Abstract
In this paper, vector complementarity problems are introduced as weak versions of vector variational inequalities in ordered Banach spaces. New dual cones are introduced and proved to be closed. In the sense of efficient point, we prove that the minimal element problem is solvable if a vector variational inequality is solvable; we also prove that any solution of a strong vector variational inequality or positive vector complementarity problem is a solution of the minimal element problem.
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Cryer, C. W., andDempster, M. A. H.,Equivalence of Linear Complementarity Problems and Linear Programs in Vector Lattice Hilbert Spaces, SIAM Journal on Control and Optimization, Vol. 18, pp. 76–90, 1980.
Riddell, R. C.,Equivalence of Nonlinear Complementarity Problems and Least-Element Problems in Banach Lattices, Mathematics of Operations Research, Vol. 6, pp. 462–474, 1981.
Giannessi, F., andNicolucci, F.,Connections between Nonlinear and Integer Programming Problems, Symposia Mathematica, Vol. 19, pp. 162–176, 1976.
Giannessi, F.,Theorems of Alternative, Quadratic Programs, and Complementarity Problems, Variational Inequalities and Complementarity Problems, Edited by R. W. Cottle, F. Giannessi, and J. L. Lions, Wiley, New York, New York, pp. 151–186, 1980.
Chen, G. Y. andChen, G. M.,Vector Variational Inequality and Vector Optimization, Lecture Notes in Economics and Mathematics Systems, Springer-Verlag, New York, New York, Vol. 285, pp. 408–416, 1986.
Chen, G. Y., andYang, X. Q.,The Complementarity Problems and Their Equivalences with the Weak Minimal Element in Ordered Spaces, Journal of Mathematical Analysis and Applications, Vol. 153, pp. 136–158, 1990.
Chen, G. Y.,Existence of Solutions for a Vector Variational Inequality: An Extension of the Hartman-Stampacchia Theorem, Journal of Optimization Theory and Applications, Vol. 74, pp. 445–456, 1992.
Jahn, J.,Existence Theorems in Vector Optimization, Journal of Optimization Theory and Applications, Vol. 50, pp. 397–406, 1986.
Jameson, G.,Ordered Linear Spaces, Lecture Notes in Mathematics, Springer-Verlag, New York, New York, Vol. 141, 1970.
Borwein, J. M.,Generalized Linear Complementarity Problems Treated without Fixed-Point Theory, Journal of Optimization Theory and Applications, Vol. 22, pp. 343–356, 1984.
Kelley, J. L., andNamioka, I.,Linear Topological Spaces, Springer-Verlag, New York, New York, 1976.
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Communicated by F. Giannessi
This work was done while the author was with the Chongqing Institute of Architecture and Engineering, Chongqing, P. R. China.
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Yang, X.Q. Vector complementarity and minimal element problems. J Optim Theory Appl 77, 483–495 (1993). https://doi.org/10.1007/BF00940446
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DOI: https://doi.org/10.1007/BF00940446