Graph-Structured Tensor Optimization for Nonlinear Density Control and Mean Field Games

IF 2.2 2区数学 Q2 AUTOMATION & CONTROL SYSTEMS SIAM Journal on Control and Optimization Pub Date : 2024-07-16 DOI:10.1137/23m1571587

Axel Ringh, Isabel Haasler, Yongxin Chen, Johan Karlsson

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Abstract

SIAM Journal on Control and Optimization, Volume 62, Issue 4, Page 2176-2202, August 2024.
Abstract. In this work we develop a numerical method for solving a type of convex graph-structured tensor optimization problem. This type of problem, which can be seen as a generalization of multimarginal optimal transport problems with graph-structured costs, appears in many applications. Examples are unbalanced optimal transport and multispecies potential mean field games, where the latter is a class of nonlinear density control problems. The method we develop is based on coordinate ascent in a Lagrangian dual, and under mild assumptions we prove that the algorithm converges globally. Moreover, under a set of stricter assumptions, the algorithm converges R-linearly. To perform the coordinate ascent steps one has to compute projections of the tensor, and doing so by brute force is in general not computationally feasible. Nevertheless, for certain graph structures it is possible to derive efficient methods for computing these projections, and here we specifically consider the graph structure that occurs in multispecies potential mean field games. We also illustrate the methodology on a numerical example from this problem class.

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非线性密度控制和均值场博弈的图结构张量优化

SIAM 控制与优化期刊》，第 62 卷第 4 期，第 2176-2202 页，2024 年 8 月。摘要在这项工作中，我们开发了一种求解凸图结构张量优化问题的数值方法。这类问题可视为具有图结构成本的多边际最优运输问题的一般化，在许多应用中都会出现。例如，不平衡最优传输和多物种势均场博弈，后者是一类非线性密度控制问题。我们开发的方法基于拉格朗日对偶中的坐标上升，在温和的假设条件下，我们证明了算法的全局收敛性。此外，在一系列更严格的假设条件下，该算法可实现 R 线性收敛。要执行坐标上升步骤，必须计算张量的投影，而用蛮力计算一般是不可行的。不过，对于某些图结构，我们可以推导出计算这些投影的高效方法，这里我们特别考虑了多物种潜在均值场博弈中出现的图结构。我们还将用这一类问题中的一个数值例子来说明这种方法。

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来源期刊

SIAM Journal on Control and Optimization 数学-应用数学

CiteScore

4.00

自引率

4.50%

发文量

143

审稿时长

12 months

期刊介绍： SIAM Journal on Control and Optimization (SICON) publishes original research articles on the mathematics and applications of control theory and certain parts of optimization theory. Papers considered for publication must be significant at both the mathematical level and the level of applications or potential applications. Papers containing mostly routine mathematics or those with no discernible connection to control and systems theory or optimization will not be considered for publication. From time to time, the journal will also publish authoritative surveys of important subject areas in control theory and optimization whose level of maturity permits a clear and unified exposition. The broad areas mentioned above are intended to encompass a wide range of mathematical techniques and scientific, engineering, economic, and industrial applications. These include stochastic and deterministic methods in control, estimation, and identification of systems; modeling and realization of complex control systems; the numerical analysis and related computational methodology of control processes and allied issues; and the development of mathematical theories and techniques that give new insights into old problems or provide the basis for further progress in control theory and optimization. Within the field of optimization, the journal focuses on the parts that are relevant to dynamic and control systems. Contributions to numerical methodology are also welcome in accordance with these aims, especially as related to large-scale problems and decomposition as well as to fundamental questions of convergence and approximation.