Proximal constrained optimization approach with time penalization

This article concerns a proximal-point algorithm with time penalization. The case where the cost of moving from one position to a better one is penalized by the time taken by the agent for the decision-making is studied and the restriction employing the penalty method is incorporated. It is shown that the method converges monotonically with respect to the minimal weighted norm to a unique minimal point under mild assumptions. The gradient method is employed for solving the objective function, and its convergence is proven. The rate of convergence of the method is also estimated by computing the optimal parameters. The effectiveness of the method is illustrated by a numerical optimization example employing continuous-time Markov chains.