Multistart Algorithm for Identifying All Optima of Nonconvex Stochastic Functions

TL;DR

This paper introduces a multistart algorithm designed to find all local minima in constrained, nonconvex stochastic optimization problems, demonstrating its effectiveness on quantum optimization tasks.

Contribution

The paper presents a novel multistart approach that asymptotically guarantees identification of all local optima in stochastic nonconvex problems under certain conditions.

Findings

Algorithm asymptotically finds all local minima with high probability.

Implementation successfully applied to quantum approximate optimization.

Finitely many local stochastic runs are almost surely sufficient.

Abstract

We propose a multistart algorithm to identify all local minima of a constrained, nonconvex stochastic optimization problem. The algorithm uniformly samples points in the domain and then starts a local stochastic optimization run from any point that is the "probabilistically best" point in its neighborhood. Under certain conditions, our algorithm is shown to asymptotically identify all local optima with high probability; this holds even though our algorithm is shown to almost surely start only finitely many local stochastic optimization runs. We demonstrate the performance of an implementation of our algorithm on nonconvex stochastic optimization problems, including identifying optimal variational parameters for the quantum approximate optimization algorithm.