Stability in Set-Valued Optimization Problems Using Asymptotic Analysis and Epi-Convergence

We study the stability of set-valued optimization problems by using epi-convergence coupled with asymptotic analysis. To do this, we recall the notion of epi-convergence and introduce the notion of total epi-convergence for set-valued maps. We characterize them by means of epi-limits and horizon epi-limits. We use these epi-limits to study the behavior of vector/set type solutions and level/colevel sets under variations of the whole data. We also introduce several stronger epi-convergence notions and use them to study the stability of solution sets and minimal solution sets. We extend and generalize various results from the literature since we deal with unbounded feasible sets and objective maps. Finally, to illustrate our main results, we apply them to various classes of set-valued maps.