Abstract
We study first- and second-order necessary and sufficient optimality conditions for approximate (weakly, properly) efficient solutions of multiobjective optimization problems. Here, tangent cone, -normal cone, cones of feasible directions, second-order tangent set, asymptotic second-order cone, and Hadamard upper (lower) directional derivatives are used in the characterizations. The results are first presented in convex cases and then generalized to nonconvex cases by employing local concepts.
Highlights
The investigation of the optimality conditions is one of the most attractive topics of optimization theory
In 4, Aghezzaf and Hachimi obtained second-order optimality conditions by means of a second-order tangent set which can be considered an extension of the tangent cone; Cambini et al 5 and Penot 6 introduced a new second-order tangent set called asymptotic second-order cone
Second-order optimality conditions for vector optimization problems have been widely studied by using these second-order tangent sets; see 7–9
Summary
The investigation of the optimality conditions is one of the most attractive topics of optimization theory. These conditions are expressed by means of tangent cone, second-order tangent set and asymptotic second-order set. Some sufficient conditions are given for local weakly approximate efficient solutions by using Hadamard upper lower directional derivatives
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