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Duality for location problems with unbounded unit balls
European Journal of Operational Research 179 (2007) 1252–1265 www.elsevier.com/locate/ejor
Duality for location problems with unbounded unit balls Gert Wanka *, Radu Ioan Botß, Emese Vargyas Faculty of Mathematics, Chemnitz University of Technology, D-09107 Chemnitz, Germany Received 15 December 2003; accepted 15 September 2005 Available online 11 April 2006
Abstract Given an optimization problem with a composite of a convex and componentwise increasing function with a convex vector function as objective function, by means of the conjugacy approach based on the perturbation theory, we determine a dual to it. Necessary and sufficient optimality conditions are derived using strong duality. Furthermore, as special case of this problem, we consider a location problem, where the ‘‘distances’’ are measured by gauges of closed convex sets. We prove that the geometric characterization of the set of optimal solutions for this location problem given by Hinojosa and Puerto in a recently published paper can be obtained via the presented dual problem. Finally, the Weber and the minmax location problems with gauges are given as applications. 2006 Elsevier B.V. All rights reserved. Keywords: Convex programming; Location; Conjugate duality; Gauges; Optimality conditions
1. Introduction Location problems play an important role in a lot of fields of applications, as they appear in many areas such as transportation planning, industrial engineering, telecommunication, computer science, etc. A lot of research has been carried out in location analysis, the results of these problems being to locate some items, to optimize transportation costs, to minimize covered distances etc. Among the large number of papers and books dealing with location analysis we mention [2,5,6,8–10,16]. This paper is based on the work of Hinojosa and Puerto [6], in which the authors introduced a location problem, where the ‘‘distances’’ were measured by gauges of closed (not necessarily bounded) convex sets. For this problem the authors obtained a characterization of the set of optimal solutions and gave some methods to solve it. The goal of our paper is to treat the problem introduced in [6] by means of duality. On the other hand, we show how it is possible to derive the optimality conditions for this optimization problem via strong duality. In order to do this we consider at first a more general optimization problem, and then we particularize the results for the location problems in [6]. The objective function of the original optimization problem we *
Corresponding author. E-mail addresses: [email protected] (G. Wanka), [email protected] (R.I. Botß), emese. [email protected] (E. Vargyas). 0377-2217/$ - see front matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2005.09.048
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consider is a composite of a convex and componentwise increasing function with a convex vector function. Applying the Fenchel–Rockafellar duality concept based on conjugacy and perturbations (cf. [3]), we construct a dual problem to it and we prove the strong duality. Then, by means of strong duality, we derive the optimality conditions for the primal optimization problem. Afterwards, we study optimization problems with monotonic gauges as objective functions as particular cases of the general problem treated before. Analogously to the general problem, we construct a dual problem and prove the strong duality, and then we derive the optimality conditions. In Section 5 we consider the optimization problem treated by Hinojosa and Puerto in [6]. The obtained optimality conditions turn out to be the same as in [6]. Sections 6 and 7 are devoted to the specialization of the Weber and the minmax problem with gauges, respectively. In the past most of the references concerning location problems have considered distances induced by norms, but recently some papers have been published that consider the use of gauges like [4,12,17]. These lead to more general models, for example to model situations where the symmetry property of a norm does not make sense. 2. Notations and preliminary results In this section, we provide some definitions and results that we shall use in the sequel. As usual, Rn denotes the n-dimensional real space, for n 2 N. Throughout this paper all the vectors are considered as column vectors belonging to Rn , unless otherwise specified. An upper indexT transposes a column vector to a row one and T T viceversa. The inner product Pnof two vectors x = (x1, . . . , xn) and y = (y1, . . . , yn) in the n-dimensional real T space is denoted by x y ¼ i¼1 xi y i . Now we recall the concepts of gauges and polar sets and we relate them with some other concepts of convex analysis. Definition 2.1. Let C Rn be a closed convex set containing the origin. The function cC defined by cC ðxÞ :¼ inffa > 0 : x 2 aCg is called the gauge of C. The set C is called the unit ball associated with cC. As usual, we set cC(x) :¼ +1, if there is no a > 0 such that x 2 aC. Definition 2.2. Let C Rn be a closed convex set containing the origin. The set given by C 0 ¼ fy 2 Rn : y T x 6 1; 8x 2 Cg is called the polar set of C. Remark 2.3. C0 is a closed convex set containing the origin. Definition 2.4. Let C Rn be a convex set. The function rC given by rC ðyÞ :¼ supfy T x : x 2 Cg is called the support function of C. Theorem 2.5 [7]. Let C be a closed convex set containing the origin. Then (i) its gauge cC is a nonnegative closed sublinear function, (ii) fx 2 Rn : cC ðxÞ 6 rg ¼ rC, for all r > 0. Proposition 2.6 [7]. Let C be a closed convex set containing the origin. Its gauge cC is the support function of the set C0, namely cC ðxÞ ¼ rC0 ðxÞ ¼ supfy T x : y 2 C 0 g.
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Corollary 2.7 [7]. Let C be a closed convex set containing the origin. Its support function rC is the gauge of C0 and is denoted by cC0 , i.e. rC ðyÞ ¼ cC0 ðyÞ ¼ inffa > 0 : y 2 aC 0 g. Proposition 2.8. The conjugate function cC : Rn ! R [ fþ1g of cC verifies 8 < 0; if y 2 C 0 ; cC ðyÞ ¼ : þ1; otherwise, where C0 is the polar set of C. Proof. By the definition of the conjugate function of cC(x) we get
cC ðyÞ ¼ sup y T x cC ðxÞ ¼ sup y T x inffa > 0 : x 2 aCg ¼ sup x2Rn
x2Rn
x2Rn
¼ sup y T x a ¼ sup y T ðazÞ a ¼ sup a sup y T z 1 a>0; x2aC
a>0; z2C
a>0
z2C
8 < :
¼
y T x þ supðaÞ a>0; x2aC
9 = ;
8 < 0;
if y 2 C 0 ;
:
otherwise.
þ1;
Remark 2.9. By Theorem 2.5 and Remark 2.3 the fact that y 2 C0 is equivalent to the inequality cC0 ðyÞ 6 1, 0; if cC0 ðyÞ 6 1; so, one can write cC ðyÞ ¼ þ1; otherwise. 3. The optimization problem with a composed convex function as objective function Let (X, k Æ k) be a normed space and X* the topological dual space of X. hx*, xi will denote the value at x 2 X of the continuous linear functional x* 2 X*. Further, let gi : X ! R; i ¼ 1; . . . ; m, be convex and continuous functions and f : Rm ! R be a convex and componentwise increasing function, i.e. for y = (y1, . . . , ym)T, T
z ¼ ðz1 ; . . . ; zm Þ 2 Rm , y i P zi ;
i ¼ 1; . . . ; m ) f ðyÞ P f ðzÞ.
The optimization problem which we consider is the following one ðP Þ
inf f ðgðxÞÞ; x2X
where g : X ! Rm , g(x) = (g1(x), . . . , gm(x))T. In this section, we find out a dual problem to (P) and prove the existence of weak and strong duality. Moreover, by means of strong duality we derive the optimality conditions for (P). The approach, we use to find a dual problem to (P), is the so-called Fenchel–Rockafellar approach and it was very well described in [3,11]. It offers the possibility to construct different dual problems to a primal optimization problem by perturbing it in different ways (cf. [13–15]). In order to find a dual problem to (P) we consider the following perturbation function W:X X Rm ! R, |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} mþ1
Wðx; q; dÞ ¼ f ððg1 ðx þ q1 Þ; . . . ; gm ðx þ qm ÞÞT þ dÞ; where q = (q1, . . . , qm) 2 X · · X and d 2 Rm are the so-called perturbation variables.
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Then the dual problem to (P), obtained by using the perturbation function W, is ðDÞ
sup k2Rm ;pi 2X ; i¼1;...;m
fW ð0; p; kÞg;
X X Rm ! R [ fþ1g is the conjugate function of W. Here, pi, i = 1, . . . , m, and k 2 Rm where W : |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} mþ1
are the dual variables. We recall that for a function h : Y ! R, Y being a Hausdorff locally convex vector space, its conjugate function h : Y ! R [ fþ1g has the form h*(y*) = supy2Y{hy*, yi h(y)}. Y* is the topological dual space to Y. Therefore, the conjugate function of W can be calculated by the following formula: ( ) m X T T W ðx ; p; kÞ ¼ sup hx ; xi þ hpi ; qi i þ k d f ððg1 ðx þ q1 Þ; . . . ; gm ðx þ qm ÞÞ þ dÞ . qi 2X ;i¼1;...;m; x2X ;d2Rm
i¼1
To treat this expression we introduce at first the new variable t instead of d and then the new variables ri instead of qi by T
t ¼ d þ ðg1 ðx þ q1 Þ; . . . ; gm ðx þ qm ÞÞ 2 Rm and r i ¼ x þ qi 2 X ; This implies
W ðx ; p; kÞ ¼
i ¼ 1; . . . ; m. (
hx ; xi þ
sup qi 2X ;i¼1;...;m; x2X ;t2Rm
¼
(
hpi ; qi iþk ðt ðg1 ðx þ q1 Þ; . . . ; gm ðx þ qm ÞÞ Þ f ðtÞ
m X T hx ; xi þ hpi ; ri xikT ðg1 ðr1 Þ; . . . ; gm ðrm ÞÞ
ri 2X ;i¼1;...;m; x2X
¼
) T
T
i¼1
sup m X
m X
i¼1
*
supfhpi ; ri i ki gi ðri Þg þ sup x
i¼1 ri 2X
¼ f ðkÞ þ
x2X
m X
*
ðki gi Þ ðpi Þ þ sup x x2X
i¼1
m X i¼1
m X
)
þ sup kT t f ðtÞ t2Rm
+ pi ; x
þ f ðkÞ
+
pi ; x .
i¼1
We have now to consider x* = 0 and, so, the dual problem of (P) has the following form: ( * +) m m X X ðDÞ sup f ðkÞ ðki gi Þ ðpi Þ þ inf pi ; x . k2Rm ;pi 2X ; i¼1;...;m
i¼1
x2X
i¼1
Pm
Pm In the objective function of (D), if i¼1 pi 6¼ 0X , there exists x0 2 X, x0 5 0X, such that i¼1 p i ; x0 < 0. But, for all a > 0 we have * + * + m m X X inf pi ; x < a p i ; x0 ; x2X
i¼1
i¼1
Pm
and this means that, in this case, inf x2X i¼1 p i ; x ¼ 1. Pm In conclusion, in order to have the supremum in (D), we must consider i¼1 pi ¼ 0. By this, the dual problem of (P) is ( ) m X ðDÞ sup f ðkÞ ðki gi Þ ðpi Þ . ð1Þ k2RmP ;pi 2X ; i¼1 m i¼1;...;m;
p ¼0 i¼1 i
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Now, let us point out a property of the conjugate of a componentwise increasing function. Proposition 3.1. Let be f : Rm ! R a componentwise increasing function. Then f*(k) = +1 for all k 2 Rm n Rm þ. Proof. Let be k 2 Rm n Rmþ . Then there exists at least one i 2 {1, . . . , m} such that ki < 0. But f ðkÞ ¼ sup kT d f ðdÞ P sup fki d i f ð0; . . . ; d i ; . . . ; 0Þg; d2Rm
d¼ð0;...;d i ;...;0Þ; d i 2R
this means that f ðkÞ P supfki d i f ð0; . . . ; d i ; . . . ; 0Þg P supfki d i f ð0; . . . ; d i ; . . . 0Þg d i 2R
d i <0
P supfki d i g f ð0; . . . ; 0Þ ¼ þ1; d i <0
i.e. f*(k) = + 1, 8k 2 Rm n Rmþ . h By Proposition 3.1, the dual problem of (P) becomes ( ) m X ðDÞ sup f ðkÞ ðki gi Þ ðpi Þ . k2Rm ;pi 2X ; þP i¼1 m i¼1;...;m;
p ¼0 i¼1 i
Let us point out that, by the Fenchel–Rockafellar approach, between (P) and (D) weak duality, i.e. inf(P) P sup(D), always holds (cf. [3]). But, we are interested in the existence of strong duality inf(P) = max(D). This can be shown by proving that the problem (P) is stable (cf. [3]). Therefore, we show that the stability criterion described in Proposition III.2.3 in [3] is fulfilled. For the beginning we need the following proposition. Proposition 3.2. The function W : X X Rm ! R, |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} mþ1 Wðx; q; dÞ ¼ f ðg1 ðx þ q1 Þ; . . . ; gm ðx þ qm ÞÞT þ d is convex. The convexity of W follows from the convexity of the functions f and g and the fact that f is a componentwise increasing function. Theorem 3.3 (strong duality for (P)). If inf(P) > 1, then the dual problem has an optimal solution and strong duality holds, i.e. infðP Þ ¼ maxðDÞ. Proof. By Proposition 3.2, we have that the perturbation function W is convex. Moreover, inf(P) is a finite number and the function ðq1 ; . . . ; qm ; dÞ ! Wð0; q1 ; . . . ; qm ; dÞ X Rm . This means that the stability criterion in Propis finite and continuous in ð0; . . . ; 0; 0Rm Þ 2 X |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} |fflfflfflffl{zfflfflfflffl} m
m
osition III.2.3 in [3] is fulfilled, which implies that the problem (P) is stable. Finally, the Propositions IV.2.1 and IV.2.2 in [3] lead to the desired conclusions. h The structure of the problem (P) looks like a scalarization of a vector optimization problem by means of the monotonic function f. The results concerning duality for the problem (P) could be used to derive duality statements in the multiobjective optimization. But, this is the subject of some of our present research.
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The last part of this section is devoted to the presentation of the optimality conditions for the primal problem (P). They are derived, by the use of the equality between the optimal values of the primal and dual problem. Theorem 3.4 (optimality conditions for (P)) (1) Let x 2 X be an optimal solution to (P). Then there exist pi 2 X , i = 1, . . . , m, and k 2 Rmþ such that ð k; p1 ; . . . ; pm Þ is an optimal solution to (D) and the following optimality conditions are satisfied: Pm ki g ðxÞ, (i) f ðgðxÞÞ þ f ð kÞ ¼ i
i¼1
(ii) ki gi ðxÞ þ ð ki gi Þ ð pi Þ ¼ h pi ; xi; i ¼ 1; . . . ; m, Pm (iii) pi ¼ 0. i¼1 (2) If x 2 X , ð k; p1 ; . . . ; pm Þ is feasible to (D) and (i)–(iii) are fulfilled, then x is an optimal solution to (P), ð k; p1 ; . . . ; pm Þ is an optimal solution to (D) and strong duality holds m X kÞ ð ki gi Þ ð pi Þ. f ðgðxÞÞ ¼ f ð i¼1
Proof . . . , m, and k 2 Rmþ , such that ðk; p1 ; . . . ; pm Þ is a (1) By Theorem 3.3 follows that there exist pi 2 X , i = 1,P m solution to (D) and inf(P) = max(D). This means that i¼1 pi ¼ 0 and f ðgðxÞÞ ¼ f ð kÞ
m X
ð ki gi Þ ð pi Þ.
ð2Þ
i¼1
The last equality is equivalent to 0 ¼ f ðgðxÞÞ þ f ð kÞ
m X
ki gi ðxÞ þ
i¼1
m X ki gi ðxÞ þ ðki gi Þ ðpi Þ hpi ; xi .
ð3Þ
i¼1
From the definition of the conjugate functions we have that the following so-called Young-inequalities f ðgðxÞÞ þ f ð kÞ P kT gðxÞ ¼
m X
ki gi ðxÞ
ð4Þ
i¼1
and ki gi ðxÞ þ ð ki gi Þ ð pi Þ P h pi ; xi;
i ¼ 1; . . . ; m;
ð5Þ
are true. By (4) and (5) all the terms of the sum in (3) must be equal to zero. In conclusion, the equalities in (i) and (ii) must hold. (2) All the calculations and transformations done within part (1) may be carried out in the inverse direction starting from the conditions (i)–(iii). Thus the equality (2) results, which is the strong duality and shows that x solves (P) and ð k; p1 ; . . . ; pm Þ solves (D). h
4. The case of monotonic gauges In this section, we give an application to the problem presented above. Therefore, let cC : Rm ! R be a monotonic gauge of a closed convex set C containing the origin. Recall that cC is a monotonic gauge on Rm (cf. [1]), if cC(u) 6 cC(v) for every u and v in Rm satisfying juij 6 jvij for each i = 1, . . . , m. As in Section 3 X is assumed to be a normed space and g : X ! Rm , g(x) = (g1(x), . . . , gm(x))T, where gi, i = 1, . . . , m, are convex and continuous functions.
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Let us introduce now the following primal problem ðP cC Þ where
cþ C
inf cþ C ðgðxÞÞ; x2X
T
þ þ þ þ þ : R ! R, cþ C ðtÞ :¼ cC ðt Þ, with t ¼ ðt 1 ; . . . ; t m Þ and t i ¼ maxf0; t i g, i = 1, . . . , m. m
m Proposition 4.1. The function cþ C : R ! R is convex and componentwise increasing. þ
T
m þ Proof. First, let us point out that the function ðÞ : Rm ! Rmþ , defined by tþ ¼ ðtþ 1 ; . . . ; t m Þ , for t 2 R , is a m convex function. This means that, for u; v 2 R and a 2 [0, 1], it holds þ
ðau þ ð1 aÞvÞ 5auþ þ ð1 aÞvþ . Here, ‘‘5’’ is the ordering induced on Rm by the cone of nonnegative elements Rmþ . By the positive sublinearity and monotonicity of the gauge cC, we have for u; v 2 Rm and a 2 [0, 1],
þ 6 cC ðauþ þ ð1 aÞvþ Þ 6 acC ðuþ Þ þ ð1 aÞcC ðvþ Þ cþ C ðau þ ð1 aÞvÞ ¼ cC ðau þ ð1 aÞvÞ þ ¼ acþ C ðuÞ þ ð1 aÞcC ðvÞ.
This means that the function cþ C is convex. In order to prove that cþ is componentwise increasing, let u; v 2 Rm be such that ui 6 vi, i = 1, . . . , m. It C þ þ þ follows uþ i 6 vi , which implies that jui j 6 jvi j, i = 1, . . . , m. cC being a monotonic gauge, we have + + þ þ þ þ þ T þ þ T cC(u ) 6 cC(v ), where u ¼ ðu1 ; . . . ; um Þ , v ¼ ðvþ 1 ; . . . ; vm Þ or, equivalently, cC ðuÞ 6 cC ðvÞ. þ Hence the function cC is componentwise increasing. h By the approach described in Section 3, a dual problem to ðP cC Þ is ( ) m X
þ ðDcC Þ sup cC ðkÞ ðki gi Þ ðpi Þ . k2Rm ;pi 2X ; þP i¼1 m i¼1;...;m;
p ¼0 i¼1 i
m þ Proposition 4.2. The conjugate function ðcþ C Þ : R ! R [ fþ1g of cC verifies 0; if k 2 Rmþ and cC0 ðkÞ 6 1; ðcþ Þ ðkÞ ¼ C þ1; otherwise,
where cC0 is the gauge of the polar set C0. Proof. For k 2 Rm n Rm þ the assertion is a consequence of Propositions 3.1 and 4.1. þ þ Let be k 2 Rm . For t 2 Rm , we have jti j P jtþ i j, i = 1, . . . , m, which implies that cC ðtÞ P cC ðt Þ ¼ cC ðtÞ and þ
þ cC ðkÞ ¼ sup kT t cC ðtÞ 6 sup kT t cþ ð6Þ C ðtÞ ¼ cC ðkÞ. t2Rm
t2Rm
On the other hand, for the conjugate of the gauge cC we have the following formula (see Remark 2.9): T 0; if cC0 ðkÞ 6 1; ð7Þ cC ðkÞ ¼ sup k t cC ðtÞ ¼ m þ1; otherwise. t2R
þ If cC0 ðkÞ > 1, we have that þ1 ¼ cC ðkÞ 6 ðcþ C Þ ðkÞ. From here, ðcC Þ ðkÞ ¼ þ1.
Let be now cC0 ðkÞ 6 1. Because k = 0, it follows that kTt 6 kTt+, for every t 2 Rm . Furthermore, by Theorem 2.5, from cC0 ðkÞ 6 1 it follows that k 2 C0 and then by Proposition 2.6 we obtain that kTt+ 6 cC(t+). From these inequalities we obtain for the conjugate function of cþ C
T Tþ þ þ 0 6 cC ðkÞ 6 cþ ðkÞ ¼ sup k t c ðt Þ 6 sup k t c ðt Þ 6 0. C C C t2Rm
Consequently, there is
ðcþ C Þ ðkÞ
t2Rm
¼ 0 and the proposition is proved.
h
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By Proposition 4.2 the dual of ðP cC Þ has the following formulation: ( ) m X ðDcC Þ sup ðki gi Þ ðpi Þ . k2Rm i¼1 Pmþ ;pi 2X ;i¼1;...;m; p ¼0;cC 0 ðkÞ61 i¼1 i
In the objective function of this dual we separate the terms for which ki > 0 from the terms for which ki = 0 and then the dual can be written as ( ) X X ðDcC Þ supP ðki gi Þ ðpi Þ ð0Þ ðpi Þ . ð8Þ m pi 2X ;i¼1;...;m; p ¼0; i¼1 i cC 0 ðkÞ61;If1;...;mg; ki >0 ði2IÞ;ki ¼0 ði62IÞ
i62I
i2I
For i 62 I, it holds
0; if pi ¼ 0; 0 ðpi Þ ¼ supfhpi ; xi 0g ¼ suphpi ; xi ¼ þ1; otherwise. x2X x2X For i 2 I there is ðki gi Þ ðpi Þ ¼ ki gi k1i pi (cf. [3]). Denoting pi :¼ k1i pi , we obtain ( ) X ðDcC Þ sup ki gi ðpi Þ
ðI;k;pÞ2Y cC
with
i2I
( Y cC ¼
T
ðI; k; pÞ : I f1; . . . ; mg; k ¼ ðk1 ; . . . ; km Þ ; p ¼ ðp1 ; . . . ; pm Þ; cC0 ðkÞ 6 1; ki > 0 ði 2 IÞ; ki ¼ 0 ði 62 IÞ;
X
) ki p i ¼ 0 .
i2I
Because infðP cC Þ is finite being greater or equal than zero and cþ C is a convex and componentwise increasing function, by Theorem 3.3 we can formulate the following strong duality theorem for the problems ðP cC Þ and ðDcC Þ. Theorem 4.3 (strong duality for ðP cC Þ). The dual problem ðDcC Þ has an optimal solution and strong duality holds, i.e. infðP cC Þ ¼ maxðDcC Þ. Similarly to the general problem (P) the optimality conditions for ðP cC Þ can be derived. Theorem 4.4 (optimality conditions for ðP cC Þ). (1) Let x be an optimal solution to ðP cC Þ. Then there exists an optimal solution ðI; k; pÞ 2 Y cC to ðDcC Þ such that the following optimality conditions are satisfied: (i) I f1; . . . ; mg; ki > 0 ði 2 IÞ; ki ¼ 0 ði 62 IÞ, P (ii) cC0 ðkÞ 6 1; i2I ki pi ¼ 0, P ki gi ðxÞ, (iii) cþ xÞÞ ¼ i2I C ðgð (iv) gi ðxÞ þ gi ðpi Þ ¼ hpi ; xi; i 2 I. pÞ 2 Y cC and (i)–(iv) are fulfilled, then x is an optimal solution to ðP cC Þ, ðI; k; pÞ 2 Y cC is an (2) If x 2 X , ðI; k; optimal solution to ðDcC Þ and strong duality holds X ki gi ðpi Þ. xÞÞ ¼ cþ C ðgð i2I
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Proof (1) By Theorem 4.3 follows that there exists an optimal solution ðI; k; pÞ 2 Y cC to ðDcC Þ such that (i)–(ii) are fulfilled and X cþ ki gi ðpi Þ. xÞÞ ¼ C ðgð i2I
This equality is equivalent to " # X X
þ þ ki ½g ðxÞ þ g ðp Þ hp ; xi. 0 ¼ c ðgðxÞÞ þ c ðkÞ ki g ðxÞ þ C
i
C
i
i2I
i
i
i
ð9Þ
i2I
Using Young’s inequality we have X
T xÞ ¼ cþ ki gi ðxÞ xÞÞ þ cþ C ðgð C ðkÞ P k gð
ð10Þ
i2I
and gi ðxÞ þ gi ðpi Þ P hpi ; xi;
i 2 I.
ð11Þ
All terms of the sum in (9) inside brackets are positive, therefore X
T x Þ ¼ cþ ki gi ðxÞ xÞÞ þ cþ C ðgð C ðkÞ ¼ k gð
ð12Þ
i2I
and gi ðxÞ þ gi ðpi Þ ¼ hpi ; xi;
i 2 I.
ð13Þ P But, by Proposition 4.2, we have that ¼ 0, and, so, cþ xÞÞ ¼ i2I ki gi ðxÞ. C ðgð In conclusion, the relations (iii)–(iv) must also hold. (2) All the calculations and transformations done within part (1) may be carried out in the inverse direction. h ðcþ C Þ ðkÞ
5. The location model with unbounded unit balls In this section, we consider the problem treated by Hinojosa and Puerto in [6]. This is a single facility location problem, where gauges of closed convex sets are used to model distances. Throughout this section let A :¼ fa1 ; . . . ; am g be a subset of Rn which represents the set of existing facilities. Each facility ai 2 A has an associated gauge uai , whose unit ball is a closed convex set C ai containing the origin. Let w ¼ fwa1 ; . . . ; wam g be a set of positive weights and let cC : Rm ! R be a monotonic gauge of a closed convex set C containing the origin. The distance from an existing facility ai 2 A to a new facility x 2 Rn is given by uai ðx ai Þ. By u0ai we denote the gauge of the polar set C 0ai . The location problem studied in [6] is ðP cC ðAÞÞ
inf cC ðwa1 ua1 ðx a1 Þ; . . . ; wam uam ðx am ÞÞ.
x2Rn
Let g : Rn ! Rm be the vector function defined by g(x) :¼ (g1(x), . . . , gm(x))T, where gi ðxÞ ¼ wai uai ðx ai Þ for all i = 1, . . . , m. Because þ cþ C ðgðxÞÞ ¼ cC ðg ðxÞÞ ¼ cC ðgðxÞÞ;
8x 2 Rn ;
ðP cC ðAÞÞ can be written in the equivalent form ðP cC ðAÞÞ
inf cþ C ðgðxÞÞ;
x2Rn
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which is a particular case of the problem studied in the previous section. We mention, that instead of the space X considered in the case of the general optimization problem, we take here analogously to [6], the space Rn . Therefore, the dual problem to ðP cC ðAÞÞ is ( ) X ðDcC ðAÞÞ sup ki gi ðpi Þ ðI;k;pÞ2Y cC ðAÞ
with
i2I
( Y cC ðAÞ ¼
ðI; k; pÞ :I f1; . . . ; mg; k ¼ ðk1 ; . . . ; km ÞT ; p ¼ ðp1 ; . . . ; pm Þ; ) X cC0 ðkÞ 6 1; ki > 0 ði 2 IÞ; ki ¼ 0 ði 62 IÞ; ki p i ¼ 0 . i2I
n As the dual space is also R , the dual variable p belongs to R Rn . This is also valid for the duals in |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} the rest of the paper. m In our case gi ðxÞ ¼ wai uai ðx ai Þ, i = 1, . . . , m, hence (cf. [3]) pi T gi ðpi Þ ¼ ðwai uai ðx ai ÞÞ ðpi Þ ¼ ðwai uai Þ ðpi Þ þ pi ai ¼ wai uai þ pTi ai . w ai ( 0; if u0ai wpai 6 1; By Remark 2.9, uai wpai ¼ and, denoting pi :¼ wpai , i 2 I, the dual problem to i i i þ1; otherwise, ðP cC ðAÞÞ becomes ( ) X T ðDcC ðAÞÞ sup ki wai pi ai
X*
ðI;k;pÞ2Y cC ðAÞ
with
n
i2I
( Y cC ðAÞ ¼
T
ðI; k; pÞ : I f1; . . . ; mg; k ¼ ðk1 ; . . . ; km Þ ; p ¼ ðp1 ; . . . ; pm Þ; u0ai ðpi Þ 6 1; ) X i 2 I; cC0 ðkÞ 6 1; ki > 0 ði 2 IÞ; ki ¼ 0 ði 62 IÞ; ki wai pi ¼ 0 . i2I
Using Theorems 4.3 and 4.4 we can present for ðP cC ðAÞÞ and ðDcC ðAÞÞ the strong duality theorem and the optimality conditions. Theorem 5.1 (strong duality for ðP cC ðAÞÞ). The dual problem ðDcC ðAÞÞ has an optimal solution and strong duality holds, i.e. infðP cC ðAÞÞ ¼ maxðDcC ðAÞÞ. Theorem 5.2 (optimality conditions for ðP cC ðAÞÞ). (1) Let x be an optimal solution to ðP cC ðAÞÞ. Then there exists an optimal solution ðI; k; pÞ 2 Y cC ðAÞ to ðDcC ðAÞÞ such that the following optimality conditions are satisfied: (i) I f1; . . . ; mg; ki > 0 ði 2 IÞ; ki ¼ 0 ði 62 IÞ, P 0 ki wai pi ¼ 0, (ii) cC0 ð kÞ 6 1; uai ðpi Þ 6 1; i 2 I; i2I P (iii) cC ðwa1 ua1 ðx a1 Þ; . . . ; wam uam ðx am ÞÞ ¼ i2I ki wai uai ðx ai Þ, (iv) uai ðx ai Þ ¼ pi T ðx ai Þ; i 2 I. (2) If x 2 X , ðI; k; pÞ 2 Y cC and (i)–(iv) are fulfilled, then x is an optimal solution to ðP cC ðAÞÞ, ðI; k; pÞ 2 Y cC ðAÞ is an optimal solution to ðDcC ðAÞÞ and strong duality holds X ki wa pT ai . cC ðwa1 ua1 ðx a1 Þ; . . . ; wam uam ðx am ÞÞ ¼ i i i2I
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Proof. Theorem 5.2 is a direct consequence of Theorem 4.4 and the fact that cþ C ðgðxÞÞ ¼ cC ðgðxÞÞ.
h
Remark 5.3. The optimality conditions obtained for the optimization problem ðP cC ðAÞÞ are the same as the conditions obtained by Hinojosa and Puerto in Lemma 7 in [6]. In the paper cited above the authors gave a geometrical description of the set of optimal solutions, but, as one can see, by means of duality one obtains the same characterization of this set. In the next two sections of this paper we present some particular cases of the problem ðP cC ðAÞÞ, namely, the Weber problem and the minmax problem with gauges of closed convex sets. 6. The Weber problem with gauges of closed convex sets The Weber problem with gauges of closed convex sets is m X ðP w ðAÞÞ infn wai uai ðx ai Þ; x2R
i¼1
where uai , i = 1, . . . , m, are gauges whose unit balls are the closed convex sets C ai , i = 1, . . . , m, which contain the origin, and w ¼ fwa1 ; . . . ; wam g is a set of the positive weights. As one can see, the problem above is equivalent to the following one: ðP w ðAÞÞ
inf l1 ðgðxÞÞ;
x2Rn
Pm where l1 : Rm ! R, l1 ðkÞ ¼ i¼1 jki j and g : Rn ! Rm is the vector function defined by g(x) :¼ (g1(x), . . . , gm(x))T, with gi ðxÞ ¼ wai uai ðx ai Þ for all i = 1, . . . , m. One may observe that the function l1 is a monotonic gauge, actually, a monotonic norm. Taking cC(k) :¼ l1(k) for all k 2 Rm , by the results obtained in the previous section, the dual problem to ðP w ðAÞÞ becomes ( ) X T ðDw ðAÞÞ sup ki wai pi ai ðI;k;pÞ2Y w ðAÞ
with
i2I
( ðI; k; pÞ : I f1; . . . ; mg; k ¼ ðk1 ; . . . ; km ÞT ; p ¼ ðp1 ; . . . ; pm Þ; u0ai ðpi Þ 6 1;
Y w ðAÞ ¼
i2
I; l01 ðkÞ
6 1; ki > 0 ði 2 IÞ; ki ¼ 0 ði 62 IÞ;
X
) ki wai pi ¼ 0 .
i2I
Remark 6.1. In case the gauge cC of a convex set C is a norm, the gauge of the polar set C0 actually becomes the dual norm. Because the dual norm of the l1-norm is l01 ðkÞ ¼ maxi¼1;...;m jki j, we obtain the following formulation for the dual problem: ( ðDw ðAÞÞ
sup ðI;k;pÞ2Y w ðAÞ
with
X
) ki wai pTi ai
i2I
( Y w ðAÞ ¼
ðI; k; pÞ : I f1; . . . ; mg; k ¼ ðk1 ; . . . ; km ÞT ; p ¼ ðp1 ; . . . ; pm Þ; u0ai ðpi Þ 6 1; i 2 I; max ki 6 1; ki > 0 ði 2 IÞ; ki ¼ 0 ði 62 IÞ; i2I
X i2I
) ki wai pi ¼ 0 .
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Let us give now the strong duality theorem and the optimality conditions for ðP w ðAÞÞ and its dual ðDw ðAÞÞ. Theorem 6.2 (strong duality for ðP w ðAÞÞ). The dual problem ðDw ðAÞÞ has an optimal solution and strong duality holds, i.e. infðP w ðAÞÞ ¼ maxðDw ðAÞÞ. Theorem 6.3 (optimality conditions for ðP w ðAÞÞ) (1) Let x be an optimal solution to ðP w ðAÞÞ. Then there exists an optimal solution ðI; k; pÞ 2 Y w ðAÞ to ðDw ðAÞÞ such that the following optimality conditions are satisfied: i > 0 ði 2 IÞ; ki ¼ 0 ði 62 IÞ, (i) I f1; . . . ; mg; k P 0 ki wai pi ¼ 0, (ii) maxi2I ki 6 1; uai ðpi Þ 6 1; i 2 I; i2I Pm P (iii) ki wa u ðx ai Þ, wa u ðx ai Þ ¼ i¼1
ai
i
i2I
i
ai
T
(iv) uai ðx ai Þ ¼ pi ðx ai Þ; i 2 I. (2) If x 2 X , ðI; k; pÞ 2 Y w ðAÞ and (i)–(iv) are fulfilled, then x is an optimal solution to ðP w ðAÞÞ; ðI; k; pÞ 2 Y w ðAÞ is an optimal solution to ðDw ðAÞÞ and strong duality holds m X X ki wai wai uai ðx ai Þ ¼ pTi ai . i¼1
i2I
Proof. Theorem 6.3 is a direct consequence of Theorem 5.2. h 7. The minmax problem with gauges of closed convex sets The optimization problem studied in this last section is the minmax problem with gauges of closed convex sets ðP m ðAÞÞ
inf max wai uai ðx ai Þ;
x2Rn i¼1;...;m
where uai , i = 1, . . . , m, and w ¼ fwa1 ; . . . ; wam g are considered like in the previous section. One can see that this problem is equivalent to the following one: ðP m ðAÞÞ
inf l1 ðgðxÞÞ;
x2Rn
where l1 : Rm ! R; l1 ðkÞ ¼ maxi¼1;...;m jki j and g : Rn ! Rm is the vector function defined by g(x) :¼ (g1(x), . . . , gm(x))T, with gi ðxÞ ¼ wai uai ðx ai Þ for all i = 1, . . . , m. One may observe that the function l1 is also a monotonic norm. Taking cC(k) :¼ l1(k) for all k 2 Rm , the dual problem to ðP m ðAÞÞ becomes ( ) X ðDm ðAÞÞ sup ki wai pTi ai ; ðI;k;pÞ2Y m ðAÞ
with
i2I
( Y m ðAÞ ¼
T
ðI; k; pÞ : I f1; . . . ; mg; k ¼ ðk1 ; . . . ; km Þ ; p ¼ ðp1 ; . . . ; pm Þ; u0ai ðpi Þ 6 1; i2
I; l01 ðkÞ
6 1; ki > 0 ði 2 IÞ; ki ¼ 0 ði 62 IÞ;
X i2I
) ki wai pi ¼ 0 .
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Remark 7.1. Because the dual norm of the l1-norm is l01 ðkÞ ¼ for the dual problem: ( ðDm ðAÞÞ
sup ðI;k;pÞ2Y m ðAÞ
X
Pm
i¼1 jki j
we obtain the following formulation
) ki wai pTi ai
;
i2I
with ( Y m ðAÞ ¼
T
ðI; k; pÞ : I f1; . . . ; mg; k ¼ ðk1 ; . . . ; km Þ ; p ¼ ðp1 ; . . . ; pm Þ; u0ai ðpi Þ 6 1; i 2 I;
m X
ki 6 1; ki > 0 ði 2 IÞ; ki ¼ 0 ði 62 IÞ;
X
) ki wai pi ¼ 0 .
i2I
i¼1
Like in the previous section we give now the strong duality theorem and the optimality conditions for ðP m ðAÞÞ and its dual ðDm ðAÞÞ. Theorem 7.2 (strong duality for ðP m ðAÞÞ). The dual problem ðDm ðAÞÞ has an optimal solution and strong duality holds, i.e. infðP m ðAÞÞ ¼ maxðDm ðAÞÞ. Theorem 7.3 (optimality conditions for ðP m ðAÞÞ). (1) Let x be an optimal solution to ðP m ðAÞÞ. Then there exists an optimal solution ðI; k; pÞ 2 Y m ðAÞ to ðDm ðAÞÞ such that the following optimality conditions are satisfied: (i) I f1; . . . ; mg; ki > 0 ði 2 IÞ; ki ¼ 0 ði 62 IÞ, P P 0 (ii) k ki wai pi ¼ 0, 6 1; u ð p Þ 6 1; i 2 I; i ai i2I i P i2I (iii) maxi¼1;...;m wai uai ðx ai Þ ¼ i2I ki wai uai ðx ai Þ, (iv) uai ðx ai Þ ¼ pi T ðx ai Þ; i 2 I. (2) If x 2 X , ðI; k; pÞ 2 Y m ðAÞ and (i)–(iv) are fulfilled, then x is an optimal solution to ðP m ðAÞÞ, ðI; k; pÞ 2 Y m ðAÞ is an optimal solution to ðDm ðAÞÞ and strong duality holds X ki wai max wai uai ðx ai Þ ¼ pTi ai . i¼1;...;m
i2I
Proof. Theorem 7.3 is a direct consequence of Theorem 5.2. h References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
F.L. Bauer, J. Stoer, C. Witzgall, Absolute and monotonic norms, Numerische Mathematik 3 (1961) 257–264. Z. Drezner, H.W. Hamacher, Facility Location, Applications and Theory, Springer-Verlag, Berlin Heidelberg, 2002. I. Ekeland, R. Temam, Convex Analysis and Variational Problems, North-Holland Publishing Company, Amsterdam, 1976. J. Fliege, Solving convex location problems with gauges in polynomial time, Studies in Locational Analysis 14 (2000) 153–172. H.W. Hamacher, S. Nickel, Classification of location models, Location Science 6 (1998) 229–242. Y. Hinojosa, J. Puerto, Single facility location problems with unbounded unit balls, Mathematical Methods of Operations Research 58 (2003) 87–104. J.B. Hiriart-Urruty, C. Lemare´chal, Convex Analysis and Minimization Algorithms, Springer, Berlin Heidelberg New York, 1993. R.F. Love, J.G. Morris, G.O. Wesolowsky, Facilities Location, Models and methods, North-Holland, New York, 1988. C. Michelot, Localization in multifacility location theory, European Journal of Operational Research 31 (1987) 177–184. S. Nickel, J. Puerto, A.M. Rodrı´guez-Chı´a, An approach to location models involving sets as existing facilities, Mathematics of Operations Research 28 (4) (2003) 693–715. R.T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, 1970.
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[12] A.M. Rodrı´guez-Chı´a, S. Nickel, J. Puerto, F.R. Ferna´ndez, A flexible approach to location problems, Mathematical Methods of Operations Research 51 (2000) 69–89. [13] G. Wanka, R.I. Botß, A new duality approach for multiobjective convex optimization problems, Journal of Nonlinear and Convex Analysis 3 (1) (2002) 41–57. [14] G. Wanka, R.I. Botß, Multiobjective duality for convex-linear problems II, Mathematical Methods of Operations Research 53 (3) (2001) 419–433. [15] G. Wanka, R.I. Botß, On the relations between different dual problems in convex mathematical programming, in: P. Chamoni, R. Leisten, A. Martin, J. Minnemann, H. Stadtler (Eds.), Operations Research Proceedings 2001, Springer, Heidelberg, 2002, pp. 255– 262. [16] G. Wanka, R.I. Botß, E. Vargyas, Duality for the multiobjective location model involving sets as existing facilities, in: P.M. Pardalos, I. Tseveendorj, R. Enkhbat (Eds.), Optimization and Optimal Control, World Scientific Publishing CO, 2003, pp. 307–333. [17] G. Wanka, U. Krallert, Duality for optimal control-approximation problems with gauges, Journal for Analysis and its Applications 18 (2) (1999) 491–504.
Duality for location problems with unbounded unit balls Gert Wanka *, Radu Ioan Botß, Emese Vargyas Faculty of Mathematics, Chemnitz University of Technology, D-09107 Chemnitz, Germany Received 15 December 2003; accepted 15 September 2005 Available online 11 April 2006
Abstract Given an optimization problem with a composite of a convex and componentwise increasing function with a convex vector function as objective function, by means of the conjugacy approach based on the perturbation theory, we determine a dual to it. Necessary and sufficient optimality conditions are derived using strong duality. Furthermore, as special case of this problem, we consider a location problem, where the ‘‘distances’’ are measured by gauges of closed convex sets. We prove that the geometric characterization of the set of optimal solutions for this location problem given by Hinojosa and Puerto in a recently published paper can be obtained via the presented dual problem. Finally, the Weber and the minmax location problems with gauges are given as applications. 2006 Elsevier B.V. All rights reserved. Keywords: Convex programming; Location; Conjugate duality; Gauges; Optimality conditions
1. Introduction Location problems play an important role in a lot of fields of applications, as they appear in many areas such as transportation planning, industrial engineering, telecommunication, computer science, etc. A lot of research has been carried out in location analysis, the results of these problems being to locate some items, to optimize transportation costs, to minimize covered distances etc. Among the large number of papers and books dealing with location analysis we mention [2,5,6,8–10,16]. This paper is based on the work of Hinojosa and Puerto [6], in which the authors introduced a location problem, where the ‘‘distances’’ were measured by gauges of closed (not necessarily bounded) convex sets. For this problem the authors obtained a characterization of the set of optimal solutions and gave some methods to solve it. The goal of our paper is to treat the problem introduced in [6] by means of duality. On the other hand, we show how it is possible to derive the optimality conditions for this optimization problem via strong duality. In order to do this we consider at first a more general optimization problem, and then we particularize the results for the location problems in [6]. The objective function of the original optimization problem we *
Corresponding author. E-mail addresses: [email protected] (G. Wanka), [email protected] (R.I. Botß), emese. [email protected] (E. Vargyas). 0377-2217/$ - see front matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2005.09.048
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consider is a composite of a convex and componentwise increasing function with a convex vector function. Applying the Fenchel–Rockafellar duality concept based on conjugacy and perturbations (cf. [3]), we construct a dual problem to it and we prove the strong duality. Then, by means of strong duality, we derive the optimality conditions for the primal optimization problem. Afterwards, we study optimization problems with monotonic gauges as objective functions as particular cases of the general problem treated before. Analogously to the general problem, we construct a dual problem and prove the strong duality, and then we derive the optimality conditions. In Section 5 we consider the optimization problem treated by Hinojosa and Puerto in [6]. The obtained optimality conditions turn out to be the same as in [6]. Sections 6 and 7 are devoted to the specialization of the Weber and the minmax problem with gauges, respectively. In the past most of the references concerning location problems have considered distances induced by norms, but recently some papers have been published that consider the use of gauges like [4,12,17]. These lead to more general models, for example to model situations where the symmetry property of a norm does not make sense. 2. Notations and preliminary results In this section, we provide some definitions and results that we shall use in the sequel. As usual, Rn denotes the n-dimensional real space, for n 2 N. Throughout this paper all the vectors are considered as column vectors belonging to Rn , unless otherwise specified. An upper indexT transposes a column vector to a row one and T T viceversa. The inner product Pnof two vectors x = (x1, . . . , xn) and y = (y1, . . . , yn) in the n-dimensional real T space is denoted by x y ¼ i¼1 xi y i . Now we recall the concepts of gauges and polar sets and we relate them with some other concepts of convex analysis. Definition 2.1. Let C Rn be a closed convex set containing the origin. The function cC defined by cC ðxÞ :¼ inffa > 0 : x 2 aCg is called the gauge of C. The set C is called the unit ball associated with cC. As usual, we set cC(x) :¼ +1, if there is no a > 0 such that x 2 aC. Definition 2.2. Let C Rn be a closed convex set containing the origin. The set given by C 0 ¼ fy 2 Rn : y T x 6 1; 8x 2 Cg is called the polar set of C. Remark 2.3. C0 is a closed convex set containing the origin. Definition 2.4. Let C Rn be a convex set. The function rC given by rC ðyÞ :¼ supfy T x : x 2 Cg is called the support function of C. Theorem 2.5 [7]. Let C be a closed convex set containing the origin. Then (i) its gauge cC is a nonnegative closed sublinear function, (ii) fx 2 Rn : cC ðxÞ 6 rg ¼ rC, for all r > 0. Proposition 2.6 [7]. Let C be a closed convex set containing the origin. Its gauge cC is the support function of the set C0, namely cC ðxÞ ¼ rC0 ðxÞ ¼ supfy T x : y 2 C 0 g.
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Corollary 2.7 [7]. Let C be a closed convex set containing the origin. Its support function rC is the gauge of C0 and is denoted by cC0 , i.e. rC ðyÞ ¼ cC0 ðyÞ ¼ inffa > 0 : y 2 aC 0 g. Proposition 2.8. The conjugate function cC : Rn ! R [ fþ1g of cC verifies 8 < 0; if y 2 C 0 ; cC ðyÞ ¼ : þ1; otherwise, where C0 is the polar set of C. Proof. By the definition of the conjugate function of cC(x) we get
cC ðyÞ ¼ sup y T x cC ðxÞ ¼ sup y T x inffa > 0 : x 2 aCg ¼ sup x2Rn
x2Rn
x2Rn
¼ sup y T x a ¼ sup y T ðazÞ a ¼ sup a sup y T z 1 a>0; x2aC
a>0; z2C
a>0
z2C
8 < :
¼
y T x þ supðaÞ a>0; x2aC
9 = ;
8 < 0;
if y 2 C 0 ;
:
otherwise.
þ1;
Remark 2.9. By Theorem 2.5 and Remark 2.3 the fact that y 2 C0 is equivalent to the inequality cC0 ðyÞ 6 1, 0; if cC0 ðyÞ 6 1; so, one can write cC ðyÞ ¼ þ1; otherwise. 3. The optimization problem with a composed convex function as objective function Let (X, k Æ k) be a normed space and X* the topological dual space of X. hx*, xi will denote the value at x 2 X of the continuous linear functional x* 2 X*. Further, let gi : X ! R; i ¼ 1; . . . ; m, be convex and continuous functions and f : Rm ! R be a convex and componentwise increasing function, i.e. for y = (y1, . . . , ym)T, T
z ¼ ðz1 ; . . . ; zm Þ 2 Rm , y i P zi ;
i ¼ 1; . . . ; m ) f ðyÞ P f ðzÞ.
The optimization problem which we consider is the following one ðP Þ
inf f ðgðxÞÞ; x2X
where g : X ! Rm , g(x) = (g1(x), . . . , gm(x))T. In this section, we find out a dual problem to (P) and prove the existence of weak and strong duality. Moreover, by means of strong duality we derive the optimality conditions for (P). The approach, we use to find a dual problem to (P), is the so-called Fenchel–Rockafellar approach and it was very well described in [3,11]. It offers the possibility to construct different dual problems to a primal optimization problem by perturbing it in different ways (cf. [13–15]). In order to find a dual problem to (P) we consider the following perturbation function W:X X Rm ! R, |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} mþ1
Wðx; q; dÞ ¼ f ððg1 ðx þ q1 Þ; . . . ; gm ðx þ qm ÞÞT þ dÞ; where q = (q1, . . . , qm) 2 X · · X and d 2 Rm are the so-called perturbation variables.
G. Wanka et al. / European Journal of Operational Research 179 (2007) 1252–1265
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Then the dual problem to (P), obtained by using the perturbation function W, is ðDÞ
sup k2Rm ;pi 2X ; i¼1;...;m
fW ð0; p; kÞg;
X X Rm ! R [ fþ1g is the conjugate function of W. Here, pi, i = 1, . . . , m, and k 2 Rm where W : |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} mþ1
are the dual variables. We recall that for a function h : Y ! R, Y being a Hausdorff locally convex vector space, its conjugate function h : Y ! R [ fþ1g has the form h*(y*) = supy2Y{hy*, yi h(y)}. Y* is the topological dual space to Y. Therefore, the conjugate function of W can be calculated by the following formula: ( ) m X T T W ðx ; p; kÞ ¼ sup hx ; xi þ hpi ; qi i þ k d f ððg1 ðx þ q1 Þ; . . . ; gm ðx þ qm ÞÞ þ dÞ . qi 2X ;i¼1;...;m; x2X ;d2Rm
i¼1
To treat this expression we introduce at first the new variable t instead of d and then the new variables ri instead of qi by T
t ¼ d þ ðg1 ðx þ q1 Þ; . . . ; gm ðx þ qm ÞÞ 2 Rm and r i ¼ x þ qi 2 X ; This implies
W ðx ; p; kÞ ¼
i ¼ 1; . . . ; m. (
hx ; xi þ
sup qi 2X ;i¼1;...;m; x2X ;t2Rm
¼
(
hpi ; qi iþk ðt ðg1 ðx þ q1 Þ; . . . ; gm ðx þ qm ÞÞ Þ f ðtÞ
m X T hx ; xi þ hpi ; ri xikT ðg1 ðr1 Þ; . . . ; gm ðrm ÞÞ
ri 2X ;i¼1;...;m; x2X
¼
) T
T
i¼1
sup m X
m X
i¼1
*
supfhpi ; ri i ki gi ðri Þg þ sup x
i¼1 ri 2X
¼ f ðkÞ þ
x2X
m X
*
ðki gi Þ ðpi Þ þ sup x x2X
i¼1
m X i¼1
m X
)
þ sup kT t f ðtÞ t2Rm
+ pi ; x
þ f ðkÞ
+
pi ; x .
i¼1
We have now to consider x* = 0 and, so, the dual problem of (P) has the following form: ( * +) m m X X ðDÞ sup f ðkÞ ðki gi Þ ðpi Þ þ inf pi ; x . k2Rm ;pi 2X ; i¼1;...;m
i¼1
x2X
i¼1
Pm
Pm In the objective function of (D), if i¼1 pi 6¼ 0X , there exists x0 2 X, x0 5 0X, such that i¼1 p i ; x0 < 0. But, for all a > 0 we have * + * + m m X X inf pi ; x < a p i ; x0 ; x2X
i¼1
i¼1
Pm
and this means that, in this case, inf x2X i¼1 p i ; x ¼ 1. Pm In conclusion, in order to have the supremum in (D), we must consider i¼1 pi ¼ 0. By this, the dual problem of (P) is ( ) m X ðDÞ sup f ðkÞ ðki gi Þ ðpi Þ . ð1Þ k2RmP ;pi 2X ; i¼1 m i¼1;...;m;
p ¼0 i¼1 i
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Now, let us point out a property of the conjugate of a componentwise increasing function. Proposition 3.1. Let be f : Rm ! R a componentwise increasing function. Then f*(k) = +1 for all k 2 Rm n Rm þ. Proof. Let be k 2 Rm n Rmþ . Then there exists at least one i 2 {1, . . . , m} such that ki < 0. But f ðkÞ ¼ sup kT d f ðdÞ P sup fki d i f ð0; . . . ; d i ; . . . ; 0Þg; d2Rm
d¼ð0;...;d i ;...;0Þ; d i 2R
this means that f ðkÞ P supfki d i f ð0; . . . ; d i ; . . . ; 0Þg P supfki d i f ð0; . . . ; d i ; . . . 0Þg d i 2R
d i <0
P supfki d i g f ð0; . . . ; 0Þ ¼ þ1; d i <0
i.e. f*(k) = + 1, 8k 2 Rm n Rmþ . h By Proposition 3.1, the dual problem of (P) becomes ( ) m X ðDÞ sup f ðkÞ ðki gi Þ ðpi Þ . k2Rm ;pi 2X ; þP i¼1 m i¼1;...;m;
p ¼0 i¼1 i
Let us point out that, by the Fenchel–Rockafellar approach, between (P) and (D) weak duality, i.e. inf(P) P sup(D), always holds (cf. [3]). But, we are interested in the existence of strong duality inf(P) = max(D). This can be shown by proving that the problem (P) is stable (cf. [3]). Therefore, we show that the stability criterion described in Proposition III.2.3 in [3] is fulfilled. For the beginning we need the following proposition. Proposition 3.2. The function W : X X Rm ! R, |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} mþ1 Wðx; q; dÞ ¼ f ðg1 ðx þ q1 Þ; . . . ; gm ðx þ qm ÞÞT þ d is convex. The convexity of W follows from the convexity of the functions f and g and the fact that f is a componentwise increasing function. Theorem 3.3 (strong duality for (P)). If inf(P) > 1, then the dual problem has an optimal solution and strong duality holds, i.e. infðP Þ ¼ maxðDÞ. Proof. By Proposition 3.2, we have that the perturbation function W is convex. Moreover, inf(P) is a finite number and the function ðq1 ; . . . ; qm ; dÞ ! Wð0; q1 ; . . . ; qm ; dÞ X Rm . This means that the stability criterion in Propis finite and continuous in ð0; . . . ; 0; 0Rm Þ 2 X |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} |fflfflfflffl{zfflfflfflffl} m
m
osition III.2.3 in [3] is fulfilled, which implies that the problem (P) is stable. Finally, the Propositions IV.2.1 and IV.2.2 in [3] lead to the desired conclusions. h The structure of the problem (P) looks like a scalarization of a vector optimization problem by means of the monotonic function f. The results concerning duality for the problem (P) could be used to derive duality statements in the multiobjective optimization. But, this is the subject of some of our present research.
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The last part of this section is devoted to the presentation of the optimality conditions for the primal problem (P). They are derived, by the use of the equality between the optimal values of the primal and dual problem. Theorem 3.4 (optimality conditions for (P)) (1) Let x 2 X be an optimal solution to (P). Then there exist pi 2 X , i = 1, . . . , m, and k 2 Rmþ such that ð k; p1 ; . . . ; pm Þ is an optimal solution to (D) and the following optimality conditions are satisfied: Pm ki g ðxÞ, (i) f ðgðxÞÞ þ f ð kÞ ¼ i
i¼1
(ii) ki gi ðxÞ þ ð ki gi Þ ð pi Þ ¼ h pi ; xi; i ¼ 1; . . . ; m, Pm (iii) pi ¼ 0. i¼1 (2) If x 2 X , ð k; p1 ; . . . ; pm Þ is feasible to (D) and (i)–(iii) are fulfilled, then x is an optimal solution to (P), ð k; p1 ; . . . ; pm Þ is an optimal solution to (D) and strong duality holds m X kÞ ð ki gi Þ ð pi Þ. f ðgðxÞÞ ¼ f ð i¼1
Proof . . . , m, and k 2 Rmþ , such that ðk; p1 ; . . . ; pm Þ is a (1) By Theorem 3.3 follows that there exist pi 2 X , i = 1,P m solution to (D) and inf(P) = max(D). This means that i¼1 pi ¼ 0 and f ðgðxÞÞ ¼ f ð kÞ
m X
ð ki gi Þ ð pi Þ.
ð2Þ
i¼1
The last equality is equivalent to 0 ¼ f ðgðxÞÞ þ f ð kÞ
m X
ki gi ðxÞ þ
i¼1
m X ki gi ðxÞ þ ðki gi Þ ðpi Þ hpi ; xi .
ð3Þ
i¼1
From the definition of the conjugate functions we have that the following so-called Young-inequalities f ðgðxÞÞ þ f ð kÞ P kT gðxÞ ¼
m X
ki gi ðxÞ
ð4Þ
i¼1
and ki gi ðxÞ þ ð ki gi Þ ð pi Þ P h pi ; xi;
i ¼ 1; . . . ; m;
ð5Þ
are true. By (4) and (5) all the terms of the sum in (3) must be equal to zero. In conclusion, the equalities in (i) and (ii) must hold. (2) All the calculations and transformations done within part (1) may be carried out in the inverse direction starting from the conditions (i)–(iii). Thus the equality (2) results, which is the strong duality and shows that x solves (P) and ð k; p1 ; . . . ; pm Þ solves (D). h
4. The case of monotonic gauges In this section, we give an application to the problem presented above. Therefore, let cC : Rm ! R be a monotonic gauge of a closed convex set C containing the origin. Recall that cC is a monotonic gauge on Rm (cf. [1]), if cC(u) 6 cC(v) for every u and v in Rm satisfying juij 6 jvij for each i = 1, . . . , m. As in Section 3 X is assumed to be a normed space and g : X ! Rm , g(x) = (g1(x), . . . , gm(x))T, where gi, i = 1, . . . , m, are convex and continuous functions.
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Let us introduce now the following primal problem ðP cC Þ where
cþ C
inf cþ C ðgðxÞÞ; x2X
T
þ þ þ þ þ : R ! R, cþ C ðtÞ :¼ cC ðt Þ, with t ¼ ðt 1 ; . . . ; t m Þ and t i ¼ maxf0; t i g, i = 1, . . . , m. m
m Proposition 4.1. The function cþ C : R ! R is convex and componentwise increasing. þ
T
m þ Proof. First, let us point out that the function ðÞ : Rm ! Rmþ , defined by tþ ¼ ðtþ 1 ; . . . ; t m Þ , for t 2 R , is a m convex function. This means that, for u; v 2 R and a 2 [0, 1], it holds þ
ðau þ ð1 aÞvÞ 5auþ þ ð1 aÞvþ . Here, ‘‘5’’ is the ordering induced on Rm by the cone of nonnegative elements Rmþ . By the positive sublinearity and monotonicity of the gauge cC, we have for u; v 2 Rm and a 2 [0, 1],
þ 6 cC ðauþ þ ð1 aÞvþ Þ 6 acC ðuþ Þ þ ð1 aÞcC ðvþ Þ cþ C ðau þ ð1 aÞvÞ ¼ cC ðau þ ð1 aÞvÞ þ ¼ acþ C ðuÞ þ ð1 aÞcC ðvÞ.
This means that the function cþ C is convex. In order to prove that cþ is componentwise increasing, let u; v 2 Rm be such that ui 6 vi, i = 1, . . . , m. It C þ þ þ follows uþ i 6 vi , which implies that jui j 6 jvi j, i = 1, . . . , m. cC being a monotonic gauge, we have + + þ þ þ þ þ T þ þ T cC(u ) 6 cC(v ), where u ¼ ðu1 ; . . . ; um Þ , v ¼ ðvþ 1 ; . . . ; vm Þ or, equivalently, cC ðuÞ 6 cC ðvÞ. þ Hence the function cC is componentwise increasing. h By the approach described in Section 3, a dual problem to ðP cC Þ is ( ) m X
þ ðDcC Þ sup cC ðkÞ ðki gi Þ ðpi Þ . k2Rm ;pi 2X ; þP i¼1 m i¼1;...;m;
p ¼0 i¼1 i
m þ Proposition 4.2. The conjugate function ðcþ C Þ : R ! R [ fþ1g of cC verifies 0; if k 2 Rmþ and cC0 ðkÞ 6 1; ðcþ Þ ðkÞ ¼ C þ1; otherwise,
where cC0 is the gauge of the polar set C0. Proof. For k 2 Rm n Rm þ the assertion is a consequence of Propositions 3.1 and 4.1. þ þ Let be k 2 Rm . For t 2 Rm , we have jti j P jtþ i j, i = 1, . . . , m, which implies that cC ðtÞ P cC ðt Þ ¼ cC ðtÞ and þ
þ cC ðkÞ ¼ sup kT t cC ðtÞ 6 sup kT t cþ ð6Þ C ðtÞ ¼ cC ðkÞ. t2Rm
t2Rm
On the other hand, for the conjugate of the gauge cC we have the following formula (see Remark 2.9): T 0; if cC0 ðkÞ 6 1; ð7Þ cC ðkÞ ¼ sup k t cC ðtÞ ¼ m þ1; otherwise. t2R
þ If cC0 ðkÞ > 1, we have that þ1 ¼ cC ðkÞ 6 ðcþ C Þ ðkÞ. From here, ðcC Þ ðkÞ ¼ þ1.
Let be now cC0 ðkÞ 6 1. Because k = 0, it follows that kTt 6 kTt+, for every t 2 Rm . Furthermore, by Theorem 2.5, from cC0 ðkÞ 6 1 it follows that k 2 C0 and then by Proposition 2.6 we obtain that kTt+ 6 cC(t+). From these inequalities we obtain for the conjugate function of cþ C
T Tþ þ þ 0 6 cC ðkÞ 6 cþ ðkÞ ¼ sup k t c ðt Þ 6 sup k t c ðt Þ 6 0. C C C t2Rm
Consequently, there is
ðcþ C Þ ðkÞ
t2Rm
¼ 0 and the proposition is proved.
h
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By Proposition 4.2 the dual of ðP cC Þ has the following formulation: ( ) m X ðDcC Þ sup ðki gi Þ ðpi Þ . k2Rm i¼1 Pmþ ;pi 2X ;i¼1;...;m; p ¼0;cC 0 ðkÞ61 i¼1 i
In the objective function of this dual we separate the terms for which ki > 0 from the terms for which ki = 0 and then the dual can be written as ( ) X X ðDcC Þ supP ðki gi Þ ðpi Þ ð0Þ ðpi Þ . ð8Þ m pi 2X ;i¼1;...;m; p ¼0; i¼1 i cC 0 ðkÞ61;If1;...;mg; ki >0 ði2IÞ;ki ¼0 ði62IÞ
i62I
i2I
For i 62 I, it holds
0; if pi ¼ 0; 0 ðpi Þ ¼ supfhpi ; xi 0g ¼ suphpi ; xi ¼ þ1; otherwise. x2X x2X For i 2 I there is ðki gi Þ ðpi Þ ¼ ki gi k1i pi (cf. [3]). Denoting pi :¼ k1i pi , we obtain ( ) X ðDcC Þ sup ki gi ðpi Þ
ðI;k;pÞ2Y cC
with
i2I
( Y cC ¼
T
ðI; k; pÞ : I f1; . . . ; mg; k ¼ ðk1 ; . . . ; km Þ ; p ¼ ðp1 ; . . . ; pm Þ; cC0 ðkÞ 6 1; ki > 0 ði 2 IÞ; ki ¼ 0 ði 62 IÞ;
X
) ki p i ¼ 0 .
i2I
Because infðP cC Þ is finite being greater or equal than zero and cþ C is a convex and componentwise increasing function, by Theorem 3.3 we can formulate the following strong duality theorem for the problems ðP cC Þ and ðDcC Þ. Theorem 4.3 (strong duality for ðP cC Þ). The dual problem ðDcC Þ has an optimal solution and strong duality holds, i.e. infðP cC Þ ¼ maxðDcC Þ. Similarly to the general problem (P) the optimality conditions for ðP cC Þ can be derived. Theorem 4.4 (optimality conditions for ðP cC Þ). (1) Let x be an optimal solution to ðP cC Þ. Then there exists an optimal solution ðI; k; pÞ 2 Y cC to ðDcC Þ such that the following optimality conditions are satisfied: (i) I f1; . . . ; mg; ki > 0 ði 2 IÞ; ki ¼ 0 ði 62 IÞ, P (ii) cC0 ðkÞ 6 1; i2I ki pi ¼ 0, P ki gi ðxÞ, (iii) cþ xÞÞ ¼ i2I C ðgð (iv) gi ðxÞ þ gi ðpi Þ ¼ hpi ; xi; i 2 I. pÞ 2 Y cC and (i)–(iv) are fulfilled, then x is an optimal solution to ðP cC Þ, ðI; k; pÞ 2 Y cC is an (2) If x 2 X , ðI; k; optimal solution to ðDcC Þ and strong duality holds X ki gi ðpi Þ. xÞÞ ¼ cþ C ðgð i2I
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Proof (1) By Theorem 4.3 follows that there exists an optimal solution ðI; k; pÞ 2 Y cC to ðDcC Þ such that (i)–(ii) are fulfilled and X cþ ki gi ðpi Þ. xÞÞ ¼ C ðgð i2I
This equality is equivalent to " # X X
þ þ ki ½g ðxÞ þ g ðp Þ hp ; xi. 0 ¼ c ðgðxÞÞ þ c ðkÞ ki g ðxÞ þ C
i
C
i
i2I
i
i
i
ð9Þ
i2I
Using Young’s inequality we have X
T xÞ ¼ cþ ki gi ðxÞ xÞÞ þ cþ C ðgð C ðkÞ P k gð
ð10Þ
i2I
and gi ðxÞ þ gi ðpi Þ P hpi ; xi;
i 2 I.
ð11Þ
All terms of the sum in (9) inside brackets are positive, therefore X
T x Þ ¼ cþ ki gi ðxÞ xÞÞ þ cþ C ðgð C ðkÞ ¼ k gð
ð12Þ
i2I
and gi ðxÞ þ gi ðpi Þ ¼ hpi ; xi;
i 2 I.
ð13Þ P But, by Proposition 4.2, we have that ¼ 0, and, so, cþ xÞÞ ¼ i2I ki gi ðxÞ. C ðgð In conclusion, the relations (iii)–(iv) must also hold. (2) All the calculations and transformations done within part (1) may be carried out in the inverse direction. h ðcþ C Þ ðkÞ
5. The location model with unbounded unit balls In this section, we consider the problem treated by Hinojosa and Puerto in [6]. This is a single facility location problem, where gauges of closed convex sets are used to model distances. Throughout this section let A :¼ fa1 ; . . . ; am g be a subset of Rn which represents the set of existing facilities. Each facility ai 2 A has an associated gauge uai , whose unit ball is a closed convex set C ai containing the origin. Let w ¼ fwa1 ; . . . ; wam g be a set of positive weights and let cC : Rm ! R be a monotonic gauge of a closed convex set C containing the origin. The distance from an existing facility ai 2 A to a new facility x 2 Rn is given by uai ðx ai Þ. By u0ai we denote the gauge of the polar set C 0ai . The location problem studied in [6] is ðP cC ðAÞÞ
inf cC ðwa1 ua1 ðx a1 Þ; . . . ; wam uam ðx am ÞÞ.
x2Rn
Let g : Rn ! Rm be the vector function defined by g(x) :¼ (g1(x), . . . , gm(x))T, where gi ðxÞ ¼ wai uai ðx ai Þ for all i = 1, . . . , m. Because þ cþ C ðgðxÞÞ ¼ cC ðg ðxÞÞ ¼ cC ðgðxÞÞ;
8x 2 Rn ;
ðP cC ðAÞÞ can be written in the equivalent form ðP cC ðAÞÞ
inf cþ C ðgðxÞÞ;
x2Rn
G. Wanka et al. / European Journal of Operational Research 179 (2007) 1252–1265
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which is a particular case of the problem studied in the previous section. We mention, that instead of the space X considered in the case of the general optimization problem, we take here analogously to [6], the space Rn . Therefore, the dual problem to ðP cC ðAÞÞ is ( ) X ðDcC ðAÞÞ sup ki gi ðpi Þ ðI;k;pÞ2Y cC ðAÞ
with
i2I
( Y cC ðAÞ ¼
ðI; k; pÞ :I f1; . . . ; mg; k ¼ ðk1 ; . . . ; km ÞT ; p ¼ ðp1 ; . . . ; pm Þ; ) X cC0 ðkÞ 6 1; ki > 0 ði 2 IÞ; ki ¼ 0 ði 62 IÞ; ki p i ¼ 0 . i2I
n As the dual space is also R , the dual variable p belongs to R Rn . This is also valid for the duals in |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} the rest of the paper. m In our case gi ðxÞ ¼ wai uai ðx ai Þ, i = 1, . . . , m, hence (cf. [3]) pi T gi ðpi Þ ¼ ðwai uai ðx ai ÞÞ ðpi Þ ¼ ðwai uai Þ ðpi Þ þ pi ai ¼ wai uai þ pTi ai . w ai ( 0; if u0ai wpai 6 1; By Remark 2.9, uai wpai ¼ and, denoting pi :¼ wpai , i 2 I, the dual problem to i i i þ1; otherwise, ðP cC ðAÞÞ becomes ( ) X T ðDcC ðAÞÞ sup ki wai pi ai
X*
ðI;k;pÞ2Y cC ðAÞ
with
n
i2I
( Y cC ðAÞ ¼
T
ðI; k; pÞ : I f1; . . . ; mg; k ¼ ðk1 ; . . . ; km Þ ; p ¼ ðp1 ; . . . ; pm Þ; u0ai ðpi Þ 6 1; ) X i 2 I; cC0 ðkÞ 6 1; ki > 0 ði 2 IÞ; ki ¼ 0 ði 62 IÞ; ki wai pi ¼ 0 . i2I
Using Theorems 4.3 and 4.4 we can present for ðP cC ðAÞÞ and ðDcC ðAÞÞ the strong duality theorem and the optimality conditions. Theorem 5.1 (strong duality for ðP cC ðAÞÞ). The dual problem ðDcC ðAÞÞ has an optimal solution and strong duality holds, i.e. infðP cC ðAÞÞ ¼ maxðDcC ðAÞÞ. Theorem 5.2 (optimality conditions for ðP cC ðAÞÞ). (1) Let x be an optimal solution to ðP cC ðAÞÞ. Then there exists an optimal solution ðI; k; pÞ 2 Y cC ðAÞ to ðDcC ðAÞÞ such that the following optimality conditions are satisfied: (i) I f1; . . . ; mg; ki > 0 ði 2 IÞ; ki ¼ 0 ði 62 IÞ, P 0 ki wai pi ¼ 0, (ii) cC0 ð kÞ 6 1; uai ðpi Þ 6 1; i 2 I; i2I P (iii) cC ðwa1 ua1 ðx a1 Þ; . . . ; wam uam ðx am ÞÞ ¼ i2I ki wai uai ðx ai Þ, (iv) uai ðx ai Þ ¼ pi T ðx ai Þ; i 2 I. (2) If x 2 X , ðI; k; pÞ 2 Y cC and (i)–(iv) are fulfilled, then x is an optimal solution to ðP cC ðAÞÞ, ðI; k; pÞ 2 Y cC ðAÞ is an optimal solution to ðDcC ðAÞÞ and strong duality holds X ki wa pT ai . cC ðwa1 ua1 ðx a1 Þ; . . . ; wam uam ðx am ÞÞ ¼ i i i2I
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Proof. Theorem 5.2 is a direct consequence of Theorem 4.4 and the fact that cþ C ðgðxÞÞ ¼ cC ðgðxÞÞ.
h
Remark 5.3. The optimality conditions obtained for the optimization problem ðP cC ðAÞÞ are the same as the conditions obtained by Hinojosa and Puerto in Lemma 7 in [6]. In the paper cited above the authors gave a geometrical description of the set of optimal solutions, but, as one can see, by means of duality one obtains the same characterization of this set. In the next two sections of this paper we present some particular cases of the problem ðP cC ðAÞÞ, namely, the Weber problem and the minmax problem with gauges of closed convex sets. 6. The Weber problem with gauges of closed convex sets The Weber problem with gauges of closed convex sets is m X ðP w ðAÞÞ infn wai uai ðx ai Þ; x2R
i¼1
where uai , i = 1, . . . , m, are gauges whose unit balls are the closed convex sets C ai , i = 1, . . . , m, which contain the origin, and w ¼ fwa1 ; . . . ; wam g is a set of the positive weights. As one can see, the problem above is equivalent to the following one: ðP w ðAÞÞ
inf l1 ðgðxÞÞ;
x2Rn
Pm where l1 : Rm ! R, l1 ðkÞ ¼ i¼1 jki j and g : Rn ! Rm is the vector function defined by g(x) :¼ (g1(x), . . . , gm(x))T, with gi ðxÞ ¼ wai uai ðx ai Þ for all i = 1, . . . , m. One may observe that the function l1 is a monotonic gauge, actually, a monotonic norm. Taking cC(k) :¼ l1(k) for all k 2 Rm , by the results obtained in the previous section, the dual problem to ðP w ðAÞÞ becomes ( ) X T ðDw ðAÞÞ sup ki wai pi ai ðI;k;pÞ2Y w ðAÞ
with
i2I
( ðI; k; pÞ : I f1; . . . ; mg; k ¼ ðk1 ; . . . ; km ÞT ; p ¼ ðp1 ; . . . ; pm Þ; u0ai ðpi Þ 6 1;
Y w ðAÞ ¼
i2
I; l01 ðkÞ
6 1; ki > 0 ði 2 IÞ; ki ¼ 0 ði 62 IÞ;
X
) ki wai pi ¼ 0 .
i2I
Remark 6.1. In case the gauge cC of a convex set C is a norm, the gauge of the polar set C0 actually becomes the dual norm. Because the dual norm of the l1-norm is l01 ðkÞ ¼ maxi¼1;...;m jki j, we obtain the following formulation for the dual problem: ( ðDw ðAÞÞ
sup ðI;k;pÞ2Y w ðAÞ
with
X
) ki wai pTi ai
i2I
( Y w ðAÞ ¼
ðI; k; pÞ : I f1; . . . ; mg; k ¼ ðk1 ; . . . ; km ÞT ; p ¼ ðp1 ; . . . ; pm Þ; u0ai ðpi Þ 6 1; i 2 I; max ki 6 1; ki > 0 ði 2 IÞ; ki ¼ 0 ði 62 IÞ; i2I
X i2I
) ki wai pi ¼ 0 .
G. Wanka et al. / European Journal of Operational Research 179 (2007) 1252–1265
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Let us give now the strong duality theorem and the optimality conditions for ðP w ðAÞÞ and its dual ðDw ðAÞÞ. Theorem 6.2 (strong duality for ðP w ðAÞÞ). The dual problem ðDw ðAÞÞ has an optimal solution and strong duality holds, i.e. infðP w ðAÞÞ ¼ maxðDw ðAÞÞ. Theorem 6.3 (optimality conditions for ðP w ðAÞÞ) (1) Let x be an optimal solution to ðP w ðAÞÞ. Then there exists an optimal solution ðI; k; pÞ 2 Y w ðAÞ to ðDw ðAÞÞ such that the following optimality conditions are satisfied: i > 0 ði 2 IÞ; ki ¼ 0 ði 62 IÞ, (i) I f1; . . . ; mg; k P 0 ki wai pi ¼ 0, (ii) maxi2I ki 6 1; uai ðpi Þ 6 1; i 2 I; i2I Pm P (iii) ki wa u ðx ai Þ, wa u ðx ai Þ ¼ i¼1
ai
i
i2I
i
ai
T
(iv) uai ðx ai Þ ¼ pi ðx ai Þ; i 2 I. (2) If x 2 X , ðI; k; pÞ 2 Y w ðAÞ and (i)–(iv) are fulfilled, then x is an optimal solution to ðP w ðAÞÞ; ðI; k; pÞ 2 Y w ðAÞ is an optimal solution to ðDw ðAÞÞ and strong duality holds m X X ki wai wai uai ðx ai Þ ¼ pTi ai . i¼1
i2I
Proof. Theorem 6.3 is a direct consequence of Theorem 5.2. h 7. The minmax problem with gauges of closed convex sets The optimization problem studied in this last section is the minmax problem with gauges of closed convex sets ðP m ðAÞÞ
inf max wai uai ðx ai Þ;
x2Rn i¼1;...;m
where uai , i = 1, . . . , m, and w ¼ fwa1 ; . . . ; wam g are considered like in the previous section. One can see that this problem is equivalent to the following one: ðP m ðAÞÞ
inf l1 ðgðxÞÞ;
x2Rn
where l1 : Rm ! R; l1 ðkÞ ¼ maxi¼1;...;m jki j and g : Rn ! Rm is the vector function defined by g(x) :¼ (g1(x), . . . , gm(x))T, with gi ðxÞ ¼ wai uai ðx ai Þ for all i = 1, . . . , m. One may observe that the function l1 is also a monotonic norm. Taking cC(k) :¼ l1(k) for all k 2 Rm , the dual problem to ðP m ðAÞÞ becomes ( ) X ðDm ðAÞÞ sup ki wai pTi ai ; ðI;k;pÞ2Y m ðAÞ
with
i2I
( Y m ðAÞ ¼
T
ðI; k; pÞ : I f1; . . . ; mg; k ¼ ðk1 ; . . . ; km Þ ; p ¼ ðp1 ; . . . ; pm Þ; u0ai ðpi Þ 6 1; i2
I; l01 ðkÞ
6 1; ki > 0 ði 2 IÞ; ki ¼ 0 ði 62 IÞ;
X i2I
) ki wai pi ¼ 0 .
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Remark 7.1. Because the dual norm of the l1-norm is l01 ðkÞ ¼ for the dual problem: ( ðDm ðAÞÞ
sup ðI;k;pÞ2Y m ðAÞ
X
Pm
i¼1 jki j
we obtain the following formulation
) ki wai pTi ai
;
i2I
with ( Y m ðAÞ ¼
T
ðI; k; pÞ : I f1; . . . ; mg; k ¼ ðk1 ; . . . ; km Þ ; p ¼ ðp1 ; . . . ; pm Þ; u0ai ðpi Þ 6 1; i 2 I;
m X
ki 6 1; ki > 0 ði 2 IÞ; ki ¼ 0 ði 62 IÞ;
X
) ki wai pi ¼ 0 .
i2I
i¼1
Like in the previous section we give now the strong duality theorem and the optimality conditions for ðP m ðAÞÞ and its dual ðDm ðAÞÞ. Theorem 7.2 (strong duality for ðP m ðAÞÞ). The dual problem ðDm ðAÞÞ has an optimal solution and strong duality holds, i.e. infðP m ðAÞÞ ¼ maxðDm ðAÞÞ. Theorem 7.3 (optimality conditions for ðP m ðAÞÞ). (1) Let x be an optimal solution to ðP m ðAÞÞ. Then there exists an optimal solution ðI; k; pÞ 2 Y m ðAÞ to ðDm ðAÞÞ such that the following optimality conditions are satisfied: (i) I f1; . . . ; mg; ki > 0 ði 2 IÞ; ki ¼ 0 ði 62 IÞ, P P 0 (ii) k ki wai pi ¼ 0, 6 1; u ð p Þ 6 1; i 2 I; i ai i2I i P i2I (iii) maxi¼1;...;m wai uai ðx ai Þ ¼ i2I ki wai uai ðx ai Þ, (iv) uai ðx ai Þ ¼ pi T ðx ai Þ; i 2 I. (2) If x 2 X , ðI; k; pÞ 2 Y m ðAÞ and (i)–(iv) are fulfilled, then x is an optimal solution to ðP m ðAÞÞ, ðI; k; pÞ 2 Y m ðAÞ is an optimal solution to ðDm ðAÞÞ and strong duality holds X ki wai max wai uai ðx ai Þ ¼ pTi ai . i¼1;...;m
i2I
Proof. Theorem 7.3 is a direct consequence of Theorem 5.2. h References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
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