Time Consistency for Multistage Stochastic Optimization Problems under Constraints in Expectation

IF 2.6 1区数学 Q1 MATHEMATICS, APPLIED

SIAM Journal on Optimization Pub Date : 2024-06-04 DOI:10.1137/22m151830x

Pierre Carpentier, Jean-Philippe Chancelier, Michel De Lara

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引用次数: 0

Abstract

SIAM Journal on Optimization, Volume 34, Issue 2, Page 1909-1936, June 2024.
Abstract. We consider sequences—indexed by time (discrete stages)—of parametric families of multistage stochastic optimization problems; thus, at each time, the optimization problems in a family are parameterized by some quantities (initial states, constraint levels, and so on). In this framework, we introduce an adapted notion of parametric time-consistent optimal solutions: They are solutions that remain optimal after truncation of the past and that are optimal for any values of the parameters. We link this time consistency notion with the concept of state variable in Markov decision processes for a class of multistage stochastic optimization problems incorporating state constraints at the final time, formulated in expectation. For such problems, when the primitive noise random process is stagewise independent and takes a finite number of values, we show that time-consistent solutions can be obtained by considering a finite-dimensional state variable. We illustrate our results on a simple dam management problem.

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期望约束下多级随机优化问题的时间一致性

SIAM 优化期刊》，第 34 卷第 2 期，第 1909-1936 页，2024 年 6 月。摘要。我们考虑以时间（离散阶段）为索引的多阶段随机优化问题的参数族序列；因此，在每个时间，族中的优化问题都由一些量（初始状态、约束水平等）参数化。在这一框架下，我们引入了参数时间一致性最优解的调整概念：它们是在截断过去后仍保持最优的解，而且对于任何参数值都是最优的。我们将这一时间一致性概念与马尔可夫决策过程中的状态变量概念联系起来，用于一类包含最终时间状态约束条件的多阶段随机优化问题，并以期望值表示。对于这类问题，当原始噪声随机过程是阶段性独立的，并且取值数量有限时，我们证明了通过考虑有限维的状态变量可以得到时间一致的解。我们用一个简单的水坝管理问题来说明我们的结果。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

SIAM Journal on Optimization 数学-应用数学

CiteScore

5.30

自引率

9.70%

发文量

101

审稿时长

6-12 weeks

期刊介绍： The SIAM Journal on Optimization contains research articles on the theory and practice of optimization. The areas addressed include linear and quadratic programming, convex programming, nonlinear programming, complementarity problems, stochastic optimization, combinatorial optimization, integer programming, and convex, nonsmooth and variational analysis. Contributions may emphasize optimization theory, algorithms, software, computational practice, applications, or the links between these subjects.