A Unified Successive Pseudo-Convex Approximation Framework

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.Comment: submitted to IEEE Transactions on Signal Processing; original title: A Novel Iterative Convex Approximation Metho

A Non-Monotone Conjugate Subgradient Type Method for Minimization of Convex Functions

We suggest a conjugate subgradient type method without any line-search for minimization of convex non differentiable functions. Unlike the custom methods of this class, it does not require monotone decrease of the goal function and reduces the implementation cost of each iteration essentially. At the same time, its step-size procedure takes into account behavior of the method along the iteration points. Preliminary results of computational experiments confirm efficiency of the proposed modification.Comment: 11 page