Characterization of the Nonemptiness and Compactness of Solution Sets in Convex and Nonconvex Vector Optimization
As a consequence of an abstract theorem proved elsewhere, a vector Weierstrass theorem for the existence of a weakly efficient solution without any convexity assumption is established. By using the notion (recently introduced in an earlier paper) of semistrict quasiconvexity for vector functions and assuming additional structure on the space, new existence results encompassing many results appearing in the literature are derived. Also, when the cone defining the preference relation satisfies some mild assumptions (but including the polyhedral and icecream cones), various characterizations for the nonemptiness and compactness of the weakly efficient solution set to convex vector optimization problems are given. Similar results for a class of nonconvex problems on the real line are established as well.
KeywordsNonconvex vector optimization quasiconvex vector functions weakly efficient solutions efficient solutions asymptotic functions asymptotic cones
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- 1.ANSARI, Q.H., Vector Equilibrium Problems and Vector Variational Inequalities, Vector Variational Inequalities and Vector Equilibria, Edited by F. Giannessi, Kluwer Academic Publishers, Dordrecht, Holland, pp. 1–16, 2000.Google Scholar
- 10.LUC, D.T., Theory of Vector Optimization, Lecture Notes in Economics and Mathematical Systems, Springer Verlag, New York, NY, Vol. 319, 1989.Google Scholar
- 15.CAMBINI, R., Generalized Concavity for Bicriteria Functions, Generalized Convexity, Generalized Monotonicity: Recent Results, Edited by J.P. Crouzeix et al., Nonconvex Optimization and Its Applications, Kluwer Academic Publishers, Dordrecht, Holland, Vol. 27, pp. 439–451, 1998.Google Scholar