A duality-based approach for distributed min-max optimization with application to demand side management

In this paper we consider a distributed optimization scenario in which a set of processors aims at minimizing the maximum of a collection of "separable convex functions" subject to local constraints. This set-up is motivated by peak-demand minimization problems in smart grids. Here, the goal is to minimize the peak value over a finite horizon with: (i) the demand at each time instant being the sum of contributions from different devices, and (ii) the local states at different time instants being coupled through local dynamics. The min-max structure and the double coupling (through the devices and over the time horizon) makes this problem challenging in a distributed set-up (e.g., well-known distributed dual decomposition approaches cannot be applied). We propose a distributed algorithm based on the combination of duality methods and properties from min-max optimization. Specifically, we derive a series of equivalent problems by introducing ad-hoc slack variables and by going back and forth from primal and dual formulations. On the resulting problem we apply a dual subgradient method, which turns out to be a distributed algorithm. We prove the correctness of the proposed algorithm and show its effectiveness via numerical computations.Comment: arXiv admin note: substantial text overlap with arXiv:1611.0916

A Decentralized Lifetime Maximization Algorithm for Distributed Applications in Wireless Sensor Networks

We consider the scenario of a Wireless Sensor Networks (WSN) where the nodes are equipped with a programmable middleware that allows for quickly deploying different applications running on top of it so as to follow the changing ambient needs. We then address the problem of finding the optimal deployment of the target applications in terms of network lifetime. We approach the problem considering every possible decomposition of an application's sensing and computing operations into tasks to be assigned to each infrastructure component. The contribution of energy consumption due to the energy cost of each task is then considered into local cost functions in each node, allowing us to evaluate the viability of the deployment solution. The proposed algorithm is based on an iterative and asynchronous local optimization of the task allocations between neighboring nodes that increases the network lifetime. Simulation results show that our framework leads to considerable energy saving with respect to both sink-oriented and cluster-oriented deployment approaches, particularly for networks with high node densities and non-uniform energy consumption or initial battery charge

Consensus Algorithms for Estimation and Discrete Averaging in Networked Control Systems

In this thesis several topics on consensus and gossip algorithms for multi-agent systems are addressed. An agent is a dynamical system that can be fully described by a state-space representation of its dynamics. A multi-agent system is a network of agents whose pattern of interactions or couplings is described by a graph. Consensus problems in multi-agent systems consist in the study of local interaction rules between the agents such that as global emergent behavior the network converges to the so called "consensus" or "agreement" state where the value of each agent's state is the same and it is possibly a function of the initial network state, for instance the average. A consensus algorithm is thus a set of local interaction rules that solve the consensus problem under some assumptions on the network topology. A gossip algorithm is a set of local state update rules between the agents that, disregarding their objective, are supposed to be implemented in a totally asynchronous way between pairs of neighboring agents, thus resembling the act of "gossiping" in a crowd of people. In this thesis several algorithms based on gossip that solve the consensus and other related problems are presented. In the �first part, several solutions to the consensus problem based on gossip under different sets of assumptions are proposed. In the fi�rst case, it is assumed that the state of the agents is discretized and represents a collection of tasks of different size. In the second case, under the same discretization assumptions of the �rst case, it is assumed that the network is represented by a Hamiltonian graph and it is shown how under this assumption the convergence speed can be improved. In the third case, a solution for the consensus problem for networks represented by arbitrary strongly connected directed graphs is proposed, assuming that the state of the agents is a real number. In the fourth case, a coordinate-free consensus algorithm based on gossip is designed and applied to a network of vehicles able to sense the relative distance between each other but with no access to absolute position information or to a common coordinate system. The proposed algorithm is then used to build in a decentralized way a common reference frame for the network of vehicles. In the second part, a novel local interaction rule based on the consensus equation is proposed together with an algorithm to estimate in a decentralized way the spectrum of the Laplacian matrix that encodes the network topology. As emergent behavior, each agent's state oscillates only at frequencies corresponding to the eigenvalues of the Laplacian matrix thus mapping the spectrum estimation problem into a signal processing problem solvable using the Fourier Transform. It is further shown that the constant component of the emergent behavior in the frequency domain solves the consensus on the average problem. The spectrum estimation algorithm is then applied to leader-follower networks of mobile vehicles to infer in a decentralized way properties such as controllability, osservability and other topological features of the network such as its topology. Finally, a fault detection and recovery technique for sensor networks based on the so called motion-probes is presented to address the inherent lack of robustness against outlier agents in networks implementing consensus algorithms to solve the distributed averaging problem

Secure rendezvous and static containment in multi-agent systems with adversarial intruders

In this paper we propose a novel distributed local interaction protocol for networks of multi-agent systems (MASs) in a multi-dimensional space under directed time-varying graph with the objective to achieve secure rendezvous or static containment within the convex hull of a set of leader agents. We consider the scenario where a set of anonymous adversarial agents may intrude the network (or may be hijacked by a cyber-attack) and show that the proposed strategy guarantees the achievement of the global objective despite the continued influence of the adversaries which cannot be detected nor identified by the collaborative agents. We characterize the convergence properties of the proposed protocol in terms of the characteristics of the underlying network topology of the multi-agent system. Numerical simulations and examples corroborate the theoretical results

Consensus in multi-agent systems with non-periodic sampled-data exchange and uncertain network topology

In this paper consensus in second-order multi-agent systems with a non-periodic sampled-data exchange among agents is investigated. The sampling is random with bounded inter-sampling intervals. It is assumed that each agent has exact knowledge of its own state at any time instant. The considered local interaction rule is PD-type. Sufficient conditions for stability of the consensus protocol to a time-invariant value are derived based on LMIs. Such conditions only require the knowledge of the connectivity of the graph modeling the network topology. Numerical simulations are presented to corroborate the theoretical results.Comment: arXiv admin note: substantial text overlap with arXiv:1407.300

Dynamic Max-Consensus and Size Estimation of Anonymous Multi-Agent Networks

In this paper we propose a novel consensus protocol for discrete-time multi-agent systems (MAS), which solves the dynamic consensus problem on the max value, i.e., the dynamic max-consensus problem. In the dynamic max-consensus problem to each agent is fed a an exogenous reference signal, the objective of each agent is to estimate the instantaneous and time-varying value of the maximum among the signals fed to the network, by exploiting only local and anonymous interactions among the agents. The absolute and relative tracking error of the proposed distributed control protocol is theoretically characterized and is shown to be bounded and by tuning its parameters it is possible to trade-off convergence time for steady-state error. The dynamic Max-consensus algorithm is then applied to solve the distributed size estimation problem in a dynamic setting where the size of the network is time-varying during the execution of the estimation algorithm. Numerical simulations are provided to corroborate the theoretical analysis

A duality-based approach for distributed min-max optimization with application to demand side management

In this paper we consider a distributed optimiza- tion scenario in which a set of processors aims at minimizing the maximum of a collection of “separable convex functions” subject to local constraints. This set-up is motivated by peak- demand minimization problems in smart grids. Here, the goal is to minimize the peak value over a finite horizon with: (i) the demand at each time instant being the sum of contributions from different devices, and (ii) the local states at different time instants being coupled through local dynamics. The min-max structure and the double coupling (through the devices and over the time horizon) makes this problem challenging in a distributed set-up (e.g., well-known distributed dual decompo- sition approaches cannot be applied). We propose a distributed algorithm based on the combination of duality methods and properties from min-max optimization. Specifically, we derive a series of equivalent problems by introducing ad-hoc slack variables and by going back and forth from primal and dual formulations. On the resulting problem we apply a dual sub- gradient method, which turns out to be a distributed algorithm. We prove the correctness of the proposed algorithm and show its effectiveness via numerical computation