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Sensitivity Analysis for Convex Multiobjective Programming in Abstract Spaces
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.
202, 645]658 Ž1996.
0339
Sensitivity Analysis for Convex Multiobjective Programming in Abstract Spaces A. Balbas ´ Departamento de Economıa, ´ Uni¨ ersidad Carlos III, Madrid, 126.28903 Getafe, Madrid, Spain
and P. Jimenez Guerra ´ Departamento de Matematicas Fundamentales, Facultad de Ciencias, ´ Uni¨ ersidad Nacional de Educacion ´ a Distancia, Senda del Rey, s r n. 28040 Madrid, Spain Submitted by Augustine O. Esogbue Received January 29, 1992
The main object of this paper is to prove that for a linear or convex multiobjective program, a dual program can be obtained which gives the primal sensitivity without any special hypothesis about the way of choosing the optimal solution in the efficient set. Q 1996 Academic Press, Inc.
INTRODUCTION As is well known, many authors have studied the properties of multiobjective optimization problems including the existence and determination of the solutions and the properties of these solutions. Consequently, important methods of resolution have been found by the use of a wide class of techniques and interesting dual programs have been obtained which characterize the primal solutions. The problem of analyzing the sensitivity of the program with respect to the changes in the vector of the right side has also been studied by several authors Žsee, for instance, w13, 14x for the linear multiobjective programming, and w10, 11x for the nonlinear case.. In this paper, we prove that for a convex multiobjective program, a dual program can be obtained which gives the primal sensitivity without any special hypothesis about the way of choosing the optimal solution in the efficient set. 645 0022-247Xr96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
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BALBAS GUERRA ´ AND JIMENEZ ´
The present work begins by introducing several concepts, such as the concepts of T-optimal solution and T-dual program, and then later defining a dual program for convex programming, which extends the dual programs stated in w4, 13x Žfor linear programming. and w17x Žfor convex scalar programming.. After introducing the concept of associated solutions, Theorem 10 proves their existence and subsequently Theorem 12 and 15 prove the surprising fact that, in the multiobjective case, the sensitivity of the program is measured by the dual solution plus the derivative of this solution or a projection of such derivative, pointing out that this result is a generalization of the scalar case. The comparative advantage of operators T taking values in any ordered Banach space of finite or infinite dimension is that the generalization can have, as a consequence, important properties for some special classes of programs for which Theorems 12 and 15 imply a much more restricted range where the derivative of the dual solution can be valued. As a special case of operator T, we consider the operator introduced in Theorem 5 of w5x, which is a topological isomorphism and then the projection onto Ker T of the derivative of the dual solution is null. It may also be viewed as a linear and continuous mapping in the real line, with a duality theory for proper optimums useful in performing their sensitivity analysis.
T-OPTIMAL SOLUTIONS AND LAGRANGIAN T-MULTIPLIERS Let X, Y, Z, and W be four Banach spaces and let us assume that Y, Z, and W are ordered vector spaces with positive cones Yq, Zq, and Wq, respectively, being the orders of Y and W antisymmetrical and the order of W verifies the infimum axiom Ži.e., for each non-empty order-bounded from below subset B of W the inf B exists, that is, there exists w 0 g W such that w 0 F w for every w g B and if there is w 1 g W verifying the w 1 F w for every w g B, then w 1 F w 0 .. From now on, the cones Yq, Zq, and Wq will be closed and Zq and Wq will have moreover non-empty T interior Ži.e., ZTq/ B and Wq / B.. Let T : Y ª W be a linear and continuous surjective mapping such that T Ž Yqy 04. ; Wqy 04 , and let us denote by D a convex subset of X and by f : D ª Y and g: D ª Z two convex functions. Consider now the program Min f Ž x . x g Dgy b
5
Ž 1 gy b .
SENSITIVITY IN CONVEX PROGRAMMING
647
with Dgy b s x g D: g Ž x . F b4 and b g Z. In the particular case of being b s 0 we will write Ž1 g . and Dg . DEFINITION 1. A point x 0 g Dg is said to be a T-optimal solution Žof the program Ž1 g .. if Tf Ž x 0 . F Tf Ž x . holds for every x g Dg Žwe will write Tf instead of T ( f, TL instead of T ( L, and so on.. It is clear that every T-optimal solution of Ž1 g . is an optimal solution of that program. We say that y g Y is a T-ideal point Žof the program Ž1 g .. if T Ž y . s inf Tf Ž x . : x g Dg 4 .
Ž 1.1.
Trivially, x 0 g D is a T-optimal solution of Ž1 g . if and only if f Ž x 0 . is a T-ideal point of this program, and it is said that the program Ž1 g . is T-bounded if it admits one Žor more. T-ideal point, which is equivalent to the order-boundedness from below of the subset Tf Ž x .: x g Dg 4 of W. DEFINITION 2. A positive Ži.e., preserving the order. linear and continuous mapping L: Z ª W Žwe will write L g Lq Ž Z, W .. is said to be a Lagrangian T-multiplier Žfor the program Ž1 g .. if the following equality holds: inf Tf Ž x . : x g Dg 4 s inf Tf Ž x . q Lg Ž x . : x g D 4 .1
Ž 2.1.
THEOREM 3. If the program Ž1 g . has a T-optimal solution, there exists an x 1 g Dg such that g Ž x 1 . g yZTq and the order of Y ¨ erifies the infimum axiom, then there exists a Lagrangian T-multiplier for the program Ž1 g .. Proof. It follows from Theorem 5 of w28x that there exists Ž x 0 , L0 . g Dg = Lq Ž Z, W . such that the inequalities Tf Ž x . q L0 g Ž x . G Tf Ž x 0 . q L0 g Ž x 0 . G Tf Ž x 0 . q Lg Ž x 0 . hold for every x g D and every L g Lq Ž Z, W .. So taking L the null operator from Z into W we obtain that Inf Tf Ž x . : x g Dg 4 G Inf Tf Ž x . q L0 g Ž x . : x g D 4 G Tf Ž x 0 . q L0 g Ž x 0 . G Tf Ž x 0 . , from where it follows immediately Žsee footnote 1. that L0 is a Lagrangian T-multiplier for the program Ž1 g .. 1
Remark that the inequality infTf Ž x . q Lg Ž x .: x g D4 F infTf Ž x .: x g Dg 4 is always satisfied for every L g Lq Ž Z, W ..
BALBAS GUERRA ´ AND JIMENEZ ´
648
THEOREM 4. If b g Z, y 0 , y b g Y are two T-ideal points of the programs Ž1 g . and Ž1 gyb ., respecti¨ ely, L0 , L b g Lq Ž Z, W . are two Lagrangian T-multipliers for the programs Ž1 g . and Ž1 gyb ., respecti¨ ely, then yL b Ž b . G T Ž y b . y T Ž y 0 . G yL0 Ž b . .
Ž 4.1.
In the particular case of being x 0 g Dg and x b g Dgyb two T-optimal solutions for the programs Ž1 g . and Ž1 gyb ., respecti¨ ely, then we ha¨ e that yL b Ž b . G Tf Ž x b . y Tf Ž x 0 . G yL0 Ž b . .
Ž 4.2.
Proof. This is an immediate consequence of Definitions 1 and 2 since for every x g Dg and every x9 g Dgyb we have that T Ž y b . F Tf Ž x . q L b Ž g Ž x . y b . F Tf Ž x . y L b Ž b . and T Ž y 0 . F Tf Ž x . q L0 g Ž x9 . F Tf Ž x9 . q L0 Ž b . , from where the result follows trivially. THEOREM 5. If W is a Banach lattice and there exists a neighborhood V of the zero ¨ ector of Z such that for e¨ ery b g V there exists a T-optimal solution x b g Dgyb and a Lagrangian T-multiplier L b g Lq Ž Z, W ., both for the program Ž1 gy b ., such that lim b ª 0 L b s L0 in the space L Ž Z, W . of the linear and continuous mappings from Z into W, endowed with the topology of the uniform con¨ ergence on the bounded subsets of Z, then the function F: V ª W, such that F Ž b . s Tf Ž x b . for e¨ ery b g V, is Frechet differentiable ´ at 0 g V and its Frechet differential at zero coincides with yL0 . ´ Proof. For every b g V y 04 , it follows from Ž4.2. that 0F
Tf Ž x b . y Tf Ž x 0 . q L0 Ž b . 5 b5
F
L0 Ž b . y L b Ž b . 5 b5
from where the result is immediately deduced since L0 Ž b . y L b Ž b . 5 b5
F 5 L0 y L b 5 .
THE DUAL PROGRAM AND ASSOCIATED SOLUTIONS From now on, let us denote by T the set of the surjective linear and continuous mappings T : Y ª W such that T Ž Yqy 04. ; Wqy 04 and Ker T has a topological supplement in Y, and for every T g T let GT be
649
SENSITIVITY IN CONVEX PROGRAMMING
the set of all the operators L g Lq Ž Z, W . such that Tf Ž x . q Lg Ž x .: x g D4 is an order-bounded from below subset of W. For every T g T and L g GT let us consider
w Ž T , L . s inf Tf Ž x . q Lg Ž x . : x g D 4 g W. If T g T and YT is a topological supplement in Y of Ker T then, since YT is closed, it follows from the open-mapping theorem that the restriction Tˆ of T to YT is an isomorphism from YT onto W. For simplifying the notation we will suppose fixed YT for every T g T Žin the case of Y being a Hilbert space, YT could be for instance Žthe orthogonal of Ker T . ŽKer T . H. . Finally, for every T g T and L g GT let
c Ž T , L . s Tˆy1w Ž T , L . g Y . DEFINITION 6. Let b g Z and the program
¦ ¥ §
Min f Ž x . xgD . gŽ x. F b
Ž 1 gy b .
For every T g T , the T-dual program of the program Ž1 gy B . will be Max w Ž T , L . y L Ž b . . L g GT
5
Ž 2 gy b .
Moreover, the dual program of the program Ž1 gy b . will be
¦
Max c Ž T , TG . y G Ž b . TgT . G g L Ž Z, Y . TG g GT
¥
§
PROPOSITION 7.
Ž 3 gy b .
If x g Dgy b , G g L Ž Z, Y ., T g T , and TG g GT , then
c Ž T , TG . y G Ž b . y f Ž x . f Yqy 0 4 ,
Ž 7.1.
which means that the dual objecti¨ e is ne¨ er greater than the primal one. Proof. If Ž7.1. does not hold then we would have that
w Ž T , TG . y TG Ž b . y Tf Ž x . g Wqy 0 4
BALBAS GUERRA ´ AND JIMENEZ ´
650 and, therefore,
Tf Ž u . q TG g Ž u . y b y Tf Ž x . g Wqy 0 4 for every u g D, and in the particular case of being u s x we have that TG g Ž x . y b g Wqy 0 4 .
Ž 7.2.
Since x g Dgy b and TG g GT ; Lq Ž Z, W ., it results that TG g Ž x . y b g yWq,
Ž 7.3.
and now it follows from Ž7.2. and Ž7.3. a contradiction. PROPOSITION 8. If x g Dgy b , G g L Ž Z, Y ., T g T , TG g GT , and c ŽT, TG . y GŽ b . s f Ž x ., then x is an optimal solution of Ž1 gy b . and ŽT, G . is an optimal solution of Ž3 gy b .. Proof. This is an immediate consequence of Proposition 7. DEFINITION 9. Under the notations of Proposition 8, x and ŽT, G . will be called associated solutions. THEOREM 10. Let T g T and x 0 g Dgyb be a T-optimal solution of the program Ž1 gy b .. If b / 0 then the following assertions are equi¨ alent: 10.1 There exists L g Lq Ž Z, W . which is a Lagrangian T-multiplier for the program Ž1 gy b .. 10.2 There exists G g L Ž Z, Y . such that ŽT, G . is an optimal solution of the program Ž3 gy b . and x 0 and ŽT, G . are associated solutions.2 Proof. First let us suppose that 10.1 holds, then Tf Ž x 0 . s inf Tf Ž x . q L g Ž x . y b : x g D 4
Ž 10.1.
and since YT is a topological supplement in Y of Ker T, then there exists y 0 g Ker T verifying that f Ž x 0 . s y 0 q Tˆy1 Tf Ž x 0 . .
Ž 10.2.
Moreover, since b / 0 we can find z9 g Z9 Ž Z9 being as usual the dual space of Z . such that z9 Ž b . s 1.
Ž 10.3.
2 Let us remark that Ž10.2. implies Ž10.1. also being b s 0, and moreover that L s TG. Also it must be pointed out that the proof of the implication Ž10.1. « Ž10.2. is a constructive one.
SENSITIVITY IN CONVEX PROGRAMMING
651
Let us define G Ž z . s Tˆy1 L Ž z . y z9 Ž z . y 0 for every z g Z. Clearly, G g L Ž Z, y . and TG s L g Lq Ž Z, W ., since y 0 g Ker T. Moreover we have that
w Ž T , TG . s w Ž T , L . s inf Tf Ž x . q Lg Ž x . : x g D 4 s inf Tf Ž x . q L g Ž x . y b : x g D 4 q L Ž b . , and then it follows from Ž10.1. that w ŽT, TG . s Tf Ž x 0 . q LŽ b .. Therefore, TG g GT and
c Ž T , TG . s Tˆy1 w Ž T , TG . s Tˆy1 Tf Ž x 0 . q Tˆy1 L Ž b . , and it follows from Ž10.3. that GŽ b . s Tˆy1 LŽ b . y y 0 and
c Ž T , TG . s Tˆy1 Tf Ž x 0 . q G Ž b . q y 0 , so it results from Ž10.2. that
c Ž T , TG . y G Ž b . s Tˆy1 Tf Ž x 0 . q y 0 s f Ž x0 . . Let us assume now that 10.2 holds and let be L s TG g Lq Ž Z, W .. Then since f Ž x 0 . s c ŽT, TG . y GŽ b ., we have that Tf Ž x 0 . s w Ž T , TG . y L Ž b . s inf Tf Ž x . y Lg Ž x . : x g D 4 y L Ž b . s inf Tf Ž x . y L g Ž x . y b : x g D 4 and L is a Lagrangian T-multiplier for the program Ž1 gy b .. SENSITIVITY ANALYSIS LEMMA 11. Let V be an open subset of Z, M a Banach space, and H : V ª L Ž Z, M . a Frechet differentiable function Ž L Ž Z, M . denotes as ´ usual the space of the linear and continuous mappings from Z into M, endowed with the topology of the uniform con¨ ergence on the bounded subsets
BALBAS GUERRA ´ AND JIMENEZ ´
652
of Z .. If F: V ª M is such that F Ž b . s Hb Ž b . with Hb s H Ž b ., for e¨ ery b g V, then F is Frechet differentiable at V and ´ F9 Ž b, z . s Hb Ž z . q H 9 Ž b, z . Ž b . for e¨ ery z g Z and e¨ ery b g V.3 Proof. In fact, if b g V and z g Z then we have that 1 5 z5
F Ž b q z . y F Ž b . y Hb Ž z . y H Ž b, z . Ž b . s F
1 5 z5 1 5 z5
Hbq z Ž b q z . y Hb Ž b . y Hb Ž z . y H Ž b, z . Ž b . Hbq z Ž b . y Hb Ž b . y H 9 Ž b, z . Ž b . q Hbqz Ž z . y Hb Ž z .
from where the result follows immediately since the function H is Frechet ´ differentiable at b g V and, therefore, it is also continuous at b g V. We are now able to prove a rather surprising result. In general, in a multiobjective convex program, the sensitivity of the optimum depends not only on the value of the dual solution but also on the differential of this dual solution and more concretely, on the projection of such a derivative onto Ker T. In the scalar programming cases Ži.e., Y s R., such a projection is clearly null and therefore, the next Theorem 12 is a generalization of the corresponding happening in the scalar programming. As we will see later on, in the particular case of the linear programming the differential of the dual solution is in Ker T Žand therefore, it coincides with its projection onto Ker T .. THEOREM 12. Let W be a Banach lattice, V an open subset of Z, T g T , x b g Dgyb a T-optimal solution of the program Ž1 gyb ., and G b g L Ž Z, Y . an operator such that ŽT, G b . is an optimal solution of the program Ž3 gyb . associated with x b , for e¨ ery b g V. If the function G : V ª L Ž Z, Y . defined by G Ž b . s G b for e¨ ery b g V, is Frechet differentiable Ž at V . then the ´ function F: V ª Y such that F Ž b . s f Ž x b . for e¨ ery b g V, is also Frechet ´ differentiable Ž at V . and moreo¨ er, the equality F9 Ž b, z . s yGb Ž z . y K Ž b, z . holds for e¨ ery z g Z and b g V, where K Ž b, z . denotes the projection of G 9Ž b, z .Ž b . onto Ker T, for e¨ ery z g Z and b g V. 3
F9Ž b, z . denotes the image of z g Z by the Frechet differential of the function F at ´ b g V. A similar notation is followed in the other cases.
,
653
SENSITIVITY IN CONVEX PROGRAMMING
Proof. For every b g V let L b s TG b g GT . It follows from the proof of Theorem 10 that L b is a Lagrangian T-multiplier for the program Ž1 gy b .. Consider now the functions w *: V ª W and c *: V ª Y such that w *Ž b . s w ŽT, L b . and c *Ž b . s c ŽT, TG b . for every b g V. Since ŽT, G b . and x 0 are associated solutions we have that F Ž b . s f Ž x b . s c Ž T , TG b . y G b Ž b .
Ž 12.1.
and Tf Ž x b . s w Ž T , L b . y L b Ž b . for every b g V, and it results from Theorem 5 and Lemma 11 that yL b Ž z . s Ž w * . 9 Ž b, z . y L b Ž z . y L 9 Ž b, z . Ž b .
Ž 12.2.
for every z g Z and b g V, where L : V ª L Ž Z, V . is the function defined by L Ž b . s L b for every b g V. Clearly, the equality Ž12.2. is equivalent to
Ž w * . 9 Ž b, z . s L 9 Ž b, z . Ž b .
Ž 12.3.
for every z g Z and b g V, and since w * s Tc *, it follows from Ž12.3. that T Ž c * . 9 Ž b, z . s L 9 Ž b, z . Ž b . for every z g Z and b g V. Moreover, since L Ž b . s T G Ž b . for every b g V, it is immediately deduced that L 9 Ž b, z . Ž b . s T G 9 Ž b, z . Ž b . , T Ž c * . 9 Ž b, z . s T G 9 Ž b, z . Ž b . and
Ž c * . 9 Ž b, z . s Tˆy1 T G 9 Ž b, z . Ž b . for every z g Z and b g V. Finally, from Ž12.1. and Lemma 11 it results that F9 Ž b, z . s yGb Ž z . y G 9 Ž b, z . Ž b . q Tˆy1 T G 9 Ž b, z . Ž b . s yGb Ž z . y K Ž b, z . for every z g Z and b g V. EXAMPLE. To illustrate easily the result stated in Theorem 12 let us consider the program Ž1 gy b . with b g Žy`, 0., X s Y s R 2 , Z s W s R, f Ž x 1 , x 2 . s Ž x 12 , 2 x 22 ., and g Ž x 1 , x 2 . s T Ž x 1 , x 2 . s x 1 q x 2 . Solving in the
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BALBAS GUERRA ´ AND JIMENEZ ´
usual way the program Min x 12 q 2 x 22 x1 q x 2 F b
5
the solution x b s Ž 23 b, 13 b . and the Lagrangian T-multiplier L b s y 43 b are obtained. Therefore, the function F: Žy`, 0. ª R 2 Žof Theorem 12. is defined by F Ž b . s f Ž 23 b, 13 b . s Ž 49 b 2 , 29 b 2 . and then F9 Ž b . s Ž 89 b, 49 b . . On the other hand, Ker T is the linear space generated by the vector Ž1, y1. and so ŽKer T . will be the linear subspace generated by Ž1, 1.. Following the construction made in the proof of Theorem 10, we obtain Tˆy1 w L b x s Ž y 23 b, y 23 b . and y b s f Ž x b . y Tˆy1 Tf Ž x b . s Ž 49 b 2 , 29 b 2 . y Tˆy1 T Ž 49 b 2 , 29 b 2 . s Ž 49 b 2 , 29 b 2 . y Tˆy1 Ž 69 b 2 . s Ž 49 b 2 , 29 b 2 . y Ž 39 b 2 , 39 b 2 . s Ž 19 b 2 , y 19 b 2 . . Taking now zXb Ž z . s Ž1rb . z for every z g R, we have that G b s Tˆy1 w L b x y
1 b
y b s Ž y 79 b, y 59 b . .
Therefore, since the projection onto Ker T is the function p : R 2 ª Ker T such that
p Ž x1 , x 2 . s
ž
x1 y x 2 2
,y
x1 y x 2 2
/
SENSITIVITY IN CONVEX PROGRAMMING
655
for every Ž x 1 , x 2 . g R 2 , with the notations of Theorem 12, we have that K Ž b . s p G 9Ž b . Ž b . s y
ž
b b , 9 9
/
and therefore, yGb y K Ž b . s Ž 89 b, 49 b . and yGb y K Ž b . s F9Ž b ., as Theorem 12 states. DUALITY AND SENSITIVITY FOR LINEAR PROGRAMS Let us now treat the important particular case of linear programs. For them, the dual program has a much simpler formulation than that given by Definition 6 Žas we will see in Theorem 14.. Of special interest is Theorem 15 which is analogous to Theorem 12, but which states that the term G 9Ž b, z .Ž b . belongs now to Ker T and, therefore, it coincides with its projection. Suppose henceforth that D is a Žnon-necessarily pointed. convex cone of X and that f g L Ž X, Y . and g g L Ž X, Z .. Then the spaces L Ž X, Y . and L Ž X, Z . will be assumed to be ordered by its natural cones Lq Ž X , Y . s j g L Ž X , Y . : j Ž x . g Yq for every x g D 4 , Lq Ž X , Z . s j g L Ž X , Z . : j Ž x . g Zq for every x g D 4 and the same will be for L Ž X, W .. LEMMA 13. Under the already established notations, if T g T then GT s L g Lq Ž Z, W . : Lg G yTf 4
4
and w ŽT, L. s 0 for e¨ ery L g GT . Proof. Let L g Lq Ž Z, W .. If Lg G yTf, then Lg Ž x . q Tf Ž x . g Wq for every x g D and, therefore, Tf Ž x . q Lg Ž x .: x g D4 ; Wq is orderbounded from below. Moreover, since 0 g D we have that
w Ž T , L . s inf Tf Ž x . q Lg Ž x . : x g D 4 s 0. If L g GT then Tf Ž x . q Lg Ž x .: x g D4 is an order-bounded from below subset of W and, therefore, w9w TF Ž x . q Lg Ž x .x: x g D4 is a bounded 4
Lg G yTf means as usual that Lg Ž x . q Tf Ž x . g Wq for every x g D.
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BALBAS GUERRA ´ AND JIMENEZ ´
X X from below subset of R for every w9 g Wq , where Wq denotes as usual the dual cone of Wq in the dual space W9 of W, and therefore, we have that
w9 TF Ž x . q Lg Ž x . G 0
Ž 13.1.
X X for every w9 g Wq and x g D Žsince if there exists wX0 g Wq and x 0 g D X such that w 0 w TF Ž x 0 . q Lg Ž x 0 .x - 0 then
lim wX0 Tf Ž a x 0 . q Lg Ž a x 0 . s y`,
aªq`
which is a contradiction.. T Now since Wq / B, it follows from Ž13.1. that Tf Ž x . q Lg Ž x . g Wq for every x g D Žsee w12x.. THEOREM 14. The dual program of Ž1 gy b . Ž under the present assumption of Ž1 gy b . being a linear program. is
¦
Max y G Ž b . TgT G g L Ž Z, Y . . TGg G yTf TG g Lq Ž Z, W .
¥
Ž 4gy b .
§
Proof. This follows immediately from Definition 6 and Lemma 13. THEOREM 15. Let W be a Banach lattice, T g T , and an open subset V of Z such that for e¨ ery b g V there exists x b g V and G b g L Ž Z, Y . such that x b is a T-optimal solution of Ž1 gyb ., ŽT, G b . is an optimal solution of Ž4gyb ., and x b and ŽT, G b . are associated solutions. Suppose that the mappings G : V ª L Ž X, Y . such that G Ž b . s G b for e¨ ery b g V, is Frechet differen´ tiable Ž at V . and let F: V ª Y be the function defined by F Ž b . s f Ž x b . for e¨ ery b g V. Then F is Frechet differentiable Ž at V . and the equalities ´ F9 Ž b, z . s yGb Ž z . y G 9 Ž b, z . Ž b .
Ž 15.1.
T G 9 Ž b, z . Ž b . s 0
Ž 15.2.
and
hold for e¨ ery z g Z and b g V. Proof. It follows from Theorem 12 that the function F is Frechet ´ differentiable Žat V . and that F9 Ž b, z . s yGb Ž z . y K Ž b, z .
Ž 15.3.
SENSITIVITY IN CONVEX PROGRAMMING
657
holds for every z g Z and b g V, where K Ž b, z . is the projection of G 9Ž b, z .Ž b . onto Ker T. Moreover, since x b and ŽT, G b . are associated solutions for every b g V, it results from Theorem 14 that F Ž b . s f Ž x b . s yGb Ž b . for every b g V, and, therefore, Ž15.1. follows from Lemma 11. Moreover, it is immediately deduced from Ž15.1. and Ž15.3. that G 9Ž b, z .Ž b . s K Ž b, z . for every z g Z and b g V and, therefore, Ž15.2. is verified.
CONCLUSIONS The dual program stated here for convex multiobjective programming extends the dual programs given in w4, 13x Žfor linear multiobjective programming. and w17x Žfor convex scalar programming.. Theorem 10 proves the existence of associated solutions which are introduced in Definition 9 and Theorems 12 establishes the surprising fact that, in the multiobjective case, the sensitivity of the program is measured by the dual solution plus the derivative of this solution or a projection of such a derivative, pointing out that this result is a generalization of the scalar case. As an important particular case linear programs are also studied since for them the results have a simpler formulation. The theory developed here in the context of Banach spaces is quite general and it may be applied in many particular situations like static, dynamic, or semi-infinite programs.
REFERENCES 1. E. J. Anderson and P. Nash, ‘‘Linear Programming in Infinite Dimensional Spaces,’’ Wiley, New York, 1987. 2. W. Alt, Stability of solutions to control-constrained nonlinear optimal control problems, Appl. Math. Optim. 21 Ž1990., 53]68. 3. W. Alt, Local stability of solutions to differentiable optimization problems in Banach spaces, J. Optim. Theory Appl. 70 Ž1991., 443]466. 4. A. Balbas ´ and A. Heras, Duality theory for infinite-dimensional multiobjective linear programming, European J. Oper. Res. 68 Ž1993., 379]388. 5. A. Balbas, Guerra, and C. Nunez, ´ P. Jimenez ´ ˜ Strongly proper optimums and maximal optimization in multiobjective programming, Re¨ . Real Acad. Cienc. Exact. Fıs. ´ Natur. Madrid 86 Ž1992., 289]295. 6. G. R. Bitran, Duality for nonlinear multiple-criteria optimization problems, J. Optim. Theory Appl. 35 Ž1981., 367]401. 7. J. M. Borwein, On the existence of Pareto efficient points, Math. Oper. Res. 8 Ž1981., 64]73.
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8. S. Brumelle, Duality for multiple objective convex programs, Math. Oper. Res. 6 Ž1981., 159]172. 9. H. W. Corley, Duality for maximizations with respect to cones, J. Math. Anal. Appl. 84 Ž1981., 560]568. 10. J. P. Dauer and M. S. A. Osman, Decomposition of the parametric space in multiobjective convex programs using the generalized Tchebycheff norm, J. Math. Anal. Appl. 107 Ž1985., 156]166. 11. A. V. Fiacco, ‘‘Introduction to Sensitivity and Stability Analysis in Nonlinear Programming,’’ Academic Press, New York, 1983. 12. J. P. Ignizio, An algorithm for solving the linear goal programming problem by solving its dual, J. Oper. Res. Soc. 36 Ž1985., 507]515. 13. H. Isermann, The relevance of duality in multiple objective linear programming, TIMS Stud. Management Sci. 6 Ž1977., 241]262. 14. H. Isermann, On some relations between a dual pair of multiple objective linear programs, Z. Oper. Res. 22 Ž1978., 33]41. 15. F. Lempio and H. Maurer, Differential stability in infinite-dimensional nonlinear programming, Appl. Math. Optim. 6 Ž1980., 139]152. 16. D. T. Luc, On duality theory in multiobjective programming, J. Optim. Theory Appl. 43 Ž1984., 557]582. 17. D. G. Luenberger, ‘‘Optimization by Vector Space Methods,’’ Wiley, New York, 1969. 18. K. Malanowski, Differentability with respect to parameters of solutions to convex programming problems, Math. Programming 33 Ž1985., 352]361. 19. S. Park, Complementary slackness theorem in multiple objective linear programming, Oper. Res. 30 Ž1982., 410]412. 20. R. T. Rockafellar, Lagrange multipliers and subderivatives of optimal value functions in nonlinear programming, Math. Programming 17 Ž1982., 28]66. 21. Y. Sawaragy and T. Tanino, Duality theorem in multiobjective programming, J. Optim. Theory Appl. 27 Ž1979., 509]529. 22. H. H. Schaefer, ‘‘Topological Vector Spaces,’’ Springer-Verlag, Berlin, 1971. 23. T. Tanino, Sensitivity analysis in multiobjective optimization, J. Optim. Theory Appl. 56 Ž1988., 479]499. 24. S. Yamamuro, Differential calculus in topological linear spaces, in ‘‘Lecture Notes in Math.,’’ Vol. 374, Springer-Verlag, Berlin, 1974. 25. M. Zeleny, ‘‘Linear Multiobjective Programming,’’ Springer-Verlag, Berlin, 1974. 26. S. Zionts, Multiple criteria mathematical programming: Overview and several approaches, in ‘‘Math. of Multiobjective Optim.’’ ŽP. Serfini, Ed.., Springer-Verlag, Berlin, 1985. 27. J. Zowe, A duality theorem for a convex programming problem in order complete vector lattices, J. Math. Anal. Appl. 50 Ž1975., 273]287. 28. J. Zowe, The saddle point theorem of Kuhn and Tucker in ordered vector spaces, J. Math. Anal. Appl. 57 Ž1977., 41]55.
202, 645]658 Ž1996.
0339
Sensitivity Analysis for Convex Multiobjective Programming in Abstract Spaces A. Balbas ´ Departamento de Economıa, ´ Uni¨ ersidad Carlos III, Madrid, 126.28903 Getafe, Madrid, Spain
and P. Jimenez Guerra ´ Departamento de Matematicas Fundamentales, Facultad de Ciencias, ´ Uni¨ ersidad Nacional de Educacion ´ a Distancia, Senda del Rey, s r n. 28040 Madrid, Spain Submitted by Augustine O. Esogbue Received January 29, 1992
The main object of this paper is to prove that for a linear or convex multiobjective program, a dual program can be obtained which gives the primal sensitivity without any special hypothesis about the way of choosing the optimal solution in the efficient set. Q 1996 Academic Press, Inc.
INTRODUCTION As is well known, many authors have studied the properties of multiobjective optimization problems including the existence and determination of the solutions and the properties of these solutions. Consequently, important methods of resolution have been found by the use of a wide class of techniques and interesting dual programs have been obtained which characterize the primal solutions. The problem of analyzing the sensitivity of the program with respect to the changes in the vector of the right side has also been studied by several authors Žsee, for instance, w13, 14x for the linear multiobjective programming, and w10, 11x for the nonlinear case.. In this paper, we prove that for a convex multiobjective program, a dual program can be obtained which gives the primal sensitivity without any special hypothesis about the way of choosing the optimal solution in the efficient set. 645 0022-247Xr96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
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The present work begins by introducing several concepts, such as the concepts of T-optimal solution and T-dual program, and then later defining a dual program for convex programming, which extends the dual programs stated in w4, 13x Žfor linear programming. and w17x Žfor convex scalar programming.. After introducing the concept of associated solutions, Theorem 10 proves their existence and subsequently Theorem 12 and 15 prove the surprising fact that, in the multiobjective case, the sensitivity of the program is measured by the dual solution plus the derivative of this solution or a projection of such derivative, pointing out that this result is a generalization of the scalar case. The comparative advantage of operators T taking values in any ordered Banach space of finite or infinite dimension is that the generalization can have, as a consequence, important properties for some special classes of programs for which Theorems 12 and 15 imply a much more restricted range where the derivative of the dual solution can be valued. As a special case of operator T, we consider the operator introduced in Theorem 5 of w5x, which is a topological isomorphism and then the projection onto Ker T of the derivative of the dual solution is null. It may also be viewed as a linear and continuous mapping in the real line, with a duality theory for proper optimums useful in performing their sensitivity analysis.
T-OPTIMAL SOLUTIONS AND LAGRANGIAN T-MULTIPLIERS Let X, Y, Z, and W be four Banach spaces and let us assume that Y, Z, and W are ordered vector spaces with positive cones Yq, Zq, and Wq, respectively, being the orders of Y and W antisymmetrical and the order of W verifies the infimum axiom Ži.e., for each non-empty order-bounded from below subset B of W the inf B exists, that is, there exists w 0 g W such that w 0 F w for every w g B and if there is w 1 g W verifying the w 1 F w for every w g B, then w 1 F w 0 .. From now on, the cones Yq, Zq, and Wq will be closed and Zq and Wq will have moreover non-empty T interior Ži.e., ZTq/ B and Wq / B.. Let T : Y ª W be a linear and continuous surjective mapping such that T Ž Yqy 04. ; Wqy 04 , and let us denote by D a convex subset of X and by f : D ª Y and g: D ª Z two convex functions. Consider now the program Min f Ž x . x g Dgy b
5
Ž 1 gy b .
SENSITIVITY IN CONVEX PROGRAMMING
647
with Dgy b s x g D: g Ž x . F b4 and b g Z. In the particular case of being b s 0 we will write Ž1 g . and Dg . DEFINITION 1. A point x 0 g Dg is said to be a T-optimal solution Žof the program Ž1 g .. if Tf Ž x 0 . F Tf Ž x . holds for every x g Dg Žwe will write Tf instead of T ( f, TL instead of T ( L, and so on.. It is clear that every T-optimal solution of Ž1 g . is an optimal solution of that program. We say that y g Y is a T-ideal point Žof the program Ž1 g .. if T Ž y . s inf Tf Ž x . : x g Dg 4 .
Ž 1.1.
Trivially, x 0 g D is a T-optimal solution of Ž1 g . if and only if f Ž x 0 . is a T-ideal point of this program, and it is said that the program Ž1 g . is T-bounded if it admits one Žor more. T-ideal point, which is equivalent to the order-boundedness from below of the subset Tf Ž x .: x g Dg 4 of W. DEFINITION 2. A positive Ži.e., preserving the order. linear and continuous mapping L: Z ª W Žwe will write L g Lq Ž Z, W .. is said to be a Lagrangian T-multiplier Žfor the program Ž1 g .. if the following equality holds: inf Tf Ž x . : x g Dg 4 s inf Tf Ž x . q Lg Ž x . : x g D 4 .1
Ž 2.1.
THEOREM 3. If the program Ž1 g . has a T-optimal solution, there exists an x 1 g Dg such that g Ž x 1 . g yZTq and the order of Y ¨ erifies the infimum axiom, then there exists a Lagrangian T-multiplier for the program Ž1 g .. Proof. It follows from Theorem 5 of w28x that there exists Ž x 0 , L0 . g Dg = Lq Ž Z, W . such that the inequalities Tf Ž x . q L0 g Ž x . G Tf Ž x 0 . q L0 g Ž x 0 . G Tf Ž x 0 . q Lg Ž x 0 . hold for every x g D and every L g Lq Ž Z, W .. So taking L the null operator from Z into W we obtain that Inf Tf Ž x . : x g Dg 4 G Inf Tf Ž x . q L0 g Ž x . : x g D 4 G Tf Ž x 0 . q L0 g Ž x 0 . G Tf Ž x 0 . , from where it follows immediately Žsee footnote 1. that L0 is a Lagrangian T-multiplier for the program Ž1 g .. 1
Remark that the inequality infTf Ž x . q Lg Ž x .: x g D4 F infTf Ž x .: x g Dg 4 is always satisfied for every L g Lq Ž Z, W ..
BALBAS GUERRA ´ AND JIMENEZ ´
648
THEOREM 4. If b g Z, y 0 , y b g Y are two T-ideal points of the programs Ž1 g . and Ž1 gyb ., respecti¨ ely, L0 , L b g Lq Ž Z, W . are two Lagrangian T-multipliers for the programs Ž1 g . and Ž1 gyb ., respecti¨ ely, then yL b Ž b . G T Ž y b . y T Ž y 0 . G yL0 Ž b . .
Ž 4.1.
In the particular case of being x 0 g Dg and x b g Dgyb two T-optimal solutions for the programs Ž1 g . and Ž1 gyb ., respecti¨ ely, then we ha¨ e that yL b Ž b . G Tf Ž x b . y Tf Ž x 0 . G yL0 Ž b . .
Ž 4.2.
Proof. This is an immediate consequence of Definitions 1 and 2 since for every x g Dg and every x9 g Dgyb we have that T Ž y b . F Tf Ž x . q L b Ž g Ž x . y b . F Tf Ž x . y L b Ž b . and T Ž y 0 . F Tf Ž x . q L0 g Ž x9 . F Tf Ž x9 . q L0 Ž b . , from where the result follows trivially. THEOREM 5. If W is a Banach lattice and there exists a neighborhood V of the zero ¨ ector of Z such that for e¨ ery b g V there exists a T-optimal solution x b g Dgyb and a Lagrangian T-multiplier L b g Lq Ž Z, W ., both for the program Ž1 gy b ., such that lim b ª 0 L b s L0 in the space L Ž Z, W . of the linear and continuous mappings from Z into W, endowed with the topology of the uniform con¨ ergence on the bounded subsets of Z, then the function F: V ª W, such that F Ž b . s Tf Ž x b . for e¨ ery b g V, is Frechet differentiable ´ at 0 g V and its Frechet differential at zero coincides with yL0 . ´ Proof. For every b g V y 04 , it follows from Ž4.2. that 0F
Tf Ž x b . y Tf Ž x 0 . q L0 Ž b . 5 b5
F
L0 Ž b . y L b Ž b . 5 b5
from where the result is immediately deduced since L0 Ž b . y L b Ž b . 5 b5
F 5 L0 y L b 5 .
THE DUAL PROGRAM AND ASSOCIATED SOLUTIONS From now on, let us denote by T the set of the surjective linear and continuous mappings T : Y ª W such that T Ž Yqy 04. ; Wqy 04 and Ker T has a topological supplement in Y, and for every T g T let GT be
649
SENSITIVITY IN CONVEX PROGRAMMING
the set of all the operators L g Lq Ž Z, W . such that Tf Ž x . q Lg Ž x .: x g D4 is an order-bounded from below subset of W. For every T g T and L g GT let us consider
w Ž T , L . s inf Tf Ž x . q Lg Ž x . : x g D 4 g W. If T g T and YT is a topological supplement in Y of Ker T then, since YT is closed, it follows from the open-mapping theorem that the restriction Tˆ of T to YT is an isomorphism from YT onto W. For simplifying the notation we will suppose fixed YT for every T g T Žin the case of Y being a Hilbert space, YT could be for instance Žthe orthogonal of Ker T . ŽKer T . H. . Finally, for every T g T and L g GT let
c Ž T , L . s Tˆy1w Ž T , L . g Y . DEFINITION 6. Let b g Z and the program
¦ ¥ §
Min f Ž x . xgD . gŽ x. F b
Ž 1 gy b .
For every T g T , the T-dual program of the program Ž1 gy B . will be Max w Ž T , L . y L Ž b . . L g GT
5
Ž 2 gy b .
Moreover, the dual program of the program Ž1 gy b . will be
¦
Max c Ž T , TG . y G Ž b . TgT . G g L Ž Z, Y . TG g GT
¥
§
PROPOSITION 7.
Ž 3 gy b .
If x g Dgy b , G g L Ž Z, Y ., T g T , and TG g GT , then
c Ž T , TG . y G Ž b . y f Ž x . f Yqy 0 4 ,
Ž 7.1.
which means that the dual objecti¨ e is ne¨ er greater than the primal one. Proof. If Ž7.1. does not hold then we would have that
w Ž T , TG . y TG Ž b . y Tf Ž x . g Wqy 0 4
BALBAS GUERRA ´ AND JIMENEZ ´
650 and, therefore,
Tf Ž u . q TG g Ž u . y b y Tf Ž x . g Wqy 0 4 for every u g D, and in the particular case of being u s x we have that TG g Ž x . y b g Wqy 0 4 .
Ž 7.2.
Since x g Dgy b and TG g GT ; Lq Ž Z, W ., it results that TG g Ž x . y b g yWq,
Ž 7.3.
and now it follows from Ž7.2. and Ž7.3. a contradiction. PROPOSITION 8. If x g Dgy b , G g L Ž Z, Y ., T g T , TG g GT , and c ŽT, TG . y GŽ b . s f Ž x ., then x is an optimal solution of Ž1 gy b . and ŽT, G . is an optimal solution of Ž3 gy b .. Proof. This is an immediate consequence of Proposition 7. DEFINITION 9. Under the notations of Proposition 8, x and ŽT, G . will be called associated solutions. THEOREM 10. Let T g T and x 0 g Dgyb be a T-optimal solution of the program Ž1 gy b .. If b / 0 then the following assertions are equi¨ alent: 10.1 There exists L g Lq Ž Z, W . which is a Lagrangian T-multiplier for the program Ž1 gy b .. 10.2 There exists G g L Ž Z, Y . such that ŽT, G . is an optimal solution of the program Ž3 gy b . and x 0 and ŽT, G . are associated solutions.2 Proof. First let us suppose that 10.1 holds, then Tf Ž x 0 . s inf Tf Ž x . q L g Ž x . y b : x g D 4
Ž 10.1.
and since YT is a topological supplement in Y of Ker T, then there exists y 0 g Ker T verifying that f Ž x 0 . s y 0 q Tˆy1 Tf Ž x 0 . .
Ž 10.2.
Moreover, since b / 0 we can find z9 g Z9 Ž Z9 being as usual the dual space of Z . such that z9 Ž b . s 1.
Ž 10.3.
2 Let us remark that Ž10.2. implies Ž10.1. also being b s 0, and moreover that L s TG. Also it must be pointed out that the proof of the implication Ž10.1. « Ž10.2. is a constructive one.
SENSITIVITY IN CONVEX PROGRAMMING
651
Let us define G Ž z . s Tˆy1 L Ž z . y z9 Ž z . y 0 for every z g Z. Clearly, G g L Ž Z, y . and TG s L g Lq Ž Z, W ., since y 0 g Ker T. Moreover we have that
w Ž T , TG . s w Ž T , L . s inf Tf Ž x . q Lg Ž x . : x g D 4 s inf Tf Ž x . q L g Ž x . y b : x g D 4 q L Ž b . , and then it follows from Ž10.1. that w ŽT, TG . s Tf Ž x 0 . q LŽ b .. Therefore, TG g GT and
c Ž T , TG . s Tˆy1 w Ž T , TG . s Tˆy1 Tf Ž x 0 . q Tˆy1 L Ž b . , and it follows from Ž10.3. that GŽ b . s Tˆy1 LŽ b . y y 0 and
c Ž T , TG . s Tˆy1 Tf Ž x 0 . q G Ž b . q y 0 , so it results from Ž10.2. that
c Ž T , TG . y G Ž b . s Tˆy1 Tf Ž x 0 . q y 0 s f Ž x0 . . Let us assume now that 10.2 holds and let be L s TG g Lq Ž Z, W .. Then since f Ž x 0 . s c ŽT, TG . y GŽ b ., we have that Tf Ž x 0 . s w Ž T , TG . y L Ž b . s inf Tf Ž x . y Lg Ž x . : x g D 4 y L Ž b . s inf Tf Ž x . y L g Ž x . y b : x g D 4 and L is a Lagrangian T-multiplier for the program Ž1 gy b .. SENSITIVITY ANALYSIS LEMMA 11. Let V be an open subset of Z, M a Banach space, and H : V ª L Ž Z, M . a Frechet differentiable function Ž L Ž Z, M . denotes as ´ usual the space of the linear and continuous mappings from Z into M, endowed with the topology of the uniform con¨ ergence on the bounded subsets
BALBAS GUERRA ´ AND JIMENEZ ´
652
of Z .. If F: V ª M is such that F Ž b . s Hb Ž b . with Hb s H Ž b ., for e¨ ery b g V, then F is Frechet differentiable at V and ´ F9 Ž b, z . s Hb Ž z . q H 9 Ž b, z . Ž b . for e¨ ery z g Z and e¨ ery b g V.3 Proof. In fact, if b g V and z g Z then we have that 1 5 z5
F Ž b q z . y F Ž b . y Hb Ž z . y H Ž b, z . Ž b . s F
1 5 z5 1 5 z5
Hbq z Ž b q z . y Hb Ž b . y Hb Ž z . y H Ž b, z . Ž b . Hbq z Ž b . y Hb Ž b . y H 9 Ž b, z . Ž b . q Hbqz Ž z . y Hb Ž z .
from where the result follows immediately since the function H is Frechet ´ differentiable at b g V and, therefore, it is also continuous at b g V. We are now able to prove a rather surprising result. In general, in a multiobjective convex program, the sensitivity of the optimum depends not only on the value of the dual solution but also on the differential of this dual solution and more concretely, on the projection of such a derivative onto Ker T. In the scalar programming cases Ži.e., Y s R., such a projection is clearly null and therefore, the next Theorem 12 is a generalization of the corresponding happening in the scalar programming. As we will see later on, in the particular case of the linear programming the differential of the dual solution is in Ker T Žand therefore, it coincides with its projection onto Ker T .. THEOREM 12. Let W be a Banach lattice, V an open subset of Z, T g T , x b g Dgyb a T-optimal solution of the program Ž1 gyb ., and G b g L Ž Z, Y . an operator such that ŽT, G b . is an optimal solution of the program Ž3 gyb . associated with x b , for e¨ ery b g V. If the function G : V ª L Ž Z, Y . defined by G Ž b . s G b for e¨ ery b g V, is Frechet differentiable Ž at V . then the ´ function F: V ª Y such that F Ž b . s f Ž x b . for e¨ ery b g V, is also Frechet ´ differentiable Ž at V . and moreo¨ er, the equality F9 Ž b, z . s yGb Ž z . y K Ž b, z . holds for e¨ ery z g Z and b g V, where K Ž b, z . denotes the projection of G 9Ž b, z .Ž b . onto Ker T, for e¨ ery z g Z and b g V. 3
F9Ž b, z . denotes the image of z g Z by the Frechet differential of the function F at ´ b g V. A similar notation is followed in the other cases.
,
653
SENSITIVITY IN CONVEX PROGRAMMING
Proof. For every b g V let L b s TG b g GT . It follows from the proof of Theorem 10 that L b is a Lagrangian T-multiplier for the program Ž1 gy b .. Consider now the functions w *: V ª W and c *: V ª Y such that w *Ž b . s w ŽT, L b . and c *Ž b . s c ŽT, TG b . for every b g V. Since ŽT, G b . and x 0 are associated solutions we have that F Ž b . s f Ž x b . s c Ž T , TG b . y G b Ž b .
Ž 12.1.
and Tf Ž x b . s w Ž T , L b . y L b Ž b . for every b g V, and it results from Theorem 5 and Lemma 11 that yL b Ž z . s Ž w * . 9 Ž b, z . y L b Ž z . y L 9 Ž b, z . Ž b .
Ž 12.2.
for every z g Z and b g V, where L : V ª L Ž Z, V . is the function defined by L Ž b . s L b for every b g V. Clearly, the equality Ž12.2. is equivalent to
Ž w * . 9 Ž b, z . s L 9 Ž b, z . Ž b .
Ž 12.3.
for every z g Z and b g V, and since w * s Tc *, it follows from Ž12.3. that T Ž c * . 9 Ž b, z . s L 9 Ž b, z . Ž b . for every z g Z and b g V. Moreover, since L Ž b . s T G Ž b . for every b g V, it is immediately deduced that L 9 Ž b, z . Ž b . s T G 9 Ž b, z . Ž b . , T Ž c * . 9 Ž b, z . s T G 9 Ž b, z . Ž b . and
Ž c * . 9 Ž b, z . s Tˆy1 T G 9 Ž b, z . Ž b . for every z g Z and b g V. Finally, from Ž12.1. and Lemma 11 it results that F9 Ž b, z . s yGb Ž z . y G 9 Ž b, z . Ž b . q Tˆy1 T G 9 Ž b, z . Ž b . s yGb Ž z . y K Ž b, z . for every z g Z and b g V. EXAMPLE. To illustrate easily the result stated in Theorem 12 let us consider the program Ž1 gy b . with b g Žy`, 0., X s Y s R 2 , Z s W s R, f Ž x 1 , x 2 . s Ž x 12 , 2 x 22 ., and g Ž x 1 , x 2 . s T Ž x 1 , x 2 . s x 1 q x 2 . Solving in the
654
BALBAS GUERRA ´ AND JIMENEZ ´
usual way the program Min x 12 q 2 x 22 x1 q x 2 F b
5
the solution x b s Ž 23 b, 13 b . and the Lagrangian T-multiplier L b s y 43 b are obtained. Therefore, the function F: Žy`, 0. ª R 2 Žof Theorem 12. is defined by F Ž b . s f Ž 23 b, 13 b . s Ž 49 b 2 , 29 b 2 . and then F9 Ž b . s Ž 89 b, 49 b . . On the other hand, Ker T is the linear space generated by the vector Ž1, y1. and so ŽKer T . will be the linear subspace generated by Ž1, 1.. Following the construction made in the proof of Theorem 10, we obtain Tˆy1 w L b x s Ž y 23 b, y 23 b . and y b s f Ž x b . y Tˆy1 Tf Ž x b . s Ž 49 b 2 , 29 b 2 . y Tˆy1 T Ž 49 b 2 , 29 b 2 . s Ž 49 b 2 , 29 b 2 . y Tˆy1 Ž 69 b 2 . s Ž 49 b 2 , 29 b 2 . y Ž 39 b 2 , 39 b 2 . s Ž 19 b 2 , y 19 b 2 . . Taking now zXb Ž z . s Ž1rb . z for every z g R, we have that G b s Tˆy1 w L b x y
1 b
y b s Ž y 79 b, y 59 b . .
Therefore, since the projection onto Ker T is the function p : R 2 ª Ker T such that
p Ž x1 , x 2 . s
ž
x1 y x 2 2
,y
x1 y x 2 2
/
SENSITIVITY IN CONVEX PROGRAMMING
655
for every Ž x 1 , x 2 . g R 2 , with the notations of Theorem 12, we have that K Ž b . s p G 9Ž b . Ž b . s y
ž
b b , 9 9
/
and therefore, yGb y K Ž b . s Ž 89 b, 49 b . and yGb y K Ž b . s F9Ž b ., as Theorem 12 states. DUALITY AND SENSITIVITY FOR LINEAR PROGRAMS Let us now treat the important particular case of linear programs. For them, the dual program has a much simpler formulation than that given by Definition 6 Žas we will see in Theorem 14.. Of special interest is Theorem 15 which is analogous to Theorem 12, but which states that the term G 9Ž b, z .Ž b . belongs now to Ker T and, therefore, it coincides with its projection. Suppose henceforth that D is a Žnon-necessarily pointed. convex cone of X and that f g L Ž X, Y . and g g L Ž X, Z .. Then the spaces L Ž X, Y . and L Ž X, Z . will be assumed to be ordered by its natural cones Lq Ž X , Y . s j g L Ž X , Y . : j Ž x . g Yq for every x g D 4 , Lq Ž X , Z . s j g L Ž X , Z . : j Ž x . g Zq for every x g D 4 and the same will be for L Ž X, W .. LEMMA 13. Under the already established notations, if T g T then GT s L g Lq Ž Z, W . : Lg G yTf 4
4
and w ŽT, L. s 0 for e¨ ery L g GT . Proof. Let L g Lq Ž Z, W .. If Lg G yTf, then Lg Ž x . q Tf Ž x . g Wq for every x g D and, therefore, Tf Ž x . q Lg Ž x .: x g D4 ; Wq is orderbounded from below. Moreover, since 0 g D we have that
w Ž T , L . s inf Tf Ž x . q Lg Ž x . : x g D 4 s 0. If L g GT then Tf Ž x . q Lg Ž x .: x g D4 is an order-bounded from below subset of W and, therefore, w9w TF Ž x . q Lg Ž x .x: x g D4 is a bounded 4
Lg G yTf means as usual that Lg Ž x . q Tf Ž x . g Wq for every x g D.
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BALBAS GUERRA ´ AND JIMENEZ ´
X X from below subset of R for every w9 g Wq , where Wq denotes as usual the dual cone of Wq in the dual space W9 of W, and therefore, we have that
w9 TF Ž x . q Lg Ž x . G 0
Ž 13.1.
X X for every w9 g Wq and x g D Žsince if there exists wX0 g Wq and x 0 g D X such that w 0 w TF Ž x 0 . q Lg Ž x 0 .x - 0 then
lim wX0 Tf Ž a x 0 . q Lg Ž a x 0 . s y`,
aªq`
which is a contradiction.. T Now since Wq / B, it follows from Ž13.1. that Tf Ž x . q Lg Ž x . g Wq for every x g D Žsee w12x.. THEOREM 14. The dual program of Ž1 gy b . Ž under the present assumption of Ž1 gy b . being a linear program. is
¦
Max y G Ž b . TgT G g L Ž Z, Y . . TGg G yTf TG g Lq Ž Z, W .
¥
Ž 4gy b .
§
Proof. This follows immediately from Definition 6 and Lemma 13. THEOREM 15. Let W be a Banach lattice, T g T , and an open subset V of Z such that for e¨ ery b g V there exists x b g V and G b g L Ž Z, Y . such that x b is a T-optimal solution of Ž1 gyb ., ŽT, G b . is an optimal solution of Ž4gyb ., and x b and ŽT, G b . are associated solutions. Suppose that the mappings G : V ª L Ž X, Y . such that G Ž b . s G b for e¨ ery b g V, is Frechet differen´ tiable Ž at V . and let F: V ª Y be the function defined by F Ž b . s f Ž x b . for e¨ ery b g V. Then F is Frechet differentiable Ž at V . and the equalities ´ F9 Ž b, z . s yGb Ž z . y G 9 Ž b, z . Ž b .
Ž 15.1.
T G 9 Ž b, z . Ž b . s 0
Ž 15.2.
and
hold for e¨ ery z g Z and b g V. Proof. It follows from Theorem 12 that the function F is Frechet ´ differentiable Žat V . and that F9 Ž b, z . s yGb Ž z . y K Ž b, z .
Ž 15.3.
SENSITIVITY IN CONVEX PROGRAMMING
657
holds for every z g Z and b g V, where K Ž b, z . is the projection of G 9Ž b, z .Ž b . onto Ker T. Moreover, since x b and ŽT, G b . are associated solutions for every b g V, it results from Theorem 14 that F Ž b . s f Ž x b . s yGb Ž b . for every b g V, and, therefore, Ž15.1. follows from Lemma 11. Moreover, it is immediately deduced from Ž15.1. and Ž15.3. that G 9Ž b, z .Ž b . s K Ž b, z . for every z g Z and b g V and, therefore, Ž15.2. is verified.
CONCLUSIONS The dual program stated here for convex multiobjective programming extends the dual programs given in w4, 13x Žfor linear multiobjective programming. and w17x Žfor convex scalar programming.. Theorem 10 proves the existence of associated solutions which are introduced in Definition 9 and Theorems 12 establishes the surprising fact that, in the multiobjective case, the sensitivity of the program is measured by the dual solution plus the derivative of this solution or a projection of such a derivative, pointing out that this result is a generalization of the scalar case. As an important particular case linear programs are also studied since for them the results have a simpler formulation. The theory developed here in the context of Banach spaces is quite general and it may be applied in many particular situations like static, dynamic, or semi-infinite programs.
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