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 Convex Framework for Epidemic Control in Networks

With networks becoming pervasive, research attention on dynamics of epidemic models in networked populations has increased. While a number of well understood epidemic spreading models have been developed, little to no attention has been paid to epidemic control strategies; beyond heuristics usually based on network centrality measures. Since epidemic control resources are typically limited, the problem of optimally allocating resources to control an outbreak becomes of interest. Existing literature considered homogeneous networks, limited the discussion to undirected networks, and largely proposed network centrality-based resource allocation strategies. In this thesis, we consider the well-known Susceptible-Infected-Susceptible spreading model and study the problem of minimum cost resource allocation to control an epidemic outbreak in a networked population. First, we briefly present a heuristic that outperforms network centrality-based algorithms on a stylized version of the problem previously studied in the literature. We then solve the epidemic control problem via a convex optimization framework on weighted, directed networks comprising heterogeneous nodes. Based on our spreading model, we express the problem of controlling an epidemic outbreak in terms of spectral conditions involving the Perron-Frobenius eigenvalue. This enables formulation of the epidemic control problem as a Geometric Program (GP), for which we derive a convex characterization guaranteeing existence of an optimal solution. We consider two formulations of the epidemic control problem -- the first seeks an optimal vaccine and antidote allocation strategy given a constraint on the rate at which the epidemic comes under control. The second formulation seeks to find an optimal allocation strategy given a budget on the resources. The solution framework for both formulations also allows for control of an epidemic outbreak on networks that are not necessarily strongly connected. The thesis further proposes a fully distributed solution to the epidemic control problem via a Distributed Alternating Direction Method of Multipliers (ADMM) algorithm. Our distributed solution enables each node to locally compute its optimum allocation of vaccines and antidotes needed to collectively globally contain the spread of an outbreak, via local exchange of information with its neighbors. Contrasting previous literature, our problem is a constrained optimization problem associated with a directed network comprising non-identical agents. For the different problem formulations considered, illustrations that validate our solutions are presented. This thesis, in sum, proposes a paradigm shift from heuristics towards a convex framework for contagion control in networked populations