Approximations of Rockafellians, Lagrangians, and Dual Functions

TL;DR

This paper explores how solutions from Rockafellian-based approximations of optimization problems can be more stable and converge to true solutions, especially in nonconvex settings, using epi-convergence analysis.

Contribution

It introduces a novel stability analysis framework for approximate solutions using Rockafellians and epi-convergence, extending beyond local minimizers.

Findings

Solutions from Rockafellian-based problems converge to true solutions under certain conditions.

The paper quantifies convergence rates, often resulting in Lipschitz stability.

Provides a rigorous alternative to classical local stability analysis.

Abstract

Solutions of an optimization problem are sensitive to changes caused by approximations or parametric perturbations, especially in the nonconvex setting. This paper shows that solutions of substitute problems, constructed from Rockafellian functions, can be less sensitive to such changes. Unlike classical stability analysis focused on local changes around (local) minimizers, we employ epi-convergence to examine whether approximating or perturbed problems suitably approach an actual (unperturbed) problem globally. \redrevvv{We demonstrate that solutions derived from the Rockafellian-based substitute problems converge to solutions of the actual optimization problem under suitable conditions, providing a rigorous alternative to potentially unstable direct approximations.} We quantify the rates of convergence that often lead to Lipschitz-kind stability properties for the substitute problems.