c R

r e l a t i o n ( f l ( x ) . . . . . f , _ l ( x ) , ~ ( x ) + e) T > L 0 holds. W e shall show that there exist tiil,e . . ., tlez,i- 1 such that

", f , : C - - * R ,

i = 1 . . . . . k,

f ( x ) = ( f l ( x ) . . . . . f k ( x ) ) T for x ~ C. I f for any e > 0, there is an unitary lower-triangular matrix L~ such that L J ( x ) + e(1 . . . . . 1) T > 0 for all x ~ C, then f is lexicographically nonnegative over C. I f C is compact and the fi's are continuous, the converse also holds. Proof. Let us s u p p o s e that the existence of matrices L~ holds, a n d let x o ~ C , i ~ ( 1 . . . . . m}. G i v e n e > 0, we c o n s i d e r the c o r r e s p o n d i n g L~ a n d a ~ , j = 1 . . . . . i - 1, the first elements of the i-th row of L,. W e have i-1

(x0) + e = E a,%(x0) +f (x0) j=l

whence f / ( x 0 ) > 0. Conversely, if f is l e x i c o g r a p h i c a l l y n o n n e g a tive over C a n d e > 0, for each i = 1 . . . . . m, the

j=l

for all x ~ C; thus, these n u m b e r s will be the first terms of the i-th row of L~. W e c o n s i d e r the set X=(x~Cift(x)=O, l=l ..... i-2, f,(x) + e < 0}. F o r a n y x ~ X we have ~ l ( x ) > 0, whence d=inf(f, l(X)lx~X}>O. If d = 0 , there is a sequence x n ~ X such that f i _ l ( x , ) ~ 0. Since C is c o m p a c t we can c h o o s e a s u b s e q u e n c e xn, ' converging to some ~ ~ C. W e o b t a i n : ft(~)=limfl(x,,,)=0

forl=l

f, l ( ~ ) = l i m ~ _ l ( x , ~ ) = l i m f ,

..... i-2, l(x,,)=0,

thus f i ( ~ ) + e > 0. But on the o t h e r h a n d , f,(ff) + e = lim f/(xn~ ) + e ~< 0, which yields a c o n t r a d i c tion. T h e r e f o r e d > 0 . Let d ' ~ ( 0 , d), c< inf{ f i ( x ) + e I x ~ C}, ae.i_l >/ max{0, - c / d ' } . F o r any x ~ X.

ti~,,_,f,. , ( x ) + f , ( x ) + e > t i y , , _ , d ' Ifx~Candfl(x)=O,

+c>O.

l = 1 . . . . . i - 2, b u t x q ~ X ,

obviously O ( ; , i _ l f / _ l ( X ) -~-L(X) q- e > 0.

T h u s we have p r o v e d that (fl(X) ..... f/_2(X),

ti;,i_lfi_l(X)

+£(X)

+ e) T

>L 0 for all x ~ C. By r e p e a t i n g this process we c a n get '-~1 ti,j (x) + scalars t i i , • . ~ d i,i 2 such that Z,= f~(x) + e > 0, w h a t was to be shown. L e m m a 0.3. Let C c R", f,. : C ~ R, i = 0, 1 . . . . . k, f ( x ) = ( f l ( x ) . . . . . f k ( x ) T for x ~ C. I f there is an l × k matrix P >~O with l <~l <~k + l such that ( f o ( X ) ) I + Pf(x)>~LO for all x ~ C, then the inequality system f ( x ) <~O, f o ( x ) < 0 is inconsistent over C. I f C and the fi's are convex and the system f ( x ) <~O, x ~ C is consistent, the converse also holds. P r o o f . If there exists a m a t r i x P with the p r o p e r ties of the statement, for a n y x ~ C such that f ( x ) <~0 we have P f ( x ) <~0 a n d ( f o ( x ) ) t + P f ( x ) > c 0, which implies f0 ( x ) > 0.

J.E. Martinez-Legaz / Lexicographicalorder and duality

344

of A by zeros. Suppose, on the contrary, that there are x ~ C and t ~ R such that

Conversely, if the system f ( x ) <~0, fo(x) < 0 is inconsistent, so is the system f ( x ) <~0, fo(x) + e t ~< 0. First, we will show that there is a matrix A >~L0 such that

A,\{fo(X)

A(f°(x)f(x)+et)

For every s o >/0, s >/0 verifying

>LO

q-et i

/=0

for all x ~ C, t ~ R. Let us consider the set

W=((Ww°)~Rk+']fo(x)+e'<~wo,

f(x)<~w

for some x ~ C, t ~ R ) . This set is convex and does not contain 0. By the Separation Theorem in [7] there exists a (k + I) × (k + I) matrix A such that A ( W>°L) 0 w

if

whence

p+l

(so) >~0.

P

Z oLiai ~ ap+l.

e'] >L0

x ~ C,

, (:oi->+e'+so) f(x) >~O,

Ap+]

Thus, by the Farkas theorem there are real numbers a 1.... , ap s u c h that

f ° r a l l ( W ° ) ~ W.

In particular,

.~( fo(x) +

we must have

i=1

Moreover, by Lemma 0.1, LpAp>~0 for some unitary lower-triangular matrix Lp. The matrix

t ~ R.

Let

us

denote

A = (ao,

a] ..... ak) T. Since

for all x ~ C , have

a

t~R,

T(f°(x)+et+s°) f(x)+s

s o>~0 and s>/0, we must

>10,

with a = ( - a] .... , - a?) verifies LAp + ] >~ 0, which again by Lemma 0.1 implies Ap+]>~L0 , contradicting the definition of p. If A = (0, A), with .~ a (k + 1) X k matrix, the system f ( x ) <~O, x ~ C would be inconsistent, since any solution x should satisfy

:1

whence

A 1 /(x)

implying a o >~ 0 or, in other words, A o >~L0, where A h = ( a o, a 1. . . . . ah) T, h = 0 , 1 , . . . , k . Let p = max(hlAh>~LO }. If p = k we can take A = A . If p < k we have

by the lexicographical nonnegativity of A, which is impossible. It shows that the first column of A is different from zero. Let us consider the matrix B obtained from A by dividing every term by the first nonzero element of the first column of A and deleting the rows l + 1. . . . . k if this element is on place l. This matrix is lexicographically nonnegative and verifies (fo(x) + et)t + Bf(x) ) L 0 for all x ~ C , t ~ R . If the l - 1 first components of (fo(x))t + Bf(x) were 0 for some x ~ C, the last one should be greater or equal than - e t for all

A Pl [fo(x) + et]>L 0 f(x) for all x ~ C , t ~ R . We shall see that this inequality is always strict, whence we can take A the matrix obtained by replacing the m - p last rows


J.E. Martinez-Legaz / Lexicographicalorder and duality

345

Pij, J :/: i, such that

t ~ R, whence nonnegative. This shows that

nonnegative vectors

(fo(x))/+

(Wi) k q- E wdPij ~L
B f ( x ) >~L O

for all x ~ C. By Lemma 0.1 we have B = L P for some unitary lower-triangular matrix L and termwise nonnegative P; moreover, given x ~ C, (fo(x))l+Bf(x)=L~p~, with L~ unitary and lower-triangular and p~ >/0. Finally,

(w a. . . . . Wk) T ~ W. (Pl . . . . . Pk ) with for

all

The

matrix

p =

k

p, = ~ pij + (1)k i=l

(fo(x)>/+ Pf(x) = ~fo(x)), + L-'Bf(x) = L-'(
, + Bf(x))

1Lxpx>~LO,

this last inequality being again a consequence of Lemma 0.1. Remark. In Lemma 0.3 we can impose l = k + 1 by adding K + 1 - l zero rows to B as its first zero rows if necessary.

1. Unconstrained problems

satisfies the conditions of the theorem. Let C c R " , ~ : C ~ R , i = 1 . . . . . k, f ( x ) = ( f l ( x ) . . . . . f k ( X ) ) T for x ~ C. We shall consider the problem consisting in finding the Pareto minima of fl . . . . . fk that is the X ~ C such that there is no x ~ C with f ( x ) < ~ f ( , Y ) and f ( x ) # tiff). Corollary 1.3. I f there is a k × k matrix P >1 0 such that the lexicographical minimum of P f ( x ) over C is attained at ~, then Y: is a Pareto minimum cf fl . . . . . fk. The converse is true if C and the fi's are convex.

In this section we shall characterize the Pareto minima of a convex unconstrained problem. First let us recall the following definition (see [1]):

Proof. A direct consequence of Proposition 1.2 applied to W = f ( C ) + Rk+ ad ~ = f l Y ) .

Definition 1.1. Let W be a subset of R k • ~ ~ W is an admissible point of W if there is no w ~ W \ ( ~ } such that w~<~.

Corollary 1.4. I f there is a k × k matrix P >_.0 and for any e > 0 there exists an unitary lower-triangular matrix L~ such that

In [1] is proved that the set A of admissible points of W verifies B c A c B with B = ( ~ WIpTw=min(pVwlw~W}} for some p = (Pl . . . . . pk)T with p , > 0 , i - 1 . . . . . k} and B the closure of B.

Proposition 1.2. L e t ~ ~ W. I f there is a k × k matrix P >_.0 such that P ~ <~L P w for all w ~ W, then ~, is an admissible point of W. I f W is convex the converse also holds.

Proof. If there exists a matrix P with the properties in the statement, from w ~ W \ ( ~ } and w~<~ we obtain P ~ > P w , whence P ~ > L P W , which is impossible. Conversely, if ~ ~ A the inequality system wj ~< ~j, j 4= i, w i < ~i, is inconsistent over W for each i. By the remark after Lemma 0.3 there are

L ~ P f ( x ) + e(1 . . . . . 1) T > L ~ P f ( f f ) for all x ~ C, then ~ is a Pareto minimum of fa . . . . . fk. The converse also holds if C is a compact convex set and the f~'s are continuous and convex. Proof. Follows from Corollary 1.3 and Lemma 0.2. A matrix P verifying the conditions of Corollary 1.3 will be called a Pareto lexicographical multiplier of f at ~. Proposition 1.5. I f P and Q are lexicographical multipliers of f at x and y respectively, they verify ( P - Q ) ( f ( x ) - f ( y ) ) <~L O. Proof. By definition we have P f ( x ) < L P f ( y ) , Q f ( Y ) ~L q ( x ) . We obtain the inequality in the statement by adding these two relations.

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.I.E. Martinez-Legaz / Lexicographicalorder and duality

2. Constrained problems The problem we shall consider in this section consists in finding the Pareto minima of f~ . . . . . fk subject to gi(x) <~O, i = 1 . . . . . m, where the f,'s and the g~'s are real-valued functions defined over C c R n. This problem will be denoted by (P): f and g will be the vector functions whose components are the f~'s and te g~'s, respectively, and the set of feasible points for (P), that is, C = { x C I g ( x ) <~0}. The following proposition states that any Pareto minimum of a convex problem is a lexicographical minimum of some associated unconstrained problem and satisfies a complementary slackness condition.

Proposition 2.1. Let ~ ~ C. I f there exists a ( k + m ) X k matrix P >_O and a ( k + m ) X m matrix Q >~0 such that the lexicographical minimum of P f ( x ) + Qg(x) over C is attained at ~ and the condition Q g ( y , ) = 0 holds, then Y~ is a Pareto minimum of (P). I f C, the fi's and the gi's are convex, the converse is true. Proof. Let P and Q be as in the statement, x ~ such that f ( x ) ~ < f ( ~ ) and f ( x ) ~ f ( Y O . Clearly, Pf(Y~) >_. P f ( x ) , Q g ( x ) < 0 = Qg(~), whence P f ( 2 ) + Qg(~) >_ P f ( x ) + Qg(x), which contradicts the lexicographical minimality of ft. To prove the converse, let us consider for i = 1 . . . . . k the inconsistent system f j ( x ) ~<~(ff), j i, f~(x) /0 such that ( f / ( x ) ) k + m + }--',f j ( x ) p , j + Q i g ( x ) j~i

>L (fi ('~))k+ m -~

E ~(~)p,j.

j~i

The matrices Q = E "/ = l Q i and P = ( p l . . . . . Pk) given by P j = E j , i P q + ( 1 ) k ÷ , , satisfy the required conditions, since by taking x = ~ in the above inequality we obtain Q~g(~)>~L 0, whence Q g ( x ) >~L0, and this implies Qg(~) = 0 by Lemma 0.1 and the fact that Q >~ 0 and g(~) ~< 0. The interest of Proposition 2.1 lies in the absence of regularity assumptions. We do not need a Slater constraint qualification for the existence of

the matrix of multipliers and the 'scalarization' of the problem by means of some P is possible for any (not necessarily properly) minimal element. In order to illustrate this point we will give some examples. The first one is a scalar optimization problem which does not satisfy the Slater constraint qualification: minimize x,

subject to

x 2~<0

(x~R).

It is easy to check that P = (0), with arbitrary a > 0 and Q = ( b ) , with b > 0 and c ~ R are the matrices with the properties stated in proposition 2.1. The second one is the problem consisting in finding the Pareto minima of f l ( x , y ) = x, f=(x, y ) =y, subject to - v~- - y ~<0. One of the these Pareto minima is the origin, but it is not a proper minimal element. Here C = ((x, y ) ~ R = I x >/0} and we can take, for example,

P =

and

Q = 0.

Of course some other problems presenting both lacks of regularities or having C = R n could be given, but the size of the matrices would increase together with the difficulty of finding them.

3. Duality Next, similarly to [6], we shall obtain the duality theorems corresponding to our theory. We define the dual problem of (P) as (D) Find a maximal element of W={y~Rkl3P,(k+m) X k matrix, P_>0, Q, ( k + m ) X m matrix, Q>~0, with P f ( x ) + Qg(x) >~L PY Vx ~ C}. By a maximal element of W we mean an y ~ W such that there is no y ~ W \ { y ) with v~
Theorem 3.1. (Weak Duality Theorem). Let y ~ W. There exists a ( k + m) × k matrix P >_0 such that PY <~L P f ( x ) for all x ~ C. Proof. By definition of W there are P >L O, Q >~ 0 verifying

Pf(x) + Og(x)

ey

for all x

C.

J.E. Martinez-Legaz / Lexicographical order and duality

347

If x ~ C from g(x)<~O and Q > / 0 we obtain Q g ( x ) <~O, whence P f ( x ) ~L P~;

~ V. By using L e m m a 3.2 we obtain the second part of the statement.

Lemma 3.2. (i) I f x ~ C and y ~ W then y >~f ( x ) implies y = f ( x ). (ii) I f f ( Y ) ~ W for some Y c ~ C then Y is a Pareto minimum of (P) and f ( Y ) is a maximal element of W.

As in Proposition 2.1, the interest of our Strong Duality Theorem lies in the absence of regularity assumptions. In [6] the maximality for the dual problem of the minimal values of the primal can only be assured if these values correspond to properly minimal solutions. But, of course, in our case the dual problem is more complicated.

Proof. (i) Let x ~ C, y ~ W, y >~f(x). Then Py <~L P f ( x ) , for P given by Theorem 3.1. But this would be impossible if y 4~f(x). (ii) This is an immediate consequence of (i). In the next theorem, we will assume that the f,'s and gj's are convex. Theorem 3.3 (Strong Duality Theorem). Let Y be a Pareto minimum of (P). Then f(Y:) is a maximal element of W. Conversely if ~ is a maximal element of W for every ~ ~ Rn, y > y, there is an x ~ such that f ( x ) <~y. I f the set V = ( y ~ R k [ y >~f ( x ) for some x ~ C } is closed, there is a Pareto minimum of (P), Y', such that f ( Y ) = y. Proof. Let ~ be a Pareto minimum of (P). By Proposition 2.1 there are P >__0, Q >~L0 such that P f ( x ) + Q g ( x ) >~L P f ( x ) for all x ~ C and Qg(~) = 0. Hence f ( Y ) ~ W and we can apply L e m m a 3.2 (ii). Conversely, let ~ be a maximal element of IV, y > 33 > ~. Let us suppose that 33 does not belong to the closure of V. By a classical separation theorem there exist a ~ R k \ ( 0 } , a ~ R , such that aT): >/a > aV33 for all y ~ V. The structure of V implies that a >/0. In particular a T f ( x ) >1a for all x ~ (7. This, by Lemma 0.3, implies the existence of an l × m matrix (~ >1 0, with 1 ~< l~< m + 1, such that ( a T f ( x ) ) , + Q g ( x ) >~L ( a ) , > L (aT33)~ for all x ~ C . Thus for the ( k + m ) x k matrix P = (0, a, /3)T, with f i t > 0 of size (k + m - l) × k, we have P >_ 0 and P f ( x ) + Q g ( x ) >L P33 for all x ~ C, where Q = ( 0 T, 0) T. Hence 33 W, but this contradicts the maximality of ~. Thus we have proved that 33 belongs to the closure of V. We can take y ~ V close enough to 33 to verify y ~ ~ we must conclude that

3. Conclusions In this paper we have developed a duality theory in multiobjective convex programming by using a lexicographical approach. In this way regularity assumptions were necessary in order to get the results. The usual scalarization had to be replaced by nonnegative linear combinations with vector coefficients of the objectives, which permits to obtain all the Pareto minima. It should be noted that the methods employed in this paper could perhaps be applied in connection with other duality theories in multiobjective programming.

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