Epi-convergence in distribution of normal integrands with applications to sets of epsilon-optimal solutions

TL;DR

This paper establishes necessary and sufficient conditions for epi-convergence in distribution of normal integrands, linking it to the convergence of epsilon-optimal solution sets and measurable selections, with implications for stochastic optimization.

Contribution

It introduces a new characterization for distributional convergence of random closed sets and connects epi-convergence of integrands to the convergence of solution sets under various topologies.

Findings

Epi-convergence in distribution of normal integrands implies convergence of epsilon-optimal solution sets.

Under boundedness and uniqueness, convergence holds in the Fell topology.

Measurable selections converge weakly to a Choquet-capacity.

Abstract

We derive necessary and sufficient conditions for epi-convergence in distribution of normal integrands. As a basic tool for the proof a new characterisation for distributional convergence of random closed sets is used. Our approach via the epi-topology allows us to show that, if a net of normal integrands epiconverges in distribution, then the pertaining sets of epsilon-optimal solutions converge in distribution in the underlying hyperspace endowed with the upper-Fell topology. Under some boundedness and uniquenss assumptions the convergence even holds for the Fell topology. Finally, measurable selections converge weakly to a Choquet-capacity.