Smoothing neural network for L₀ regularized optimization problem with general convex constraints.

Abstract

In this paper, we propose a neural network modeled by a differential inclusion to solve a class of discontinuous and nonconvex sparse regression problems with general convex constraints, whose objective function is the sum of a convex but not necessarily differentiable loss function and L₀ regularization. We construct a smoothing relaxation function of L₀ regularization and propose a neural network to solve the considered problem. We prove that the solution of proposed neural network with any initial point satisfying linear equality constraints is global existent, bounded and reaches the feasible region in finite time and remains there thereafter. Moreover, the solution of proposed neural network is its slow solution and any accumulation point of it is a Clarke stationary point of the brought forward nonconvex smoothing approximation problem. In the box-constrained case, all accumulation points of the solution own a unified lower bound property and have a common support set. Except for a special case, any accumulation point of the solution is a local minimizer of the considered problem. In particular, the proposed neural network has a simple structure than most existing neural networks for solving the locally Lipschitz continuous but nonsmooth nonconvex problems. Finally, we give some numerical experiments to show the efficiency of proposed neural network.