The vector complementary problem and its equivalences with the weak minimal element in ordered spaces. (English) Zbl 0712.90083

Let (X,C) and (Y,P) be ordered Banach spaces with a nonempty interior int P. We define the weak dual cone with respect to P as \(C_ p^{w+}=\{\ell \in L(X,Y):\) (\(\ell,x)\nless 0\), all \(x\in C\}\). Let T be a map from X to L(X,Y). The following problem, \[ \text{find \(x\in C,\) such that }(T(x),x)\ngtr 0,\;T(x)\in C_ p^{w+}, \] may be called the vector complementarity problem. We prove the equivalence of the vector complementarity problem, the vector variational inequality, the vector extremum problem, the weak minimal element problem, and the vector unilateral minimization problem in ordered spaces.
Reviewer: Guangya Chen


90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)
49J40 Variational inequalities
90C29 Multi-objective and goal programming
90C48 Programming in abstract spaces
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[1] Benson, H. P., Efficiency and proper Efficiency in vector maximization with respect to cones, J. Math. Anal. Appl., 93, 273-289 (1983) · Zbl 0519.90080
[2] Browder, F., Existence and approximation of solution of nonlinear variational inequality, (Proc. Natl. Acad. Sci. U.S.A., 56 (1966)), 1080-1086 · Zbl 0148.13502
[3] Borwein, J. M., Generalized linear complementary problems treated without fixed point theory, J. Optim. Theory Appl., 43, 343-356 (1984) · Zbl 0518.90087
[4] Chen Guang-Ya and Chen Ging-Minin; Chen Guang-Ya and Chen Ging-Minin
[5] Cottle, R. W.; Pang, J. S., On solving linear complementary problem as linear programs, Math. Programming Stud., 7, 88-107 (1978) · Zbl 0381.90072
[6] Cryer, C. W.; Dempster, A. H., Equivalence of linear complementary problems and linear program in vector lattice Hilbert space, SIAM. J. Control Optim., 18 (1980) · Zbl 0428.90077
[7] Borwein, J. M., Multivalued convexity and optimization: A unified approaches to inequality and equality constraint, Math. Programming, 13, 183-199 (1977) · Zbl 0375.90062
[8] Ding, G.-G., Introduction to Banach Spaces (1984), Sci. Press: Sci. Press China
[9] Ekeland, I., Nonconvex minimization problems, Bull. Amer. Math. Soc., 1, 442-474 (1974)
[10] Elster, K. H.; Nehse, R., Optimality condition for some non-convex problems, (“Optimization Techniques,” Part 2. “Optimization Techniques,” Part 2, Lecture Notes in Control and Information Sci., Vol. 23 (1980), Springer: Springer New York), 1-9 · Zbl 0445.90096
[11] Giannessi, F., (Cottle, R. W.; Giannessi, F.; Lions, J.-L., Theorems of Alternative, Quadratic Programs and Complementary Problems, Variational Inequalities and Complementary Problems (1980), Wiley: Wiley New York), 151-186 · Zbl 0484.90081
[12] Jahn, J., Existence theorem in vector optimization, J. Optim. Theory Appl., 50, 397-406 (1986) · Zbl 0577.90078
[13] Jameson, G., Ordered linear spaces, (Lecture Notes in Math. (1970), Springer-Verlag: Springer-Verlag New York/Berlin) · Zbl 0196.13401
[14] Karamardian, S., Generalized complementary problem, J. Optim. Theory Appl., 8, 161-168 (1971) · Zbl 0218.90052
[15] Lassonde, M., On the use of KKM multifunctions in fixed point theory and related topics, J. Math. Anal. Appl., 97, 151-201 (1983) · Zbl 0527.47037
[16] Riddell, R. C., Equivalence of nonlinear complementary problems and least element problems in Banach lattice, Math. Oper. Res., 6 (1981) · Zbl 0494.90077
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