A Relaxation Method for Nonsmooth Nonlinear Optimization with Binary Constraints

TL;DR

This paper introduces a novel relaxation algorithm for nonsmooth binary optimization problems, leveraging DC reformulation and Moreau envelopes, with proven complexity guarantees and superior empirical performance.

Contribution

It develops the DCRA algorithm using DC reformulation and Moreau envelopes, providing global complexity guarantees and improved accuracy for nonsmooth binary optimization.

Findings

DCRA achieves better accuracy than existing methods.

The algorithm has explicit bounds on optimality gap.

Numerical results confirm superior scalability.

Abstract

We study binary optimization problems of the form \( \min_{x\in\{-1,1\}^n} f(Ax-b) \) with possibly nonsmooth loss \(f\). Following the lifted rank-one semidefinite programming (SDP) approach\cite{qian2023matrix}, we develop a majorization-minimization algorithm by using the difference-of-convexity (DC) reformuation for the rank-one constraint and the Moreau envelop for the nonsmooth loss. We provide global complexity guarantees for the proposed \textbf{D}ifference of \textbf{C}onvex \textbf{R}elaxation \textbf{A}lgorithm (DCRA) and show that it produces an approximately feasible binary solution with an explicit bound on the optimality gap. Numerical experiments on synthetic and real datasets confirm that our method achieves superior accuracy and scalability compared with existing approaches.