A Proximal Variable Smoothing for Minimization of Nonlinearly Composite Nonsmooth Function -- Maxmin Dispersion and MIMO Applications

Abstract: We propose a proximal variable smoothing algorithm for a nonsmooth optimization problem whose cost function is the sum of three functions including a weakly convex composite function. The proposed algorithm has a single-loop structure inspired by a proximal gradient-type method. More precisely, the proposed algorithm consists of two steps: (i) a gradient descent of a time-varying smoothed surrogate function designed partially with the Moreau envelope of the weakly convex function; (ii) an application of the proximity operator of the remaining function not covered by the smoothed surrogate function. We also present a convergence analysis of the proposed algorithm by exploiting a novel asymptotic approximation of a gradient mapping-type stationarity measure. Numerical experiments demonstrate the effectiveness of the proposed algorithm in two scenarios: (i) maxmin dispersion problem and (ii) multiple-input-multiple-output (MIMO) signal detection.