Convergence Analysis of Penalty Decomposition Algorithm for Cardinality Constrained Convex Optimization in Hilbert Spaces

Abstract

The paper examines an algorithm for finding approximate sparse solutions of convex cardinality constrained optimization problem in Hilbert spaces. The proposed algorithm uses the penalty decomposition (PD) approach and solves sub-problems on each iteration approximately. We examine the convergence of the algorithm to a stationary point satisfying necessary optimality conditions. Unlike other similar works, this paper discusses the properties of PD algorithms in infinite-dimensional (Hilbert) space. The results showed that the convergence property obtained in previous works for cardinality constrained optimization in Euclidean space also holds for infinite-dimensional (Hilbert) space. Moreover, in this paper we established a similar result for convex optimization problems with cardinality constraint with respect to a dictionary (not necessarily the basis).