Further extensions on the successive approximation method for hierarchical optimal control problems and its application to learning

In this paper, further extensions of the result of the paper "A successive approximation method in functional spaces for hierarchical optimal control problems and its application to learning, arXiv:2410.20617 [math.OC], 2024" concerning a class of learning problem of point estimations for modeling of high-dimensional nonlinear functions are given. In particular, we present two viable extensions within the nested algorithm of the successive approximation method for the hierarchical optimal control problem, that provide better convergence property and computationally efficiency, which ultimately leading to an optimal parameter estimate. The first extension is mainly concerned with the convergence property of the steps involving how the two agents, i.e., the "leader" and the "follower," update their admissible control strategies, where we introduce augmented Hamiltonians for both agents and we further reformulate the admissible control updating steps as as sub-problems within the nested algorithm of the hierarchical optimal control problem that essentially provide better convergence property. Whereas the second extension is concerned with the computationally efficiency of the steps involving how the agents update their admissible control strategies, where we introduce intermediate state variable for each agent and we further embed the intermediate states within the optimal control problems of the "leader" and the "follower," respectively, that further lend the admissible control updating steps to be fully efficient time-parallelized within the nested algorithm of the hierarchical optimal control problem.