Optimality conditions for locally Lipschitz optimization with $$l_0$$-regularization

Abstract

This paper mainly investigates the locally Lipschitz optimization problem (LLOP) with $$l_0$$-regularization in a finite dimensional space, which is generally NP-hard but highly applicable in statistics, compressed sensing and deep learning. First, we introduce two classes of stationary points for this problem: subdifferential-stationary point and proximal-stationary point. Secondly, based on these two concepts, we analyze the first-order necessary/sufficient optimality conditions for the LLOP with $$l_0$$-regularization. Finally, we present two examples to illustrate the validity of the proposed optimality conditions.