Isolated minimizers and proper efficiency for [formula omitted] constrained vector optimization problems

Abstract

We consider the vector optimization problem min C f ( x ) , g ( x ) ∈ − K , where f : R n → R m and g : R n → R p are C 0 , 1 (i.e. locally Lipschitz) functions and C ⊆ R m and K ⊆ 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 Dini 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.