确定性与随机最优控制问题的景观：一次性优化与动态规划
The landscape of deterministic and stochastic optimal control problems: One-shot Optimization versus Dynamic Programming
AI总结：
本文研究离散时间有限时域最优控制中一次性优化与动态规划两种方法的优化景观关系，证明在确定性情形下二者局部极小点相互对应，在参数化策略下动态规划局部极小点对应一次性优化局部极小点但反之不一定成立，并在唯一局部最小策略条件下二者得到相同局部解。
摘要：
最优控制问题可以通过一次性（单次）优化或利用动态规划（DP）进行一系列优化来求解。然而，其全局最优解的计算往往面临 NP 困难，因此最多只能获得局部最优解。本文考虑离散时间有限时域最优控制问题在确定性与随机两种情形下的情况，研究与两种不同方法（一次性优化和动态规划）相关的优化景观。在确定性情形下，我们证明一次性优化的每个局部极小点都对应于由动态规划的某个局部最小控制策略所诱导的控制输入，反之亦然。然而，在参数化策略方法下，我们证明确定性与随机情形都表现出如下良好性质：动态规划的每个局部极小点都对应于一次性优化的某个局部极小点，但反过来不一定成立。尽管如此，在确定性与随机情形各自不同的技术假设下，如果只存在唯一的局部最小控制策略，那么一次性优化和动态规划将得到相同的局部解。这些结果为理解局部搜索方法在最优控制中的性能和稳定性奠定了基础。
英文摘要：
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.
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