The landscape of deterministic and stochastic optimal control problems: One-shot Optimization versus Dynamic Programming

TL;DR

This paper explores the relationship between one-shot optimization and dynamic programming in deterministic and stochastic finite-horizon optimal control problems, revealing conditions under which local minima correspond and differ.

Contribution

It provides theoretical insights into the optimization landscapes of one-shot and dynamic programming approaches, clarifying their equivalences and differences in various scenarios.

Findings

Local minimizers of one-shot optimization relate to those of DP in deterministic cases.

In stochastic cases, local minima of DP may not correspond to one-shot solutions.

Under certain conditions, both methods identify the same local solutions.

Abstract

Optimal control problems can be solved via a one-shot (single) optimization or a sequence of optimization using dynamic programming (DP). However, the computation of their global optima often faces NP-hardness, and thus only locally optimal solutions may be obtained at best. In this work, we consider the discrete-time finite-horizon optimal control problem in both deterministic and stochastic cases and study the optimization landscapes associated with two different approaches: one-shot and DP. In the deterministic case, we prove that each local minimizer of the one-shot optimization corresponds to some control input induced by a locally minimum control policy of DP, and vice versa. However, with a parameterized policy approach, we prove that deterministic and stochastic cases both exhibit the desirable property that each local minimizer of DP corresponds to some local minimizer of the one-shot optimization, but the converse does not necessarily hold. Nonetheless, under different technical assumptions for deterministic and stochastic cases, if there exists only a single locally minimum control policy, one-shot and DP turn out to capture the same local solution. These results pave the way to understand the performance and stability of local search methods in optimal control.