Non-convex optimization via strongly convex majoirziation-minimization

Abstract

In this paper, we introduce a class of nonsmooth nonconvex least square optimization problem using convex analysis tools and we propose to use the iterative minimization-majorization (MM) algorithm on a convex set with initializer away from the origin to find an optimal point for the optimization problem. For this, first we use an approach to construct a class of convex majorizers which approximate the value of non-convex cost function on a convex set. The convergence of the iterative algorithm is guaranteed when the initial point $x^{(0)}$ is away from the origin and the iterative points $x^{(k)}$ are obtained in a ball centred at $x^{(k-1)}$ with small radius. The algorithm converges to a stationary point of cost function when the surregators are strongly convex. For the class of our optimization problems, the proposed penalizer of the cost function is the difference of $\ell_1$-norm and the Moreau envelope of a convex function, and it is a generalization of GMC non-separable penalty function previously introduced by Ivan Selesnick in \cite{IS17}.