A Unified Successive Pseudo-Convex Approximation Framework

TL;DR

This paper introduces a unified iterative framework for solving complex nonconvex optimization problems by successively refining pseudo-convex approximations, leading to easier computations and faster convergence.

Contribution

It develops a general successive pseudo-convex approximation method that includes existing algorithms and introduces new algorithms with improved implementation and convergence properties.

Findings

Framework includes gradient and Jacobi methods as special cases

Proposes a novel line search for nondifferentiable problems

Demonstrates effectiveness in MIMO, energy maximization, and sparse recovery

Abstract

In this paper, we propose a successive pseudo-convex approximation algorithm to efficiently compute stationary points for a large class of possibly nonconvex optimization problems. The stationary points are obtained by solving a sequence of successively refined approximate problems, each of which is much easier to solve than the original problem. To achieve convergence, the approximate problem only needs to exhibit a weak form of convexity, namely, pseudo-convexity. We show that the proposed framework not only includes as special cases a number of existing methods, for example, the gradient method and the Jacobi algorithm, but also leads to new algorithms which enjoy easier implementation and faster convergence speed. We also propose a novel line search method for nondifferentiable optimization problems, which is carried out over a properly constructed differentiable function with the benefit of a simplified implementation as compared to state-of-the-art line search techniques that directly operate on the original nondifferentiable objective function. The advantages of the proposed algorithm are shown, both theoretically and numerically, by several example applications, namely, MIMO broadcast channel capacity computation, energy efficiency maximization in massive MIMO systems and LASSO in sparse signal recovery.