A Unified Successive Pseudoconvex Approximation Framework

In this paper, we propose a successive pseudoconvex 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, pseudoconvexity. 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 that 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 the 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, multi-input multi-output (MIMO) broadcast channel capacity computation, energy efficiency maximization in massive MIMO systems, and ℓ
₁
-regularized least square problems in sparse signal recovery.