A quadratic regularization of optimal transport problems and its application to bilevel optimization
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Date
2023
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Abstract
This thesis consists of two parts, in each of which a quadratic regularization
is applied to an optimal transport problem and its effect on a prototypical
bilevel optimization problem is investigated.
In the first part, we use the mentioned quadratic regularization in combination
with a smoothing of the marginals to improve certain properties of the
well-known Kantorovich problem, which is a linear optimization problem defined
on infinite-dimensional spaces. In this way we obtain, for example, the
uniqueness of the optimal solution and an associated optimality system containing
(non-unique) dual variables. We then use these improved properties
of the problem to regularize a bilevel optimization problem whose constraints
require to solve the Kantorovich problem. We then show that the regularized
bilevel problem has a solution and that we can, under certain conditions, approximate
solutions of the non-regularized bilevel problem by solutions of the
regularized one. We conclude the first part with a brief overview of possible
applications of this regularization approach.
In the second part, we apply the same regularization approach to the also
well-known Hitchcock problem, which we introduce as a finite-dimensional
special case of the Kantorovich problem. Due to the structure of this problem,
however, we can dispense with the additional smoothing of the boundary
conditions. Similar to the first part, we regularize a bilevel problem whose
constraints require the solution of the Hitchcock problem. We again show
the existence of solutions to the regularized bilevel problem and that we can
use this to approximate solutions to the non-regularized bilevel problem, in
certain cases. By introducing a further regularization of the Lagrangian dual
problem, we enforce the uniqueness of the dual variables from the optimality
system. This enables us to calculate derivatives of the marginal-to-transportplan
mapping and, in turn, to establish an implicit programming approach for
the solution of the regularized bilevel problem. To conclude the second part,
we test our findings numerically by means of an transportation identification
problem.
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Keywords
Optimal transport, Kantorovich problem, Bilevel optimization, Quadratic regularization