Isolated minimizers and proper efficiency for C0,1 constrained vector optimization problems

We consider the vector optimization problem min(C) f (x), g(x) is an element of - K, where f:R-n -> R-m and g: R-n -> R-p are C-0,C-1 (i.e. locally Lipschitz) functions and C subset of R-m and K subset of R-p are closed convex cones. We give several notions of solution (efficiency concepts), among them the notion of properly efficient point (p-minimizer) of order k and the notion of isolated minimizer of order k. We show that each isolated minimizer of order k >= 1 is a p-minimizer of order k. The possible reversal of this statement in the case k = 1 is studied through first order necessary and sufficient conditions in terms of Dim derivatives. Observing that the optimality conditions for the constrained problem coincide with those for a suitable unconstrained problem, we introduce sense I solutions (those of the initial constrained problem) and sense II solutions (those of the unconstrained problem). Further, we obtain relations between sense I and sense II isolated minimizers and p-minimizers. (c) 2005 Elsevier Inc. All rights reserved.