Probably Approximately Correct Nash Equilibrium Learning

We consider a multi-agent noncooperative game with agents' objective functions being affected by uncertainty. Following a data driven paradigm, we represent uncertainty by means of scenarios and seek a robust Nash equilibrium solution. We treat the Nash equilibrium computation problem within the realm of probably approximately correct (PAC) learning. Building upon recent developments in scenario-based optimization, we accompany the computed Nash equilibrium with a priori and a posteriori probabilistic robustness certificates, providing confidence that the computed equilibrium remains unaffected (in probabilistic terms) when a new uncertainty realization is encountered. For a wide class of games, we also show that the computation of the so called compression set - a key concept in scenario-based optimization - can be directly obtained as a byproduct of the proposed solution methodology. Finally, we illustrate how to overcome differentiability issues, arising due to the introduction of scenarios, and compute a Nash equilibrium solution in a decentralized manner. We demonstrate the efficacy of the proposed approach on an electric vehicle charging control problem.Comment: Preprint submitted to IEEE Transactions on Automatic Contro

Gradient-Bounded Dynamic Programming with Submodular and Concave Extensible Value Functions

We consider dynamic programming problems with finite, discrete-time horizons and prohibitively high-dimensional, discrete state-spaces for direct computation of the value function from the Bellman equation. For the case that the value function of the dynamic program is concave extensible and submodular in its state-space, we present a new algorithm that computes deterministic upper and stochastic lower bounds of the value function similar to dual dynamic programming. We then show that the proposed algorithm terminates after a finite number of iterations. Finally, we demonstrate the efficacy of our approach on a high-dimensional numerical example from delivery slot pricing in attended home delivery.Comment: 6 pages, 2 figures, accepted for IFAC World Congress 202

Regularized Jacobi iteration for decentralized convex optimization with separable constraints

We consider multi-agent, convex optimization programs subject to separable constraints, where the constraint function of each agent involves only its local decision vector, while the decision vectors of all agents are coupled via a common objective function. We focus on a regularized variant of the so called Jacobi algorithm for decentralized computation in such problems. We first consider the case where the objective function is quadratic, and provide a fixed-point theoretic analysis showing that the algorithm converges to a minimizer of the centralized problem. Moreover, we quantify the potential benefits of such an iterative scheme by comparing it against a scaled projected gradient algorithm. We then consider the general case and show that all limit points of the proposed iteration are optimal solutions of the centralized problem. The efficacy of the proposed algorithm is illustrated by applying it to the problem of optimal charging of electric vehicles, where, as opposed to earlier approaches, we show convergence to an optimal charging scheme for a finite, possibly large, number of vehicles