On a multi-stage nonlinear programming problem
On a multi-stage nonlinear programming problem
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关于多阶段非线性规划问题
DOI：
10.1016/0022-247x(67)90173-4
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发表时间：
1967
期刊：
影响因子：
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通讯作者：
S. Arimoto
中科院分区：
文献类型：
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作者：
S. Arimoto
在多阶段过程或离散时间控制过程的优化问题中，多篇论文提出了某些类型的最优性必要条件，例如连续时间控制过程的极大值原理。在最早的论文[l]中，张提出了最优性的必要条件，并将其称为“数字化极大值原理”。 Katz [2] 也得到了类似的结果。 Butkovsky [3] 除了卡茨定理的反例之外，还提供了局部极大值原理，这意味着哈密顿量在最优控制的邻域内达到最大值。然而，离散时间过程的局部极大值原理一般并不成立，如本文最后一节将给出的反例所示。最近，Halkin [4]、Jordan 和 Polak [5] 的两篇重要论文发表。 Halkin 展示了最优性必要条件的几何方面，Jordan 和 Polak 建立了局部最大值或平稳原理。 在本文中，我们将考虑比上述引用的参考文献中更一般的多阶段过程优化问题。第 2 节中所述的问题是为连续时间控制系统制定的 Berkovitz 问题 [6] 的离散版本。它也被认为是贝尔曼[7]首先讨论的多阶段生产过程中非线性瓶颈型规划问题的推广。第 4 节将证明最优性的必要条件。第 5 节和第 6 节讨论该问题的特殊情况。在第 5 节中，定理 2 中将给出局部最优的充分条件。在第 6 节中，定理 3 中将在附加条件下提出全局极大值原理，该附加条件类似于 Fillipov [8] 给出的证明连续时间系统最优控制存在性的条件。在我们之前的工作 [9] 中，我们提出了与定理 3 161 类似的定理
In optimization problems of multi-stage processes or discrete-time control processes, some types of necessary conditions for optimality were proposed in several papers, like the maximum principle for continuous-time control processes. In the earliest paper [l], Chang proposed a necessary condition for optimality and called it “the digitized maximum principle.” Similar results were also obtained by Katz [2]. Butkovsky [3], in addition to a counter-example to Katz’s theorem, offered the local maximum principle which implies that the Hamiltonian attains the maximum value in a neighborhood of the optimal control. However, the local maximum principle for discrete-time processes does not hold in general as shown by a counterexample which will be given in the last section of this paper. Recently, the two important papers by Halkin [4], Jordan and Polak [5] were published. Halkin showed geometric aspects of necessary conditions for optimality, and Jordan and Polak established the local maximum or stationary principle.In the present paper we shall consider more general optimization problems for multi-stage processes than those in the above cited references. The problem stated in Section 2 is a discrete version of Berkovitz’s problem [6] formulated for continuous-time control systems. It is also regarded as a generalization of nonlinear bottleneck-type programming problems in multi-stage production processes first discussed by Bellman [7]. In Section 4 a necessary condition for optimality will be proved. Sections 5 and 6 treat a special case of the problem. In Section 5 a sufficient condition for local optimality will be given in Theorem 2. In Section 6, a global maximum principle will be proposed in Theorem 3 under an additional condition which is analogous to that given by Fillipov [8] for the proof of the existence of an optimal control for continuous-time systems. In our previous work [9] we proposed the analogous theorem to Theorem 3 161