On reverse Stackelberg game and optimal mean field control for a large population of thermostatically controlled loads

Li, Sen; Zhang, Wei; Lian, Jianming; Kalsi, Karanjit

doi:10.1109/acc.2016.7525463

Cited by 8 publications

(7 citation statements)

References 14 publications

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“…8]. In our paper, when we say that the social welfare optimization problem has strong duality, it indicates the problem already satisfies certain constraint qualifications so that the solution to (36) exists.…”

Section: Connections Under Strong Dualitymentioning

confidence: 97%

“…It includes both discrete-time [9], [23] and continuous-time system [4] as special cases, and addresses both deterministic and stochastic cases. In the cost function (3), the inner product term F (x) · x i can be either interpreted as the price multiplied by quantity [10], [36], [37] or part of the quadratic penalty of the deviation of the system state from the population mean [4], [26]. This structure is very common in the literature.…”

Section: The Mean-field Gamementioning

confidence: 99%

“…where D * is the optimal value of the dual problem. When the dual problem (36) admits a solution, and the optimal value of the dual problem (36) coincides with that of the primal problem (32), then we say the optimization problem (30) has strong duality. Formally, we define it as follows:…”

Section: Connections Under Strong Dualitymentioning

confidence: 99%

“…Note that the definition of strong duality not only requires the duality gap to be zero, but also requires the dual problem to have a finite solution λ * . This is slightly stronger than only requiring zero duality gap between the primal problem (32) and the dual problem (36). In general, the existence of a finite multiplier to (36) can be easily guaranteed under mild constraint qualifications (e.g., Slater's condition) [42,Chap.…”

Section: Connections Under Strong Dualitymentioning

confidence: 99%

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Connections between mean-field game and social welfare optimization

Zhang

Zhao

2019

Automatica

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This paper studies the connection between a class of mean-field games and a social welfare optimization problem. We consider a mean-field game in function spaces with a large population of agents, and each agent seeks to minimize an individual cost function. The cost functions of different agents are coupled through a mean-field term that depends on the mean of the population states. We show that although the mean-field game is not a potential game, under some mild condition the -Nash equilibrium of the mean-field game coincides with the optimal solution to a social welfare optimization problem, and this is true even when the individual cost functions are non-convex. The connection enables us to evaluate and promote the efficiency of the mean-field equilibrium. In addition, it also leads to several important implications on the existence, uniqueness, and computation of the mean-field equilibrium. Numerical results are presented to validate the solution, and examples are provided to show the applicability of the proposed approach.

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Section: Connections Under Strong Dualitymentioning

confidence: 97%

Section: The Mean-field Gamementioning

confidence: 99%

Section: Connections Under Strong Dualitymentioning

confidence: 99%

Section: Connections Under Strong Dualitymentioning

confidence: 99%

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Connections between mean-field game and social welfare optimization

Zhang

Zhao

2019

Automatica

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“…As the problem is defined in the general functional space, it provides a unifying formulation that includes both discrete-time [11], [13] and continuoustime system [4] as special cases, and addresses both deterministic and stochastic cases. This class of problems frequently arises in various applications [4], [5], [9], [11], [13], [17], [19], [20], [21], where F (m) · x i can be either interpreted as the price multiplied by quantity [7], [22] or part of the quadratic penalty of the deviation of the system state from the population mean [4], [17].…”

Section: A the Mean Field Gamementioning

confidence: 99%

On social optima of non-cooperative mean field games

Zhang

Zhao

et al. 2016

2016 IEEE 55th Conference on Decision and Control (CDC)

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Abstract-This paper studies the connections between meanfield games and the social welfare optimization problems. We consider a mean field game in functional spaces with a large population of agents, each of which seeks to minimize an individual cost function. The cost functions of different agents are coupled through a mean field term that depends on the mean of the population states. We show that under some mild conditions any -Nash equilibrium of the mean field game coincides with the optimal solution to a convex social welfare optimization problem. The results are proved based on a general formulation in the functional spaces and can be applied to a variety of mean field games studied in the literature. Our result also implies that the computation of the mean field equilibrium can be cast as a convex optimization problem, which can be efficiently solved by a decentralized primal dual algorithm. Numerical simulations are presented to demonstrate the effectiveness of the proposed approach.

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Transactive Energy Systems: The Market-Based Coordination of Distributed Energy Resources

Lian

Conejo

et al. 2020

IEEE Control Syst.

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On reverse Stackelberg game and optimal mean field control for a large population of thermostatically controlled loads

Cited by 8 publications

References 14 publications

Connections between mean-field game and social welfare optimization

Connections between mean-field game and social welfare optimization

On social optima of non-cooperative mean field games

Transactive Energy Systems: The Market-Based Coordination of Distributed Energy Resources

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