Mixed integer optimal compensation: Decompositions and mean-field approximations

Zhu

Başar

2016

J Optim Theory Appl

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

Section: Highlights Of the Main Results And Relationship With The Relmentioning

confidence: 99%

Section: Highlights Of the Main Results And Relationship With The Relmentioning

confidence: 99%

Section: Highlights Of the Main Results And Relationship With The Relmentioning

confidence: 99%

See 1 more Smart Citation

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

Zhu

Başar

2016

J Optim Theory Appl

“…[21], one can deal with a robust approximation which would appear in the form of a min-max program [4]. Observe that the reverse dwell time conditions (2) can be described by linear constraints of the form:…”

Section: Preliminary Result: Cost Boundsmentioning

confidence: 99%

Impulsively-controlled systems and reverse dwell time: A linear programming approach

Nonlinear Analysis: Hybrid Systems

2015

We present a receding horizon algorithm that converges to the exact solution in polynomial time for a class of optimal impulse control problems with uniformly distributed impulse instants and governed by so-called reverse dwell time conditions. The cost has two separate terms, one depending on time and the second monotonically decreasing on the state norm. The obtained results have both theoretical and practical relevance. From a theoretical perspective we prove certain geometrical properties of the discrete set of feasible solutions. From a practical standpoint, such properties reduce the computational burden and speed up the search for the optimum thus making the algorithm suitable for on-line implementation in real-time problems. Our approach consists in approximating the optimal impulse control problem via a binary linear programming problem with a totally unimodular constraint matrix. Hence, solving the binary linear programming problem is equivalent to solving its linear relaxation. Then, given the feasible solution from the linear relaxation, we find the optimal solution via receding horizon and local search. Numerical illustrations of a queueing system are performed.

“…Mean field game formulations apply also to multi-inventory systems with quadratic cost and additional set up costs, as discussed in [29]. Decision problems with mean field coupling terms have also been formalized and studied in [11]. An example from production engineering has been first introduced by [17] (see also [31]).…”

Section: Introductionmentioning

confidence: 99%

Mean-Field Games and Dynamic Demand Management in Power Grids

Bagagiolo

2013

Dyn Games Appl

This paper applies mean field game theory to dynamic demand management. For a large population of electrical heating or cooling appliances (called agents), we provide a mean field game that guarantees desynchronization of the agents thus improving the power network resilience. Second, for the game at hand, we exhibit a mean field equilibrium, where each agent adopts a bang-bang switching control with threshold placed at a nominal temperature. At the equilibrium, through an opportune design of the terminal penalty, the switching control regulates the mean temperature (computed over the population) and the mains frequency around the nominal value. To overcome Zeno phenomena we also adjust the bang-bang control by introducing a thermostat. Third, we show that the equilibrium is stable in the sense that all agents' states, initially at different values, converge to the equilibrium value or remain confined within a given interval for an opportune initial distribution.