### 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.

Original language | English (US) |
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Pages (from-to) | 1-25 |

Number of pages | 25 |

Journal | Journal of Optimization Theory and Applications |

DOIs | |

State | Accepted/In press - Feb 17 2016 |

### Fingerprint

### Keywords

- Mean-field games
- Mixed integer optimization
- Optimal control

### ASJC Scopus subject areas

- Applied Mathematics
- Control and Optimization
- Management Science and Operations Research

### Cite this

*Journal of Optimization Theory and Applications*, 1-25. https://doi.org/10.1007/s10957-016-0881-6

**Decomposition and Mean-Field Approach to Mixed Integer Optimal Compensation Problems.** / Bauso, Dario; Zhu, Quanyan; Başar, Tamer.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Decomposition and Mean-Field Approach to Mixed Integer Optimal Compensation Problems

AU - Bauso, Dario

AU - Zhu, Quanyan

AU - Başar, Tamer

PY - 2016/2/17

Y1 - 2016/2/17

N2 - 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.

AB - 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.

KW - Mean-field games

KW - Mixed integer optimization

KW - Optimal control

UR - http://www.scopus.com/inward/record.url?scp=84958763528&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84958763528&partnerID=8YFLogxK

U2 - 10.1007/s10957-016-0881-6

DO - 10.1007/s10957-016-0881-6

M3 - Article

SP - 1

EP - 25

JO - Journal of Optimization Theory and Applications

JF - Journal of Optimization Theory and Applications

SN - 0022-3239

ER -