Iterative Hard Thresholding Methods for $l_0$ Regularized Convex Cone Programming

In this paper we consider l0 regularized convex cone programming problems. In particular, we first propose an iterative hard thresholding (IHT) method and its variant for solving l0 regularized box constrained convex programming. We show that the sequence generated by these methods converges to a local minimizer. Also, we establish the iteration complexity of the IHT method for finding an -local-optimal solution. We then propose a method for solving l0 regularized convex cone programming by applying the IHT method to its quadratic penalty relaxation and establish its iteration complexity for finding an -approximate local minimizer. Finally, we propose a variant of this method in which the associated penalty parameter is dynamically updated, and show that every accumulation point is a local minimizer of the problem.