Large-scale nonconvex optimization: randomization, gap estimation, and numerical resolution

TL;DR

This paper studies large-scale nonconvex optimization with an aggregative term, proposing a randomized relaxation, a stochastic method for approximate solutions, and a stochastic Frank-Wolfe algorithm with convergence guarantees, validated by numerical experiments.

Contribution

It introduces a novel randomized relaxation for large-scale nonconvex problems and develops a stochastic Frank-Wolfe algorithm with proven convergence properties.

Findings

Relaxation gap converges to zero as N increases.

The stochastic Frank-Wolfe algorithm converges in expectation and probability.

Numerical experiments demonstrate the method's efficiency on mixed-integer quadratic problems.

Abstract

We address a large-scale and nonconvex optimization problem, involving an aggregative term. This term can be interpreted as the sum of the contributions of N agents to some common good, with N large. We investigate a relaxation of this problem, obtained by randomization. The relaxation gap is proved to converge to zeros as N goes to infinity, independently of the dimension of the aggregate. We propose a stochastic method to construct an approximate minimizer of the original problem, given an approximate solution of the randomized problem. McDiarmid's concentration inequality is used to quantify the probability of success of the method. We consider the Frank-Wolfe (FW) algorithm for the resolution of the randomized problem. Each iteration of the algorithm requires to solve a subproblem which can be decomposed into N independent optimization problems. A sublinear convergence rate is obtained for the FW algorithm. In order to handle the memory overflow problem possibly caused by the FW algorithm, we propose a stochastic Frank-Wolfe (SFW) algorithm, which ensures the convergence in both expectation and probability senses. Numerical experiments on a mixed-integer quadratic program illustrate the efficiency of the method.