This paper considers decentralized control and optimization methodologies for large populations of systems, consisting of several agents with different individual behaviors, constraints and interests, and affected by the aggregate behavior of the overall population. For such large-scale systems, the theory of aggregative and mean field games has been established and successfully applied in various scientific disciplines. While the existing literature addresses the case of unconstrained agents, we formulate deterministic mean field control problems in the presence of heterogeneous convex constraints for the individual agents, for instance arising from agents with linear dynamics subject to convex state and control constraints. We propose several model-free feedback iterations to compute in a decentralized fashion a mean field Nash equilibrium in the limit of infinite population size. We apply our methods to the constrained linear quadratic deterministic mean field control problem and to the constrained mean field charging control problem for large populations of plug-in electric vehicles.

We analyse deterministic aggregative games, with large but finite number of players, that are subject to both local and coupling constraints. Firstly, we derive sufficient conditions for the existence of a generalized Nash equilibrium, by using the theory of variational inequalities together with the specific structure of the objective functions and constraints. Secondly, we present a coordination scheme, belonging to the class of asymmetric projection algorithms, and we prove that it converges R-linearly to a generalized Nash equilibrium. To this end, we extend the available results on asymmetric projection algorithms to our setting. Finally, we show that the proposed scheme can be implemented in a decentralized fashion and it is suitable for the analysis of large populations. Our theoretical results are applied to the problem of charging a fleet of plug-in electric vehicles, in the presence of capacity constraints coupling the individual demands.

We consider the framework of aggregative games, in which the cost function of each agent depends on his own strategy and on the average population strategy. As first contribution, we investigate the relations between the concepts of Nash and Wardrop equilibria. By exploiting a characterization of the two equilibria as solutions of variational inequalities, we bound their distance with a decreasing function of the population size. As second contribution, we propose two decentralized algorithms that converge to such equilibria and are capable of coping with constraints coupling the strategies of different agents. Finally, we study the applications of charging of electric vehicles and of route choice on a road network.

Constrained charging control of large populations of Plug-in Electric Vehicles (PEVs) is addressed using mean field game theory. We consider PEVs as heterogeneous agents, with different charging constraints (plug-in times and deadlines). The agents minimize their own charging cost, but are weakly coupled by the common electricity price. We propose an iterative algorithm that, in the case of an infinite population, converges to the Nash equilibrium associated with a related decentralized optimization problem. In this way we approximate the centralized optimal solution, which in the unconstrained case fills the overnight power demand valley, via a decentralized procedure. The benefits of the proposed formulation in terms of convergence behavior and overall charging cost are illustrated through numerical simulations

As relational datasets modeled as graphs keep increasing in size and their data-acquisition is permeated by uncertainty, graph-based analysis techniques can become computationally and conceptually challenging. In particular, node centrality measures rely on the assumption that the graph is perfectly known -a premise not necessarily fulfilled for large, uncertain networks. Accordingly, centrality measures may fail to faithfully extract the importance of nodes in the presence of uncertainty. To mitigate these problems, we suggest a statistical approach based on graphon theory: we introduce formal definitions of centrality measures for graphons and establish their connections to classical graph centrality measures. A key advantage of this approach is that centrality measures defined at the modeling level of graphons are inherently robust to stochastic variations of specific graph realizations. Using the theory of linear integral operators, we define degree, eigenvector, Katz and PageRank centrality functions for graphons and establish concentration inequalities demonstrating that graphon centrality functions arise naturally as limits of their counterparts defined on sequences of graphs of increasing size. The same concentration inequalities also provide high-probability bounds between the graphon centrality functions and the centrality measures on any sampled graph, thereby establishing a measure of uncertainty of the measured centrality score.

Systems biology rests on the idea that biological complexity can be better unraveled through the interplay of modeling and experimentation. However, the success of this approach depends critically on the informativeness of the chosen experiments, which is usually unknown a priori. Here, we propose a systematic scheme based on iterations of optimal experiment design, flow cytometry experiments, and Bayesian parameter inference to guide the discovery process in the case of stochastic biochemical reaction networks. To illustrate the benefit of our methodology, we apply it to the characterization of an engineered light-inducible gene expression circuit in yeast and compare the performance of the resulting model with models identified from nonoptimal experiments. In particular, we compare the parameter posterior distributions and the precision to which the outcome of future experiments can be predicted. Moreover, we illustrate how the identified stochastic model can be used to determine light induction patterns that make either the average amount of protein or the variability in a population of cells follow a desired profile. Our results show that optimal experiment design allows one to derive models that are accurate enough to precisely predict and regulate the protein expression in heterogeneous cell populations over extended periods of time. T he use of quantitative mathematical models to investigate biochemical reaction networks is nowadays common practice. Typically, models are built based on the available biological knowledge and used to generate hypotheses, which are then refined or invalidated through experimentation. For this process to be successful, it is of paramount importance to design and perform experiments that yield the information required to identify the model under consideration. Optimal experiment design techniques have been extensively studied for ordinary differential equation models (1-5), which are typically used to describe the average behavior of cell populations (6-8). With the development of high-throughput measurement techniques, such as flow cytometry, it has, however, become evident that restricting the attention only to the average population behavior neglects the potentially valuable information contained in the full population distribution (9-11). This additional information can be captured by stochastic models. Recently, methods for parameter inference (12-15) and optimal experiment design (16, 17) for stochastic models have been developed and applied to a number of biological systems (12, 18). However, a systematic characterization procedure that exploits the information gained from each performed experiment has not yet been fully developed or experimentally validated.Here, we provide the first study, to our knowledge, in which a noisy biochemical reaction network is characterized and ultimately also controlled through iterations of optimally designed flow cytometry experiments and stochastic modeling. Specifically, we consider a gene expression circuit in yeast that has been eng...

We consider quasi-aggregative games for large populations of heterogeneous agents, whose interaction is determined by an underlying communication network. Specifically, each agent minimizes a quadratic cost function, which depends on its own strategy and on a convex combination of the strategies of its neighbors, and is subject to heterogeneous convex constraints. We suggest two distributed algorithms that can be implemented to steer the best responses of the rational agents to a Nash equilibrium configuration. The convergence of these schemes is guaranteed under different sufficient conditions depending on the matrices defining the agents' cost functions and on the communication network

We consider the framework of average aggregative games, where the cost function of each agent depends on his own strategy and on the average population strategy. We focus on the case in which the agents are coupled not only via their cost functions, but also via constraints coupling their strategies. We propose a distributed algorithm that achieves an almost-Nash equilibrium by requiring only local communications of the agents, as specified by a sparse communication network. The proof of convergence of the algorithm relies on the auxiliary class of network aggregative games and exploits a novel result of parametric convergence of variational inequalities, which is applicable beyond the context of games. We apply our theoretical findings to a multi-market Cournot game with transportation costs and maximum market capacity.