Decomposition and Mean-Field Approach to Mixed Integer Optimal Compensation Problems

Dario Bauso,Tamer Başar,Quanyan Zhu

doi:10.1007/s10957-016-0881-6

Abstract

Mixed integer optimal compensation deals with optimization problems with integer- and real-valued control variables to compensate disturbances in dynamic systems. The mixed integer nature of controls could lead to intractability in problems of large dimensions. To address this challenge, we introduce a decomposition method which turns the original n-dimensional optimization problem into n independent scalar problems of lot sizing form. Each of these problems can be viewed as a two-player zero-sum game, which introduces some element of conservatism. Each scalar problem is then reformulated as a shortest path one and solved through linear programming over a receding horizon, a step that mirrors a standard procedure in mixed integer programming. We apply the decomposition method to a mean-field coupled multi-agent system problem, where each agent seeks to compensate a combination of an exogenous signal and the local state average. We discuss a large population mean-field type of approximation and extend our study to opinion dynamics in social networks as a special case of interest.

Highlights

Mixed integer optimal compensation arises when optimizing a mix of integer- and real-valued control variables in order to compensate for disturbances in dynamic systems
It is known that new structural properties of the problem play important roles in mixed integer control; as an example, see multimodularity presented as the counterpart of convexity in discrete action spaces [5]
We have proposed a robust decomposition method which brings an ndimensional hybrid optimization problem into n independent tractable scalar problems of lot sizing form

Summary

Introduction

Mixed integer optimal compensation arises when optimizing a mix of integer- and real-valued control variables in order to compensate for disturbances in dynamic systems. Mixed integer control can be viewed as a specific subfield of optimal hybrid control [1], addressed recently in a receding horizon framework [2]. Optimal integer control problems have been receiving growing attention and are often categorized under different names (e.g., alphabet control [3,4]). Handling integer control requires more than standard convex optimization techniques. It is known that new structural properties of the problem play important roles in mixed integer control; as an example, see multimodularity presented as the counterpart of convexity in discrete action spaces [5]. We should note that there is vast literature on mixed integer programming [6], and it is in this context that we cast the problem addressed in this paper. Mixed integer optimal control has been dealt with in [8,9,10,11]

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