Large-Scale Nonconvex Optimization: Randomization, Gap Estimation, and Numerical Resolution
J. Frédéric Bonnans, Kang Liu, Nadia Oudjane, Laurent Pfeiffer, Cheng Wan · SIAM Journal on Optimization · 2023
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 [Formula: see text] agents to some common good, with [Formula: see text] large. We investigate a relaxation of this problem, obtained by randomization. The relaxation gap is proved to converge to zeros as [Formula: see text] 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 [Formula: see text] 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 FW) 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.