Generalized Convexity and Nonsmooth Problems of Vector Optimization

Abstract

In this paper it is shown that every generalized Kuhn-Tucker point of a vector optimization problem involving locally Lipschitz functions is a weakly efficient point if and only if this problem is KT- pseudoinvex in a suitable sense. Under a closedness assumption (in particular, under a regularity condition of the constraint functions) it is pointed out that in this result the notion of generalized Kuhn–Tucker point can be replaced by the usual notion of Kuhn–Tucker point. Some earlier results in (Martin (1985), The essence of invexity, J. Optim. Theory Appl., 47, 65–76. Osuna-Gómez et al., (1999), J. Math. Anal. Appl., 233, 205–220. Osuna-GGómez et al., (1998), J. Optim. Theory Appl., 98, 651–661. Phuong et al., (1995) J. Optim. Theory Appl., 87, 579–594) results are included as special cases of ours. The paper also contains characterizations of HC-invexity and KT- invexity properties which are sufficient conditions for KT- pseudoinvexity property of nonsmooth problems.