A Unified Successive Pseudoconvex Approximation Framework

Abstract

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.

A Parallel Algorithm for Energy Efficiency Maximization in Massive MIMO Networks

“…The computational complexity of each variable step is thus much lower than that of [5]. Based on the line of analysis in [11], we show that the proposed algorithm is guaranteed to converge. The advantage of the proposed algorithm is also illustrated numerically.…”

“…Similarly, since P c + N n=1 K j=1 p k,n in the denominator is linear in p and thus left intact. Furthermore, the division operator in the original problem (2) is kept in the approximate function (11). Although it will destroy the concavity of the approximate functionf (p; p t ) in (11) is not a concave function 1 , it presents the pseudo-concavity as we show in two steps.…”

“…In this section, we give a brief review of the unified successive convex approximation method proposed in [11] to solve the following optimization problem:…”

“…The algorithm is summarized in Algorithm 1. It is shown in [11,Theorem 1] that if: (A4) The solution set S(x t ) is nonempty for t = 1, 2, . .…”

“…The algorithm is based on the idea of successive convex approximation, recently advanced in [9], [10] and further generalized in [11]. In each iteration, the proposed approximate function only needs to be pseudo-convex, a weak form of convexity, and this weak assumption makes it possible to preserve as much structure available in the original EE function as possible, e.g., the partial concavity in the numerator function and the division operator.…”