A Proximal Variable Smoothing for Nonsmooth Minimization Involving Weakly Convex Composite with MIMO Application
Keita Kume, Isao Yamada
TL;DR
The paper tackles nonsmooth, nonconvex optimization of the form $\min_x (h+g\circ \mathfrak{S}+\phi)(x)$ where $g$ is weakly convex and prox-friendly. It introduces a proximal variable smoothing algorithm, a time-varying forward–backward method that optimizes a smoothed surrogate $f_n = h+{}^{\mu_n}g\circ \mathfrak{S}$ via gradient steps and uses $\mathrm{prox}_{\gamma_n\phi}$ for the backward step, with a carefully chosen diminishing smoothing sequence $\mu_n$. A new stationarity measure $\mathcal{M}_{\gamma}^{f,\phi}$ and its smooth approximation drive convergence analysis, establishing that cluster points are stationary and $\liminf_{n\to\infty} \mathcal{M}_{\bar{\gamma}}^{f_n,\phi}(x_n)=0$ under mild Lipschitz assumptions. The framework is applied to MU-MIMO PSK detection, reformulating the problem in polar coordinates with a contrastive regularizer $\psi_{\lambda_r,\lambda_\theta}$, and is shown to outperform standard baselines in both convergence speed and bit-error-rate across various SNR and antenna configurations. Overall, the proximal variable smoothing approach extends prior variable-smoothing methods to broader weakly convex composites and demonstrates practical gains in high-dimensional signal processing tasks.
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.