A Proximal Variable Smoothing for Minimization of Nonlinearly Composite Nonsmooth Function -- Finite-Max Minimization and MIMO Applications

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.

For the proposed algorithm, we present a subsequential convergence guarantee in terms of a stationary point, and a convergence rate ${O}(\epsilon^{-3})$ for achieving an $\epsilon$-stationary point.

Numerical experiments demonstrate the effectiveness of the proposed algorithm in two scenarios: (i) robust target localization and (ii) multiple-input-multiple-output (MIMO) signal detection.