Isolated and Proper Efficiencies in Semi-Infinite Vector Optimization Problems

Optimality conditions in convex multiobjective SIP

“…Proof Let α i ≥ 0, i ∈ I, with p i=1 α i = 1, and some β t ≥ 0, t ∈ T ( x), with β t = 0 for finitely many indexes.Due to (7), we can find some…”

Weak sharp efficiency in multiobjective optimization

“…Moreover, we also obtain some sufficient optimality conditions respectively for the local and global weak sharp efficient solutions of (MOP) by employing the approximate projection theorem and some appropriate convexity and affineness conditions. Our results about the weak sharp efficiency for (MOP) are more general and extensive, and the methods used in this paper are different form those in [1][2][3][4].…”

“…Recall from [1][2][3][4] that a pointx ∈ S is said to be a local sharp efficient solution for (MOP) iff there exists a neighborhood U ofx and a real number η > 0 such that…”

“…Subsequently, by means of the Mordukhovich generalized differentiation and the normal cone, Chuong [3] obtained a nonsmooth version of Fermat rule for the local sharp efficient solution of (MOP). For more details, we refer to [4,14].…”

“…Comparing and contrasting the local weak sharp efficient solution, defined in Definition 1.1, and the local sharp efficient solution, defined in [1][2][3][4], it is easy to verify that the local sharp efficient solutionx ∈ S is isolated in the solution set S. While, the local weak sharp efficient solutionx ∈ S may not be isolated in S. Simultaneously, if x ∈ S is a local weak sharp efficient solution for (MOP), then it must be also a local sharp efficient solution wheneverx is isolated in S. Thus, the weak sharp efficiency not only solves the isolation of the sharp efficiency, which is rigorous in real applications, but also provides a more general and easier accessible form for our analysis. Moreover, characterizations on the weak sharp efficiency for (MOP), similar to the scalar case, are very important and useful to study the metric subregularity property and growth condition for vector-valued functions, and the stability analysis for the solution set-valued mapping to parametric vector optimization problems.…”