Bilevel Optimization of the Kantorovich Problem and its Quadratic Regularization Part II: Convergence Analysis

TL;DR

This paper analyzes the convergence of solutions in a bilevel optimization framework involving the Kantorovich optimal transportation problem, using quadratic regularization to address non-smoothness, with a focus on weak-* convergence.

Contribution

It provides a convergence analysis of the quadratic regularization approach for bilevel Kantorovich problems, extending previous existence results to solution convergence.

Findings

Established weak-* convergence of regularized solutions to original problem solutions

Demonstrated the effectiveness of quadratic regularization in handling non-smoothness

Extended the theoretical framework for bilevel optimal transportation problems

Abstract

This paper is concerned with an optimization problem that is constrained by the Kantorovich optimal transportation problem. This bilevel optimization problem can be reformulated as a mathematical problem with complementarity constraints in the space of regular Borel measures. Because of the non-smoothness induced by the complementarity relations, problems of this type are frequently regularized. Here we apply a quadratic regularization of the Kantorovich problem. As the title indicates, this is the second part in a series of three papers. While the existence of optimal solutions to both the bilevel Kantorovich problem and its regularized counterpart were shown in the first part, this paper deals with the (weak-*) convergence of solutions to the regularized bilevel problem to solutions of the original bilevel Kantorovich problem.