On the Convergence of Projected Alternating Maximization for Equitable and Optimal Transport

TL;DR

This paper provides the first convergence analysis of the projected alternating maximization algorithm for equitable and optimal transport, introducing a new rounding procedure and an extrapolation-enhanced variant to improve solution quality and computational performance.

Contribution

It offers the first convergence proof for PAM in EOT, introduces a novel primal construction method, and proposes an extrapolation-based variant for better efficiency.

Findings

Convergence of PAM is established for EOT.

A new rounding procedure constructs primal solutions.

Extrapolation improves PAM's numerical performance.

Abstract

This paper studies the equitable and optimal transport (EOT) problem, which has many applications such as fair division problems and optimal transport with multiple agents etc. In the discrete distributions case, the EOT problem can be formulated as a linear program (LP). Since this LP is prohibitively large for general LP solvers, Scetbon \etal \cite{scetbon2021equitable} suggests to perturb the problem by adding an entropy regularization. They proposed a projected alternating maximization algorithm (PAM) to solve the dual of the entropy regularized EOT. In this paper, we provide the first convergence analysis of PAM. A novel rounding procedure is proposed to help construct the primal solution for the original EOT problem. We also propose a variant of PAM by incorporating the extrapolation technique that can numerically improve the performance of PAM. Results in this paper may shed lights on block coordinate (gradient) descent methods for general optimization problems.