An efficient implementable inexact entropic proximal point algorithm for a class of linear programming problems

TL;DR

This paper presents an efficient inexact entropic proximal point algorithm (iEPPA) for large-scale structured linear programming problems, especially effective for optimal transport and tomography applications, with improved stability and convergence.

Contribution

The paper introduces a novel implementable inexact entropic proximal point algorithm with a practical stopping condition and demonstrates its effectiveness for large-scale LP problems including capacity constrained optimal transport.

Findings

iEPPA is efficient and robust for large-scale CMOT problems.

The algorithm avoids numerical instability common in entropic regularization.

Numerical experiments confirm the method's effectiveness and modeling potential.

Abstract

We introduce a class of specially structured linear programming (LP) problems, which has favorable modeling capability for important application problems in different areas such as optimal transport, discrete tomography and economics. To solve these generally large-scale LP problems efficiently, we design an implementable inexact entropic proximal point algorithm (iEPPA) combined with an easy-to-implement dual block coordinate descent method as a subsolver. Unlike existing entropy-type proximal point algorithms, our iEPPA employs a more practically checkable stopping condition for solving the associated subproblems while achieving provable convergence. Moreover, when solving the capacity constrained multi-marginal optimal transport (CMOT) problem (a special case of our LP problem), our iEPPA is able to bypass the underlying numerical instability issues that often appear in the popular entropic regularization approach, since our algorithm does not require the proximal parameter to be very small in order to obtain an accurate approximate solution. Numerous numerical experiments show that our iEPPA is efficient and robust for solving large-scale CMOT problems. The experiments on the discrete tomography problem also highlight the potential modeling power of our model.