Bahman Gharesifard

Distributed Continuous-Time Convex Optimization on Weight-Balanced Digraphs

This technical note studies the continuous-time distributed optimization of a sum of convex functions over directed graphs. Contrary to what is known in the consensus literature, where the same dynamics works for both undirected and directed scenarios, we show that the consensus-based dynamics that solves the continuous-time distributed optimization problem for undirected graphs fails to converge when transcribed to the directed setting. This study sets the basis for the design of an alternative distributed dynamics which we show is guaranteed to converge, on any strongly connected weight-balanced digraph, to the set of minimizers of a sum of convex differentiable functions with globally Lipschitz gradients. Our technical approach combines notions of invariance and cocoercivity with the positive definiteness properties of graph matrices to establish the results.

When does a digraph admit a doubly stochastic adjacency matrix

Digraphs with doubly stochastic adjacency matrices play an essential role in a variety of cooperative control problems including distributed averaging, optimization, and gossiping. In this paper, we fully characterize the class of digraphs that admit an edge weight assignment that makes the digraph adjacency matrix doubly stochastic. As a by-product of our approach, we also unveil the connection between weight-balanced and doubly stochastic adjacency matrices. Several examples illustrate our results.

Distributed convergence to Nash equilibria in two-network zero-sum games

This paper considers a class of strategic scenarios in which two networks of agents have opposing objectives with regard to the optimization of a common objective function. In the resulting zero-sum game, individual agents collaborate with neighbors in their respective network and have only partial knowledge of the state of the agents in the other network. For the case when the interaction topology of each network is undirected, we synthesize a distributed saddle-point strategy and establish its convergence to the Nash equilibrium for the class of strictly concave-convex and locally Lipschitz objective functions. We also show that this dynamics does not converge in general if the topologies are directed. This justifies the introduction, in the directed case, of a generalization of this distributed dynamics which we show converges to the Nash equilibrium for the class of strictly concave-convex differentiable functions with globally Lipschitz gradients. The technical approach combines tools from algebraic graph theory, nonsmooth analysis, set-valued dynamical systems, and game theory.

Stability properties of infected networks with low curing rates

In this work, we analyze the stability properties of a recently proposed dynamical system that describes the evolution of the probability of infection in a network. We show that this model can be viewed as a concave game among the nodes. This characterization allows us to provide a simple condition, that can be checked in a distributed fashion, for stabilizing the origin. When the curing rates at the nodes are low, a residual infection stays within the network. Using properties of Hurwitz Mertzel matrices, we show that the residual epidemic state is locally exponentially stable. We also demonstrate that this state is globally asymptotically stable. Furthermore, we investigate the problem of stabilizing the network when the curing rates of a limited number of nodes can be controlled. In particular, we characterize the number of controllers required for a class of undirected graphs. Several simulations demonstrate our results.

Graph Controllability Classes for the Laplacian Leader-Follower Dynamics

In this paper, we consider the problem of obtaining graph-theoretic characterizations of controllability for the Laplacian-based leader-follower dynamics. Our developments rely on the notion of graph controllability classes, namely, the classes of essentially controllable, completely uncontrollable, and conditionally controllable graphs. In addition to the topology of the underlying graph, the controllability classes rely on the specification of the control vectors; our particular focus is on the set of binary control vectors. The choice of binary control vectors is naturally adapted to the Laplacian dynamics, as it captures the case when the controller is unable to distinguish between the followers and, moreover, controllability properties are invariant under binary complements. We prove that the class of essentially controllable graphs is a strict subset of the class of asymmetric graphs and provide numerical results that suggests that the ratio of essentially controllable graphs to asymmetric graphs increases as the number of vertices increases. Although graph symmetries play an important role in graph-theoretic characterizations of controllability, we provide an explicit class of asymmetric graphs that are completely uncontrollable, namely the class of block graphs of Steiner triple systems. We prove that for graphs on four and five vertices, a repeated Laplacian eigenvalue is a necessary condition for complete uncontrollability but, however, show through explicit examples that for eight and nine vertices, a repeated eigenvalue is not necessary for complete uncontrollability. For the case of conditional controllability, we give an easily checkable necessary condition that identifies a class of binary control vectors that result in a two-dimensional controllable subspace. Several constructive examples demonstrate our results.

Continuous-time distributed convex optimization on weight-balanced digraphs

This paper studies the continuous-time distributed optimization of a sum of convex functions over directed graphs. Contrary to what is known in the consensus literature, where the same dynamics works for both undirected and directed scenarios, we show that the consensus-based dynamics that solves the continuous-time distributed optimization problem for undirected graphs fails to converge when transcribed to the directed setting. This study sets the basis for the design of an alternative distributed dynamics which we show is guaranteed to converge, on any strongly connected weight-balanced digraph, to the set of minimizers of a sum of convex differentiable functions with globally Lipschitz gradients. Our technical approach combines notions of invariance and cocoercivity with the positive definiteness properties of graph matrices to establish the results.

Distributed strategies for making a digraph weight-balanced

A digraph is weight-balanced if, at each node, the sum of the weights of the incoming edges (in-degree) equals the sum of the weights of the outgoing edges (out-degree). Weight-balanced digraphs play an important role in a variety of cooperative control problems, including formation control, distributed averaging and optimization. We call a digraph weight-balanceable if it admits an edge weight assignment that makes it weight-balanced. It is known that semiconnectedness is a necessary and sufficient condition for a digraph to be weight-balanceable. However, to our knowledge, the available approaches to compute the appropriate set of weights are centralized. In this paper, we propose a distributed algorithm running synchronously on a directed communication network that allows individual agents to balance their in- and out-degrees. We also develop a systematic centralized algorithm for constructing a weight-balanced digraph and compute its time complexity. Finally, we modify the distributed procedure to design an algorithm which is distributed over the mirror digraph and has a time complexity much smaller than the centralized algorithm.

Price-based distributed control for networked plug-in electric vehicles

We introduce a framework for controlling the charging and discharging processes of plug-in electric vehicles (PEVs) via pricing strategies. Our framework consists of a hierarchical decision-making setting with two layers, which we refer to as aggregator layer and retail market layer. In the aggregator layer, there is a set of aggregators that are requested (and will be compensated for) to provide certain amount of energy over a period of time. In the retail market layer, the aggregator offers some price for the energy that PEVs may provide; the objective is to choose a pricing strategy to incentivize the PEVs so as they collectively provide the amount of energy that the aggregator has been asked for. The focus of this paper is on the decision-making process that takes places in the retail market layer, where we assume that each individual PEV is a price-anticipating decision-maker. We cast this decision-making process as a game, and provide conditions on the pricing strategy of the aggregator under which this game has a unique Nash equilibrium. We propose a distributed consensus-based iterative algorithm through which the PEVs can seek for this Nash equilibrium. Numerical simulations are included to illustrate our results.

Saddle-Point Dynamics: Conditions for Asymptotic Stability of Saddle Points

This paper considers continuously differentiable functions of two vector variables that have (possibly a continuum of) min-max saddle points. We study the asymptotic convergence properties of the associated saddle-point dynamics (gradient-descent in the first variable and gradient-ascent in the second one). We identify a suite of complementary conditions under which the set of saddle points is asymptotically stable under the saddle-point dynamics. Our first set of results is based on the convexity-concavity of the function defining the saddle-point dynamics to establish the convergence guarantees. For functions that do not enjoy this feature, our second set of results relies on properties of the linearization of the dynamics, the function along the proximal normals to the saddle set, and the linearity of the function in one variable. We also provide global versions of the asymptotic convergence results. Various examples illustrate our discussion.

Stability of epidemic models over directed graphs

We study the stability properties of a susceptible-infected-susceptible (SIS) diffusion model, so-called the