A Proximal Variable Smoothing for Nonsmooth Minimization Involving Weakly Convex Composite with MIMO Application

TL;DR

This paper introduces a novel proximal variable smoothing algorithm for nonsmooth weakly convex optimization problems, with convergence analysis and an application to MIMO signal detection, showing promising numerical results.

Contribution

It develops a time-varying forward-backward splitting algorithm using Moreau envelope smoothing for weakly convex functions, with convergence guarantees and practical MIMO application.

Findings

Algorithm converges to stationary points.

Effective in MIMO signal detection.

Numerical results demonstrate improved performance.

Abstract

We propose a proximal variable smoothing algorithm for nonsmooth optimization problem with sum of three functions involving weakly convex composite function. The proposed algorithm is designed as a time-varying forward-backward splitting algorithm with two steps: (i) a time-varying forward step with the gradient of a smoothed surrogate function, designed with the Moreau envelope, of the sum of two functions; (ii) the backward step with a proximity operator of the remaining function. For the proposed algorithm, we present a convergence analysis in terms of a stationary point by using a newly smoothed surrogate stationarity measure. As an application of the target problem, we also present a formulation of multiple-input-multiple-output (MIMO) signal detection with phase-shift keying. Numerical experiments demonstrate the efficacy of the proposed formulation and algorithm.