SCALARIZATION APPROACHES FOR GENERALIZED VECTOR VARIATIONAL INEQUALITIES

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1 Nonlinear Analysis Forum 12(1), pp , 2007 SCALARIZATION APPROACHES FOR GENERALIZED VECTOR VARIATIONAL INEQUALITIES Zhi-bin Liu, Nan-jing Huang and Byung-Soo Lee Department of Applied Mathematics Southwest Petroleum University Chengdu, Sichuan , P. R. China liuzhibinswpi@vip.sina.com Department of Mathematics Sichuan University Chengdu, Sichuan , P. R. China nanjinghuang@hotmail.com Department of Mathematics Kyungsung University Busan , Korean bslee@ks.ac.kr Abstract. In this paper, by using the scalarization approach of Konnov, four kinds of generalized scalar variational inequalities are introduced for studying generalized vector variational inequalities with set-valued mappings. Some relationships between these variational inequalities are then investigated under suitable conditions. 1. Introduction and Preliminaries The notion of vector variational inequality (for short, VVI) was introduced first by Giannessi [4] in finite-dimensional spaces. Since then, VVI have been investigated extensively by many authors (see, for example, [1, 2, 3, 5, 6, 8, 10, 11, 12, 13, 14, 15] and the references therein). Let Y be a real Banach space. A nonempty subset P of Y is said to be a cone if λp P for all λ>0. P is said to be a convex cone if P is a cone and P + P = P. P is called a pointed cone if P is a cone and P { P } = {0}. An ordered Banach space (Y,P) is a real Banach space Y with an order defined by a closed, convex and pointed cone P Y with apex at the origin, in the form of x y x y P, x, y Y, Received Jun Mathematics Subject Classification: 49J40, 69K10, 90C29. Key words and phrases: Generalized vector variational inequality, set-valued mapping, scalarization approach. This work was supported by the National Natural Science Foundation of China ( ), the Specialized Research Fund for the Doctoral Program of Higher Education ( ) and the Joint Research Fund (07H479).

2 120 Zhi-bin Liu, Nan-jing Huang and Byung-Soo Lee and x y x y P, x, y Y. If the interior of P, say intp, is nonempty, then a weak order in X is also defined by y<x x y intp, x, y Y, and y x x y intp, x, y Y. Let X be a real Banach space and X its dual space. Let K be a nonempty, closed and convex set of X. Denote by L(X, Y ) the space of all the continuous linear mappings from X to Y,andby l, x the value of l L(X, Y )atx X. Let T : K 2 L(X,Y ) be a set-valued mapping, and f : K Y a mapping. In this paper, we consider the following vector variational inequalities: Generalized Stampacchia Vector Variational Inequality (for short, GSVVI) is the problem of finding x K such that t T (x ): t,y x + f(y) f(x ) 0, y K; Generalized Vector Variational Inequality (for short, GVVI) is the problem of finding x K such that y K, t T (x ): t,y x + f(y) f(x ) 0. We denote by S GSV and S GV the solution sets of GSVVI and GVVI, respectively. Clearly, S GSV S GV. If f(x) u for all x K, then GSVVI and GVVI reduce to corresponding set-valued vector variational inequalities (for short, SVVI), which have been investigated by many authors (see, for example, [5, 6, 8, 15]). By using a scalarization approach, Konnov [8] converted SVVI into equivalent scalar variational inequalities. Inspired by the work of Konnov, in this paper, we introduce four kinds of generalized scalar variational inequalities for studying GSVVI and GVVI, and then we study some relationships between these problems under suitable conditions. and 2. Scalarization Approaches for GSVVI and GVVI In this section, we set Y = R n, P = R n + = {λ =(λ 1,,λ n ) R n : λ i 0,,,n}, intp =intr n + = {λ =(λ 1,,λ n ) R n : λ i > 0,,,n}, T (x) = n T i (x), where T i : K 2 X. We define a set-valued mapping T 0 : K 2 X as follows: ( n ) T 0 (x) = conv T i (x),

3 Scalarization approaches for generalized vector variational inequalities 121 where the conv{t i (x)},,n denotes the convex hull of n T i (x). Let f(x) =(f 1 (x),,f n (x)) for any x K, and B = {λ =(λ 1,,λ n ) R n + : λ i =1}. It is obvious that B is nonempty compact and convex subsets of R n. We consider the following four kinds of scalar variational inequalities: Generalized Stampacchia Variational Inequality 1 (for short, GSVI 1 ): find x K, such that e T 0 (x ): e,y x + λ, f(y) f(x ) 0, y K, λ B; Generalized Variational Inequality (for short, GVI 1 ): find x K, such that y K, e T 0 (x ): e,y x + λ, f(y) f(x ) 0, λ B; Generalized Stampacchia Variational Inequality 2 (for short, GSVI 2 ): find x K, such that e T 0 (x ), λ B: e,y x + λ, f(y) f(x ) 0, y K; Generalized Variational Inequality 2 (for short, GVI 2 ): find x K, such that y K, e T 0 (x ), λ B: e,y x + λ, f(y) f(x ) 0. We denote by SGS 1, S1 G, S2 GS and S2 G the solution set of GSVI1, GVI 1,GSVI 2 and GVI 2, respectively. The following result follows immediately from the above definitions. Theorem 2.1. Then the following conclusions hold: (i) S 1 GS S1 G ; (ii) S 2 GS S2 G ; (iii) S 1 GS S2 GS ; (iv) S 1 G S2 G. Lemma 2.1. (see [7]). Let A be a nonempty convex set in a vector space and let B be a nonempty compact convex set in a Hausdorff topological vector space. Suppose that g is a real-valued function on A B such that for each fixed a A, g(a, ) is lower semicontinuous and convex on B, and for each fixed b B, g(,b) is concave on A. Then min g(a, b) = sup min g(a, b). sup b B a A a A b B Theorem 2.2. Assume that, for each x K, T i (x) is nonempty, convex and weakly* compact and f i is convex for each i =1,,n. Then one has S 2 GS = S2 G. Proof. From Theorem 2.1, it suffices to prove SG 2 S2 GS holds. Since T i(x), i = 1,,n, are nonempty, convex and weakly* compact and B is nonempty compact and convex, so are T 0 (x) and T 0 (x) B. Let x SG 2. Then x K, for each y K, there exist e T 0 (x ) and λ =(λ 1,,λ n ) Bsuch that e,y x + λ i (f i (y) f i (x )) = e,y x + λ, f(y) f(x )

4 122 Zhi-bin Liu, Nan-jing Huang and Byung-Soo Lee Set 0. g(a, b, α) = b, x a + α, f(x ) f(a) = b, x a + α i (f i (x ) f i (a)), A = K and B = T 0 (x ) B, where α =(α 1,,α n ) B. Then it follows that sup min g(a, b, α) = sup min { b, (b,α) B (b,α) T 0(x ) B x a + α i (f i (x ) f i (a))} 0. a A a K It is clear that, for each fixed a A, g(a,, ) is continuous (with respect to the second argument in the weak* topology of X, and with respect to the third one in the norm topology of R n ) and convex on B, and for each fixed (b, α) B, g(,b,α) is concave on A since f i is convex for each i =1,,n. By Lemma 2.1, we have min sup (b,α) T 0(x ) B a K sup (b,α) B a A = min = sup 0. min a A (b,α) B g(a, b, α) g(a, b, α) Thus, there are h T 0 (x ) and δ Bsuch that or equivalently, { b, x a + α, f(x ) f(a) } h,x y + δ, f(x ) f(y) 0, y K, h,y x + δ, f(y) f(x ) 0, y K, and so x SGS 2. This completes the proof. The following conclusion follows immediately from Theorems 2.1 and 2.2. Corollary 2.1. Assume that, for each x K, T i (x) is nonempty, convex and weakly* compact and f i is convex for each i =1,,n. Then S 1 GS S 1 G S 2 GS = S 2 G. Now, we characterize the solution sets of GSVVI and GVVI via GSVI and GVI. Theorem 2.3. Assume that, for each x K, T i (x) is a nonempty, convex and weakly* compact and f i is convex for each i =1,,n. Then the following assertions hold: (i) S GSV S GV SGS 2 = S2 G. (ii) If, for each x K, T 0 (x) = n T i (x), then SGS 1 S GSV and SG 1 S GV.

5 Scalarization approaches for generalized vector variational inequalities 123 Proof. (i) From Corollary 2.1, we only need to prove S GV SG 2 holds. Let x S GV. Then for each y K, we have t,y x + f(y) f(x ) 0 for some t T (x ), i.e., for some i 0 {1,,n}, there is t i 0 T i0 (x ) such that t i 0,y x + f i0 (y) f i0 (x ) 0. Set e = t i 0 and γ =(γ 1,,γ n ) with γ i0 = 1 and γ j = 0 for each j i 0. Then, e T 0 (x ), γ B, and consequently, e,y x + γ,f(y) f(x ) 0, which implies that x SG 2. (ii) Assume that, for each x K, T 0 (x) = n T i (x). Let x SGS 1. Then x K, such that e T 0 (x ): e,y x + λ, f(y) f(x ) 0, y K, λ B. Since for each x K, T 0 (x) = n T i (x), there is i 0 {1,,n} such that e T i0 (x ). Set λ =(λ 1,,λ n ) with λ i0 = 1 and λ j = 0 for each j i 0. From the above inequality, we have e,y x + f i0 (y) f i0 (x ) 0, y K, Choose arbitrary elements t j T j (x ) for each j i 0, set t i 0 = e and t = (t 1,,t n ) T (x ). It follows that t,y x + f(y) f(x ) 0, y K, which implies that x S GSV. Similarly, we can prove inclusion SG 1 S GV. This completes the proof. References [1] G. Y. Chen, Existence of solutions for a vector variational inequality: an extension of Hartmann-Stampacchia theorem, J. Optim. Theory Appl. 74 (1992), [2] G. Y. Chen, X. X. Huang and X. Q. Yang, Vector Optimization: Set-Valued and Variational Analysis, Springer-Verlag, Berlin, Heidelberg, [3] Y. P. Fang and N. J. Huang, Variational-like inequalities with generalized monotone mappings in Banach spaces, J. Optim. Theory and Appl. 118 (2003), [4] F. Giannessi, Theorem of alternative, quadratic programs, and complementarity problems, In Variational Inequality and Complementarity Problems, (Edited by R. W. Cottle, F. Giannessi, and J. L. Lions), John Wiley and Sons, Chichester, England, pp , (1980). [5] F. Giannessi (ed.), Vector Variational Inequalities and Vector Equilibrium, Kluwer Academic Publishers, Dordrecht, Boston, London, (2000). [6] N. Hadjisavvas and S. Schaible, Quasimonotonicity and pesudomonotonicity in variational inequalities and equilibrium problems, InGeneralized Convexity, Generalized Monotonicity, (Edited by J. P. Crouzeix, J. E. Martinez-Legaz and M. Volle), Kluwer, Academic Publishers, Dordrecht, pp , (1998). [7] H. Kneser, Sur un thèoréme fondamental de la théorie des jeux, C. R. Acad. Sci. Paris 234 (1952), [8] I. V. Konnov, A scalarization approach for vector variational inequalities with applications, J. Global Optim. 32 (2005),

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