Francesca Parise

Decentralized Convergence to Nash Equilibria in Constrained Deterministic Mean Field Control

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 influenced 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.

Mean field constrained charging policy for large populations of Plug-in Electric Vehicles

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.

Iterative experiment design guides the characterization of a light-inducible gene expression circuit

Significance System identification addresses the problem of identifying unknown model parameters from measured data of a real system. In the case of biochemical reaction networks, the available measurements are typically sparse because of technical and/or economic reasons. Therefore, it is of paramount importance to maximize the information that can be gained by each experiment. Here, we apply a systematic design scheme for single-cell experiments based on information theoretic criteria. For the considered light-inducible gene expression circuit, we show that this scheme allows one to precisely identify model parameters that were practically unidentifiable from data measured in random experiments. This result provides evidence that optimal experiment design is a key requirement for the successful identification of biochemical reaction networks. 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.

Distributed computation of generalized Nash equilibria in quadratic aggregative games with affine coupling constraints

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.

A Mean Field control approach for demand side management of large populations of Thermostatically Controlled Loads

This paper presents a Mean Field (MF) control approach for demand side management of large populations of flexible electric loads, such as electrical cooling/heating appliances, called Thermostatically Controlled Loads (TCLs). We model the switching dynamics of each individual TCL as the solution of a local optimization problem, characterized by individual cost function, comfort constraints, cooling/heating rates and external temperature. We consider that a central utility company broadcasts macroscopic incentives to steer the overall TCL population towards a convenient equilibrium, to avoid power demand peaks due to possible synchronization of the TCL duty cycles. To find such pricing schemes we propose an iterative algorithm where, at every step, a simple model-free feedback law is used to update the incentives, given the current aggregate demand of the TCL population only. The convergence of such algorithm is ensured for any population size, even in the presence of heterogeneous convex constraints. We illustrate our MF control approach via numerical analysis.

Constrained linear quadratic deterministic mean field control: Decentralized convergence to Nash equilibria in large populations of heterogeneous agents

This paper considers the linear quadratic deterministic mean field control problem for large populations of heterogeneous agents, subject to convex state and input constraints, and coupled via a quadratic cost function which depends on the average population state. To control the optimal responses of the rational agents to a Nash equilibrium, we propose feedback iterative solutions based on operator theory arguments. Contrary to the state of the art, global convergence is ensured, under mild sufficient conditions on the matrices defining the cost functions, and not on the convex constraints.

On constrained mean field control for large populations of heterogeneous agents: Decentralized convergence to Nash equilibria

We consider mean field games in a large population of heterogeneous agents subject to convex constraints and coupled by a quadratic cost, which depends on the average population behavior. The problem of steering such population to a Nash equilibrium is usually addressed in the (mean field control) literature by formulating an iterative game between the agents and a central coordinator, that broadcasts at every step the average population behavior. Here we generalize this approach by allowing the central operator to filter such signal using a feedback mapping. We propose different classes of feedback mappings and we derive sufficient conditions guaranteeing convergence to a ε-Nash equilibrium, even for cases when the standard approach fails. We show that the deviation ε of each agent from its optimal cost, decreases at least linearly to zero with the increase of the population size. Contrary to the state of the art, the proposed approach guarantees convergence in the presence of heterogeneous convex constraints for the agents. Finally, we show how these results can be applied to regulate in a decentralized fashion the charging process of a large population of plug-in electric vehicles. Our findings give theoretical support and extend previous literature results.

On the reachable set of the controlled gene expression system

In this paper we investigate the reachable set of a standard stochastic model of controlled gene expression. Specifically, we explore what values of the protein mean and variance are achievable using the available external input, that is, the mRNA production rate. We proceed by constructing invariant sets in two-dimensional projections of the state space. We then use these sets to construct an outer approximation of the reachable region for the protein mean and variance. This can be computed solving a one-dimensional optimization problem and is tight enough to show that it is not possible, using such control input, to arbitrarily reduce the variance while maintaining a high mean. The lower bound on the variance derived with our approach turns out to be much higher than the one available in the literature for the case when both the mRNA production and degradation rate are controlled.

Grey-box techniques for the identification of a controlled gene expression model

The aim of this paper is to propose a computationally efficient technique for the identification of stochastic biochemical networks, involving only zero and first order reactions, from distribution measurements of the cell population. The moments of the species in such networks evolve according to an affine system, hence the use of grey-box identification methods is suggested. The performance of existing methods and of a new method, based on the transfer function computation, is compared using as benchmark a standard gene expression model. The developed discussion is of interest for the general grey-box identification problem.

On the use of hyperplane methods to compute the reachable set of controlled stochastic biochemical reaction networks

A fundamental question in the study of stochastic biochemical reaction networks is what values of mean and variance of the species present in the network are obtainable by perturbing the system with an external input. Here, we propose a computationally efficient technique to answer this question, for networks involving zero and first order reactions. Specifically, we adopt the hyperplane method to compute inner and outer approximations of the reachable set of the linear system describing the moments evolution. A remarkable feature of this approach is that it allows one to easily compute projections of the reachable set for pairs of species of interest, without requiring the computation of the full reachable set, which can be prohibitive for large networks. To illustrate the benefits of this method we consider a standard controlled gene expression model involving two species: the mRNA and the corresponding protein. We verify that the proposed approach leads to estimates of the reachable set, for the protein mean and variance, that are more accurate than those available in the literature and that are consistent with experimental data.