Weak Sharp Minima in Set-Valued Optimization Problems

and Applied Analysis 3 x0, y0 ∈ L StrD F, S resp., x0, y0 ∈ LW MinD F, S , if there exists a neighborhood U of x0 in X such that y0 ∈ StrD F U ∩ S ( resp., y0 ∈ W minD F U ∩ S ) , that is, ∀x ∈ S∩U, F x − y0 ∩ −D\{0} ∅ resp. ∀x ∈ S∩U, F x − y0 ∩ − intD ∅. 2.4 We will say that x0, y0 is a global strict global weak minimizers when U X. The set of all global strict minimizers resp., weak minimizers is denoted by StrD F, S resp., W MinD F, S . Definition 2.3. Let ψ : 0, ∞ → 0, ∞ be a nondecreasing function with the property ψ t 0 ⇔ t 0 such a family of functions is denoted by Ψ and x0 ∈ S. We say that a point pair x0, y0 ∈ GrF ∩ S × Y is a weak ψ-sharp local Pareto minimizer for 2.3 , denoted by x0, y0 ∈ WSL ψ, F, S , if there exists a constant α > 0 and U ∈ N x0 such that F x D ∩ By0, αψ dist x,W ) ∅, ∀x ∈ S ∩U \W, 2.5