Estimates of Error Bounds for Some Sets of Efficient Solutions of a Set-Valued Optimization Problem

Abstract

In this paper, we establish some estimates of the global/local error bounds for the sets $S^{\mathrm{Pareto}}_{\bar{y}}$, $S^{\mathrm{W}}_{\le \bar{y}}$ and $S^{\mathrm{W}}$, where $S^{\mathrm{Pareto}}_{\bar{y}}$ is the set of efficient solutions of a unconstrained set-valued optimization problem ($\mathcal {SP}$) corresponding to an efficient value $\bar{y}$ of a unconstrained set-valued optimization problem ($\mathcal {SP}$), $S^{\mathrm{W}}_{\le \bar{y}}$ is the set of weakly efficient solutions of ($\mathcal {SP}$) corresponding to weakly efficient values smaller than a weakly efficient value $\bar{y}$ and $S^{\mathrm{W}}$ is the set of all weakly efficient solutions of ($\mathcal {SP}$). These estimates are expressed in terms of the approximate coderivative, the limiting Fréchet/basic coderivatives and the coderivative of convex analysis. Thus, we establish conditions ensuring the existence of weak sharp minima for ($\mathcal {SP}$). We also extend the concept of the good asymptotic behavior to a convex or cone-convex set-valued map.