Bilevel Optimization of the Kantorovich problem and its quadratic regularization Part:II Convergence Analysis
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Date
2022-11
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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.
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Keywords
optimal transport, Kantorovich problem, bilevel optimization, quadratic regularization