Solving the cost to go with time penalization using the Lagrange optimization approach

This paper suggests a new method for solving the cost to go with time penalization. We consider the Lagrange approach in order to incorporate the restrictions of the problem and to solve the convex structured minimization problems. The solution is based on an improved version of the proximal method in which the regularization term that asymptotically disappear involves a penalization time parameter. By assuming that the set of equilibria is non-empty, we show that the proximal algorithm involving time penalization generates a sequence that converges to a saddle point of the Lagrange functional. This method improves the traditional approach of proximal algorithms and its applications. Our approach is implemented using continuous-time Markov chains. We analyze a queueing model with applications to waiting time of customers in the branch of a bank, for showing the effectiveness of the method.