A level-set approach to the control of state-constrained McKean-Vlasov equations: application to renewable energy storage and portfolio selection

IF 1.1 Q2 MATHEMATICS, APPLIED

Numerical Algebra Control and Optimization Pub Date : 2021-12-21 DOI:10.3934/naco.2022033

Maximilien Germain, H. Pham, X. Warin

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引用次数: 2

Abstract

We consider the control of McKean-Vlasov dynamics (or mean-field control) with probabilistic state constraints. We rely on a level-set approach which provides a representation of the constrained problem in terms of an unconstrained one with exact penalization and running maximum or integral cost. The method is then extended to the common noise setting. Our work extends (Bokanowski, Picarelli, and Zidani, SIAM J. Control Optim. 54.5 (2016), pp. 2568--2593) and (Bokanowski, Picarelli, and Zidani, Appl. Math. Optim. 71 (2015), pp. 125--163) to a mean-field setting. The reformulation as an unconstrained problem is particularly suitable for the numerical resolution of the problem, that is achieved from an extension of a machine learning algorithm from (Carmona, Lauri{\`e}re, arXiv:1908.01613 to appear in Ann. Appl. Prob., 2019). A first application concerns the storage of renewable electricity in the presence of mean-field price impact and another one focuses on a mean-variance portfolio selection problem with probabilistic constraints on the wealth. We also illustrate our approach for a direct numerical resolution of the primal Markowitz continuous-time problem without relying on duality.

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状态约束McKean-Vlasov方程控制的水平集方法:在可再生能源存储和投资组合选择中的应用

我们考虑具有概率状态约束的McKean-Vlasov动力学控制(或称平均场控制)。我们依赖于一种水平集方法，它将约束问题表示为具有精确惩罚和运行最大或积分成本的无约束问题。然后将该方法推广到常见的噪声设置。我们的工作扩展了(Bokanowski, Picarelli, and Zidani, SIAM J. Control Optim. 54.5 (2016)， pp. 2568—2593)和(Bokanowski, Picarelli, and Zidani, apple .)。数学。Optim. 71 (2015)， pp. 125—163)到平均场设置。作为无约束问题的重新表述特别适合于问题的数值解决，这是通过扩展(Carmona, Lauri{\ ' e}re, arXiv:1908.01613)的机器学习算法实现的，并出现在Ann中。达成。概率。, 2019)。第一个应用涉及平均场价格影响下可再生电力的存储，另一个应用侧重于对财富有概率约束的平均方差投资组合选择问题。我们还说明了不依赖对偶的原始马科维茨连续时间问题的直接数值解决方法。

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来源期刊

Numerical Algebra Control and Optimization MATHEMATICS, APPLIED-

CiteScore

3.10

自引率

0.00%

发文量

期刊介绍： Numerical Algebra, Control and Optimization (NACO) aims at publishing original papers on any non-trivial interplay between control and optimization, and numerical techniques for their underlying linear and nonlinear algebraic systems. Topics of interest to NACO include the following: original research in theory, algorithms and applications of optimization; numerical methods for linear and nonlinear algebraic systems arising in modelling, control and optimisation; and original theoretical and applied research and development in the control of systems including all facets of control theory and its applications. In the application areas, special interests are on artificial intelligence and data sciences. The journal also welcomes expository submissions on subjects of current relevance to readers of the journal. The publication of papers in NACO is free of charge.