A dynamical approach to constrained nonsmooth convex minimization problem coupling with penalty function method in Hilbert space

This article is concerned with a class of nonsmooth constrained convex optimization in a real Hilbert space. Coupling with the penalty method, we propose an automatic system (AS) and a nonautomatic system (NS) modeled by differential inclusions. Under a suitable assumption on the feasible region and a proper condition on the objective and constrained functions, some valuable convergence properties of (AS) are obtained. In order to obtain strong convergence result in general cases, based on evolution differential inclusion, we propose a nonautomatic system (NS). When the control item (t) of (NS) satisfies some basal conditions, global and unique existence of the solution, finite time convergence to the feasible region and slow solution choice are obtained. Moreover, under different conditions of (t), we give some strong convergence results of (NS). Furthermore, we end the article by numerical experiments to illustrate the efficiency and good performance of the proposed systems in this article.