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

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 Communicated by

Dynamic Coordination Games with Activation Costs

“…We study the cases under deterministic and stochastic disturbances. All results can be extended to the vector case by using the robust decomposition approach in [4,Section 3]. Applications arise in coordinated replenishment [8], and opportunistic maintenance [7].…”