Stochastic optimization by message passing

TL;DR

This paper presents a message passing algorithm based on the cavity method for solving stochastic optimization problems, including matching and independent set problems, with applications in communication networks.

Contribution

It provides a detailed derivation and implementation of a message passing approach for stochastic optimization, extending previous work to more complex problems.

Findings

The algorithm effectively solves stochastic matching problems.

It outperforms greedy algorithms in the tested scenarios.

Extensions to multi-stage problems are discussed.

Abstract

Most optimization problems in applied sciences realistically involve uncertainty in the parameters defining the cost function, of which only statistical information is known beforehand. In a recent work we introduced a message passing algorithm based on the cavity method of statistical physics to solve the two-stage matching problem with independently distributed stochastic parameters. In this paper we provide an in-depth explanation of the general method and caveats, show the details of the derivation and resulting algorithm for the matching problem and apply it to a stochastic version of the independent set problem, which is a computationally hard and relevant problem in communication networks. We compare the results with some greedy algorithms and briefly discuss the extension to more complicated stochastic multi-stage problems.