Bilevel Optimization of the Kantorovich Problem and its Quadratic Regularization Part I: Existence Results

TL;DR

This paper investigates the existence of solutions in a bilevel optimization framework based on the Kantorovich optimal transportation problem, utilizing quadratic regularization to handle non-smoothness, as a foundational step for subsequent convergence analysis.

Contribution

It establishes the existence of optimal solutions for the bilevel Kantorovich problem and its quadratic regularization, laying groundwork for future convergence studies.

Findings

Existence of solutions for the bilevel Kantorovich problem.

Existence of solutions for the quadratic regularization of the problem.

Foundation for convergence analysis in subsequent work.

Abstract

This paper is concerned with an optimization problem governed by the Kantorovich optimal transportation problem. This gives rise to a bilevel optimization problem, which 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 first part in a series of three papers. It addresses the existence of optimal solutions to the bilevel Kantorovich problem and its quadratic regularization, whereas part II and III are dedicated to the convergence analysis for vanishing regularization.