ε variational inequalities for vector approximation problems

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εVariational inequalities for vectorapproximation problemsE.-CHR. Henkel a & CHR. Tammer aa Department of Mathematics and Informatics , Martin-Luther-University Hulle-Wittenberg , Heideallee 1, Halle, D-06099, GermunyPublished online: 20 Mar 2007.

To cite this article: E.-CHR. Henkel & CHR. Tammer (1996) εVariational inequalities for vector approximationproblems, Optimization: A Journal of Mathematical Programming and Operations Research, 38:1, 11-21, DOI:10.1080/02331939608844233

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.+VARIATIONAL INEQUALITIES FOR VECTOR APPROXIMATION PROBLEMS

E.-CHR. HENKEL and CHR. TAMMER

Martin-Luther-Unicersity Halle-Wittenberg, Department of Mathematics and Infornzatics, Heideallee 1 , 0-06099 Halle, Germany

(Received 19 March 1995; in jinal form I 1 December 1995)

From an Ekeland-type variational principle for vector optimization problems we derive &-variational inequalities for approximate solutions of approximation problems using the subdifferential calculus.

KEY WORDS: Vectorial approximation problems. &-variational inequalities, variational principle.

Mathematics Subject Classification 1991: Primary: 49540; Secondary: 90C29, 65K10.

1. INTRODUCTION

Ekeland's variational principle [8] is a very deep assertion about the existence of an exact solution of a slightly perturbed optimization problem in a neighbourhood of an approximate solution of the original problem. It can be used in order to derive necessary conditions for solutions and approximate solutions of optimization problems. Many authors have published extensions and applications of Ekeland's variational principle as well as equivalent statements.

Extensions of Ekeland's variational principle to vector optimization were given by several authors. Loridan (1984) [20] has presented a vector-valued variational principle for the finite-dimensional case using a scalarization and Ekeland's original result. Further, Nemeth (1986) [23], Khanh (1986) [18], Tammer (1992) [26], [30], Den- tscheva and Helbig (1994) [5] and Isac (1994) [I61 have derived vector-valued variational principles for an objective function, which takes its values in general spaces.

We will consider a general vector-valued approximation problem. For special cases of this problem Jahn (1986) [17], Wanka (1994) [35] and Oettli (1995) [22] derived necessary and sufficient conditions for (weakly or properly) efficient elements. In Tammer (1993) [29], Tammer and Henkel (1994) [13] we have shown &-variational inequalities (or E-Kolmogorov conditions) for approximate solutions of this general approximation problem using a vector-valued variational principle [26] and the directional derivative of the vector-valued norm.

The aim of our paper is to derive necessary conditions for approximate solutions of a general approximation problem using the variational principle introduced by Tammer (1994) [30] and the subdifferential calculus for convex vector-valued functions. So we get a sharper result than in Tammer (1993) [29].

12 E.-CHR. HENKEL AND CHR. TAMMER

Let us assume:

(Vl): (X , . . ) is a Banach space, Y is a linear topological space, K c Y is a cone with kf int K, B c Yis a convex cone with int B # @ such that el B + (K\{O)) c int B.

(V2): f : X + Y i s lower semicontinuous with respect to k0 and B in the sense that M , = {.x~Xlf(x)erkOl - el B) is closed for each r e R and bounded from below, i.e. f [XI cj + B for a certain ~ E Y

In the following we denote the topological interior of a set C by int C, the topological boundary of C by bd C and the topological closure of C by cl C.

If K c Yis a c o y then K*: = {y*~Y*ly*(x)bO V~EK} is called the dual cone for K. Moreover, the sef q-intK": = {y*~Y*ly*(y)>O VyeK\{O)) is called the quasi-interior of the dual cone for K, where Y* is the dual space of linear continuous functionals defined on Y

Now, we consider the following vector optimization problem to determine the efficient point set off [XI with respect to B:

(P): Compute the set Eff( f [X],B), where

It is well known, that the efficient point set may be empty in the general noncom- pact case (see [12], [17], [21]). In order to derive also in this general case certain existence results we introduce in the following definition the set of approximately efficient elements (compare 1251, [27], [33]).

Definition 1: An element f(x,)~flX] is called an approximately efficient point o j AX] with respect to B, kOe int K and E > 0, if

The approximately efJicient point set off [XI with respect to B, k0 and E is denoted by Eff(f [XI, B,,,), where B,,, : = &kO + B.

2. A VARIATIONAL PRINCIPLE

Beginning with the paper of Loridan [20] various authors proved variational principles of Ekeland's type for &-efficient solutions of multicriteria optimization problems by scalarization. Tammer [26] has proved a vector-valued variational principle by using a separation theorem for nonconvex sets and Ekeland's original result. Here we will use the following variational principle which was shown in [30] and which is a sharper result than in [26].

Theorem 1: Assume (VI) and (V2). Then for any &>0, ).>O and any f(x,)~Eff(fiX],B,,,) there exist an element X,EX

&-VARIATIONAL INEQUALITIES

Figure 1

3. APPLICATIONS TO APPROXIMATION PROBLEMS

Location and approximation problems have been studied by many authors from the theoretical as well as the computational point of view (Kuhn [19], Durier and Michelot [7], Idrissi and Lefebvre and Michelot [14], Idrissi and Loridan and Michelot [I51 and many others). Such problems play an important role in optimization theory and many practical problems can be described as location or approximation problems. Besides problems with one objective function several authors investigated even vector-valued location and approximation problems. In some papers they presented theoretical results (duality assertions- Gerth and Pohler [lo], Chr. Tammer and K. Tammer [31], Wanka [34]; necessary optimality conditions-Durier [6], Wanka[35], Oettli [22]) and developed algo- rithms (Wendell and Hurter and Lowe [37], Chalmet and Francis and Gerth and Pohler [lo]).

In this section we will consider a general vectorial control approximation problem and derive necessary conditions for approximate solutions of this problem.

In the whole section we assume additionally to (Vl) that (X, . . ) , ( I : . , ) and (Z,.Il,) are real reflexive Banach spaces. Moreover, we suppose that K c Y is a pointed closed convex cone with int K # @ having the Daniel1 - property, which means that every decreasing net (i.e. i < j implies x,<xi) having a lower bound, converges to its infimum (see Jahn [17]). Further, we suppose that K has a weakly compact base. In order to formulate our vectorial control approximation problem, we will introduce a vector-valued norm (compare Jahn [17]):

1 . 1 : Z + K is called a vector-valued norm if V z,z, ,z,~Z, V RER it holds:

In the following we assume that ( I ( . ( ( ( is continuous. The set of linear continuous mappings from X to Yis denoted by L(X,Y). Suppose that CeL(X,Y), Ai E L(X,Z) and ri 3 0 (i = 1,. . . , n). A* denotes the adjoint operator to A,.

Then we consider for xeX and aieZ (i = I,. . . ,n) the vector-valued function

Now, we will introduce the following vectorial control approximation problem:

(PI): Compute the set Eff(f [XI, K).

Remarks : The problem (PI ) contains the following practically important special cases:

1. Vector-valued optimal control problems of the form (see Benker and Kossert C21):

where HI and H, are Hilbert spaces, AeL(H,,H,), UEH,, U c H , is a nonempty closed convex set and R: denotes the usual ordering cone in R2. Here u denotes the so called control variable, the image z = Au denotes the state variable. 2. Scalar location and approximation problems (Y= R, C r 0):

where 1.1 is a norm in Z. 3. Vectorial approximation and location problems (C = 0, n = 1, cf. Jahn [17],

Gerth and Pohler [lo], Wanka [34], Oettli [22]). 4. Linear vector optimization problems (r, = 0 for all i = 1,. . . ,n). 5. Surrogate problems for linear vector optimization problems with an objective

functionf(x): = C(x) subject to XEX and Ax = a, for which the feasible set is empty.

E-VARIATIONAL INEQUALITIES 15

6. Perturbed linear vector optimization problems. 7. Tykhonov-Regularization for linear vector optimization problems.

In the following theorem we will derive necessary conditions for approximately efficient solutions of Eff( f[X],B,,,) using the subdifferential calculus. Here we use the dual norm 1.1, defined by

x * * : = sup Ix*(x)l. [ x ; [ x - = l

Applying the subdifferential calculus we derive a sharper result than in Theorem 2 in C131.

Firstly, we introduce the following definition of K-comexity (compare Jahn [17] and Luc [21]).

Definition 2: A functionf X + Y is called K-convex on X iffor x, , x , E X , t ~ [ 0 , 1 ] ,

Lemma 1: The function f l ( x ) is K-conaex.

Proof: Using the linearity of the operators C and Ai (i = 1,. . .,n) and the definition of the vector-valued norm we can conclude that for all x , , x , ~ X and t ~ [ 0 , 1 ] it holds

Definition 3: The set

is called the subdiferential of the convex function f : X + Y at x , E X .

Jahn [17], Example 2.22, has shown the structure of the subdifferential of the vector-valued norm I1I.III: Z+ Y

The following results will be used in the proof of our main theorem.

Lemma 2 (Jahn [17]): Let S be a nonempty subset of a partially ordered reflexice Banach space Y with a pointed nontrivial ordering cone K . If the set S + K is concex and has a norrempty topological interior, then for each efJicient element Y E S of the set S there exists a linear functional y * ~ K * \ { 0 } with the property

y*(L;) < y*(y) for all ye Y

Lemma 3 (Jahn [17]): Let (X,/I ./I,) and ( I'll . I y ) be real reflexice Banach spaces, and let K be a concex cone in Y with a \veakly compact base. I f f : X - t Y is a comex mapping which is continuous at some ? E X , then

Now, we formulate our main result.

Theorem 2: Under the assumptions of this section for alzj E > 0, i > O and any approximatelj efficient element f,(xO)€Eff( fl [XI, B,,,) there exist an element x,, a functioizal y*€K*\{O), linear continuous ~nappilzgs M,, E L(Z, Y) with

and arz open set D c Y~vith K\${O) c D, 0 ~ b d D and el D + (K\{O)) c D, such that

Proof: We assumef(xo) E Eff ( fl [X],B,,,,). Under the given assumptions Theorem 1 implies the existence of an element x, with

x') f1(x, )~f l (x0) - jV x0 - xJX kO - clB, f l (x , )~Eff(f l [X],D,,,), where D is an open subset of Y with K\{O) c D, OebdD and clD + (K\{O}) c D,

b') I . Y O - X , X ~ E / ~ - ,

7') fl ,k"~,)~Eff(fl /k" [ X I m where f,,,, (x) : = f,(x) + j. x - x,lx kO.

Because of the K-convexity off, and by assertion 7') it is possible to conclude from Lemma 2 that there exists a functional y*€K*\{O) with

This means that x, minimizes the function

Then the subdifferential calculus of convex functionals (cf. Aubin and Ekeland [I], Corollary 4.3.6) implies that

where E denotes the (closed) unit ball in I.: So it follows immediately that there is a linear continuous functional 1,: X+R belonging to the subdifferential of the scala- rized function y* 0 fl at the point X,EX with

Further, we have the following equation

The rule of sums for subdifferentials yields the relation

Moreover, from Lemma 3 and Corollary 4.3.6 in Aubin and Ekeland [I] we get the following equation

Applying Example 2.22 in Jahn [17] (compare (1)) statement (3) implies

IllzlII - Mi,(z)eK VzeZ ( i = 1 ,..., n).

From (2), (3) and (4) we get the desired inequality

Remarks:

1. It is possible to use the functional w: Y+R

\c(y): = inf {teRIyeclB + tko)

in the proof of Theorem 2 instead of the linear continuous functional J.*EK*',{O). In Tammer [26] it was shown that the functional w has the property ]c(xko) = cc for all ~ E R under the given assumptions. Moreover, under the additional assumption that bd B is a linear subspace of Y this functional is linear (compare Weidner [36]). So we get in assertion ( 7 ) the condition

2. Obviously, if we use a scalarization of the approximation problem (PI) with linear continuous functionals y*eK*\{O) and Ekeland's original result [8] then we can show in the same way as in Theorem 2 that for any 8 > 0, i. > 0 and

any approximate solution x0 E X with y* ( fl ( xo ) ) inf,., y* ( fl ( x ) ) + E, y* E

K*',{O), there exists an element x, and linear continuous mappings M,E L(Z, Y) with

Illzlll - M i , ( z ) ~ K V Z E Z (i = 1 ,..., n), such that

But the assertion ( x ) in Theorem 2 is a sharper results as (x"). 3. The assertion (y) of Theorem 2 is a sharper result as the assertion in Theorem 2

of [13], because the mappings M,(i = 1, ..., n) in Theorem 2 not depend on a direction C E X .

4. Obviously, from ( r ) it follows t h a t f , ( x , ) ~ E f f ( f , [X],K,,,) since K\{O) c D.

In our work [I31 we have considered an extension of Theorem 5.4.3. of Au- bin/Ekeland [I] for real-valued functions to vector-valued functions. Now we want to apply this result to the vector-valued approximation problem (PI ) . Here we will use the following set

domf: = { x ~ X ( 3 y * ~ q - i n t K * with y * ( f ( x ) ) < x).

Theorem 3: Under the assumptions of this section the set of points where.fl is sub- diferentiahle is dense in dom f l , i. e., for each 2 E dom fl there exists a sequence {x , ) , EN, with:

( x ) x,-x, (b) y*( fl (x,)) +J* ( fl (2)) for a linear strictly K-monotone functional y*. (7) Vi= 1 ,..., n ~ M , , E L ( Z , Y ) with M, , (A,(x , ) -a"=IIA,(x , ) -a ' / / ,

lllzlll - M i k ( z ) e K V Z E Z and C + aiA"*Mik&"( f l ( xk ) ) # O V k . ( 5 ) i = 1

Proof: For the proof of the assertions ( x ) and ( P ) we refer to the corresponding relations in our work [13] (Theorem 6). The assertion ( y ) is a consequence of assertion (7) in Theorem 6 in [13].

4. SPECIAL CASES

In the following we will study some practically important special cases of the general approximation problem (PI ) .

Let us now assume, that the space Y is the space of real numbers R. Suppose that C E L ( X , R) , A,€ L ( X , 2) and xi 3 0 (i = I,. . .,n). Then we consider for x e X and a i € Z (i = l,...,n) the following real-valued approximation problem:

If we put c = 0 and A, = I V i = 0,. . ., n we get the special case of the real-valued location problem, i.e.

In the following corollaries we will see that for the special approximation problems (P2) and (P3) the assertions of Theorem 2 and 3 take an easier form.

Corollary 1: We consider the real-calued problein (P2), which is a special case o f ( P I ) if we put k" = 1 and K = B = { X E R lx 3 0 ) .

Then for any E > 0,;" > 0 and any approximate solution xo with f2(x0) < infXGxf2(x) + E

there exist an element x, and linear continuous funtionals E,EZ* with

such that

Corollary 2: We consider the real-valued location problem (P3), which is a special case of ( P I ) i f we additionally put k' = 1 and K = B = { x ~ R l x 2 0) . Then for any E > 0, 3. > 0 and any approximately solution xo with f3(x0) < inf,,, f3(x) + e there exist an element x , and linear continuous functionals ~ ,EZ* with

such that

Corollary 3: Consider the scalar optimization problem (P2), which is a special case of the problem ( P I ) if we additionall): put k" = 1 and K = B = { x ~ R l x 2 0 ) . Then the

assertion ( 7 ) of Theorem 3 takes the following form:

Corollary 4: Consider the scalar location problem (P3), which is a special cuse of the problem ( P I ) i f we additionallj put k c = 1 and K = B = { x ~ R j x 3 0 ) . Then the assertion ( 7 ) of Theorem 3 takes the following form:

( 7 ) : 3 I,,EZ* with lin(xn - ai) = 1 ) xn - a'11, 1 1 l in 1 ) * = 1, (i = 1,. . ., n)

References

[I] Aubin; J. P. and Ekeland, I. (1984) Applied Nonlinear Analysis., John Wiley, New York 121 Benker, H. and Kossert, S. (1982) Remarks on quadratic optimal control problems in Hilbert

spaces. Zeitschrift fur Analysis und ihre Anwendungen 1 (31, 13-21 [3] Borwein, J. (1982) Continuity and differentiability properties of convex operators. Proc. London

Math. Soc., 44. 420-444 [4] Brumelle, S., Duality for multiple objective convex programs. Mathenlatics of Operations Research,

6 (2), 159-1 72 [5] Dentscheva, D. and Helbig, S. (1994) "On variational principles in vector optimization", Lecture on

the 4th Workshop Multicriteria Decision, Holzhau, Germany [6] Durier, R. (1990) O n Pareto Optima, the Fermat-Weber Problem, and Polyhedral Gauges. Mathe-

nzatical Programming, 47, 65-79 [7] Durier, R., Michelot, C. (1985) Geometrical Properties of the Fermat-Weber Problem. European

Journal of Operational Research, 20, 332-343 [8] Ekeland, I (1974) O n the variational principle. J . Math. Anal. Appl. 47, 324-353 [9] Elster, K.-H., Nehse, R., Konjugierte Operatoren und Subdifferentiale. Math. Operationsforsch. Sta-

tist. Ser. optim., 6, 641-657 [lo] Gerth (Tammer), Chr., Pohler, K. (1988) Dualitat und algorithmische Anwendung beim vektoriellen

Standortproblem. Optimization, 19, 491-512 [l I] Gerth (Tammer), Chr. and Weidner, P. (1990) Nonconvex separation theorems and some applica-

tions in vector optimization. Journ. Optimization Theory Applications, 67 (2), 297-320 1121 Gopfert. A. and Nehse, R. (1990) Vektoroptimierung. Theorie, Verfahren und Anwendungen. BSB

B. G. Teubner Verlagsgesellschaft [13] Henkel. E.-Chr., Tammer. Chr (1996) &-Variational Inequalities in Partially Ordered Spaces. Opti-

mization. 36, 105-1 18 [14] Idrissi, H., Lefebvre, O., Michelot, C. (1988) A Primal-Dual Algorithm for a Constrained Fermat-

Weber Problem Involving Mixed Norms. Recherche operationnelle!Opevarions Research, 22, 313- 330

[I51 Idrissi, H., Loridan, P.. Michelot, C. (1988) Approximation of Solutions for Location Problems. Journ. Opt. Theory Appl., 56; 127-143

[16] Isac, G. (1994) A variant of Ekeland's principle for Pareto &-efficiency. In preparation [17] Jahn, J. (1986) Mathematical Vector Optimization in Partially Ordered Spaces. Lang Verlag Frank-

furt, Bern. New York [I81 Khanh, P. Q. (1986) O n Caristi-Kirk's theorem and Ekeland's variational principle for Pareto

extrema. Institute of Mathenzatics, Polish Acadenz), of Sciences, Preprint 357

8-VARIATIONAL INEQUALITIES

[19] Kuhn, H. W. (1973) A Note on Fermat's Problem. Matlze~natical Programming, 4, 98-107 [20] Loridan, P. (1984) &-solutions in vector minimization problems. Journ. Optimization Theorj Applica-

tions, 43 (2), 265-276 [21] Luc. D. T. (1989) Theory of Vector Optimization. Lecture .Votes in ,Mathematics. 319, Springer

Verlag Berlin Heidelberg [22] Oettli, W. (1995) Kolmogorov conditions for vectorial optimization problems. O R Spektrunz, 17,

227-229 [23] Nemeth, A. B. (1986) A Nonconvex Vector Minimization Problem. Nonlin. Anal. Theory, Methods

and Applications, 10 (7). 669-678 [24] Rockafellar, R. T. (1979) Directionally Lipschitzian functions and subdifferential calculus. Proc.

London Math. Soc., 39, 331-355 [25] Staib, T. (1988) On generalizations of Pareto minimality. Journ. Optimization Theorj Applications,

59, 289-306 [26] Tammer, Chr. (1992) A generalization of Ekeland's variational principle. Optimization, 25, 129- 141 [27] Tammer. Chr. (1993) Existence results and necessary conditions for c-efficient elements. In:

Brosowski, Ester, Helbig, Nehse (Eds) Multicriteria Decision-Proceedings of the 14th Meeting of the German Working Group "Mehrkriterielle Entsch.", Lang Verlag Frankfurt:Main Ber, 97-1 10

[28] Tammer, Chr. (1993) Erweiterungen und Anwendungen des Variationsprinzips von Ekeland. ZA,MM, 73, 832-826

[29] Tammer, Chr. (1993) Necessary conditions for approximately efficient solutions of vector approximation. Proceedings of the 2nd Conference on Approximation and Optimization, Havana

[30] Tammer, Chr. (1994) A variational principle and applications for vectorial control approximation problems. To appear: Math. Journ. Unic. Bacau (Romania).

[31] Tammer. Chr.; Tammer, K. (1991) Generalization and Sharpening of Some Duality Ralations for a class of Vector Optimization Problems. Z O R , 35, 249-265

[32] Thierfelder, J., Subdifferentiale und Vektoroptimierung. Wiss Zeitschrift der TH Ilmenau, 37 (3), 89-100

[33] Valyi, I. (1986) On approximate vector optimization. Working Paper 86-7 IIASA Laxenburg [34] Wanka. G. (1991) Duality in Vectorial Control Approximation Problems with Inequality Restric-

tions. Optimization, 22, 755-764 [35] Wanka. G. (1994) Kolmogorov Conditions for Vectorial Approximation Problems. To appear in:

OR Spektrum [36] Weidner, P. (1990) Comparision of six types of separating functionals. In: Sebastian, H.-J.:

Tammer, K. (eds.) System modelling and optimization. Lecture .\'ores in Control and Irformation Sciences. 143, 234-243

[37] Wendell, R. E., Hurter, A. P., Lowe, T. J. (1973) Efficient Points in Location Problems. AIEE Transactions, 9. 238-246

ε variational inequalities for vector approximation problems

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