We consider a network of agents that are cooperatively solving a global optimization problem, where the objective function is the sum of privately known local objective functions of the agents and the decision variables are coupled via linear constraints. Recent literature focused on special cases of this formulation and studied their distributed solution through either subgradient based methods with O(1/ √ k) rate of convergence (where k is the iteration number) or Alternating Direction Method of Multipliers (ADMM) based methods, which require a synchronous implementation and a globally known order on the agents. In this paper, we present a novel asynchronous ADMM based distributed method for the general formulation and show that it converges at the rate O (1/k).

We consider a network of agents that are cooperatively solving a global unconstrained optimization problem, where the objective function is the sum of privately known local objective functions of the agents. Recent literature on distributed optimization methods for solving this problem focused on subgradient based methods, which typically converge at the rate O 1 √ k , where k is the number of iterations. In this paper, we introduce a new distributed optimization algorithm based on Alternating Direction Method of Multipliers (ADMM), which is a classical method for sequentially decomposing optimization problems with coupled constraints. We show that this algorithm converges at the rate O 1 k .

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Most existing work uses dual decomposition and first-order methods to solve Network Utility Maximization (NUM) problems in a distributed manner, which suffer from slow rate of convergence properties. This paper develops an alternative distributed Newton-type fast converging algorithm for solving NUM problems with self-concordant utility functions. By using novel matrix splitting techniques, both primal and dual updates for the Newton step can be computed using iterative schemes in a decentralized manner. We propose a stepsize rule and provide a distributed procedure to compute it in finitely many iterations. The key feature of our direction and stepsize computation schemes is that both are implemented using the same distributed information exchange mechanism employed by first order methods. We show that even when the Newton direction and the stepsize in our method are computed within some error (due to finite truncation of the iterative schemes), the resulting objective function value still converges superlinearly in terms of primal iterations to an explicitly characterized error neighborhood. Simulation results demonstrate significant convergence rate improvement of our algorithm relative to the existing first-order methods based on dual decomposition.

Methods for distributed optimization have received significant attention in recent years owing to their wide applicability in various domains including machine learning, robotics and sensor networks. A distributed optimization method typically consists of two key components: communication and computation. More specifically, at every iteration (or every several iterations) of a distributed algorithm, each node in the network requires some form of information exchange with its neighboring nodes (communication) and the computation step related to a (sub)-gradient (computation). The standard way of judging an algorithm via only the number of iterations overlooks the complexity associated with each iteration. Moreover, various applications deploying distributed methods may prefer a different composition of communication and computation.Motivated by this discrepancy, in this work we propose an adaptive cost framework which adjusts the cost measure depending on the features of various applications. We present a flexible algorithmic framework, where communication and computation steps are explicitly decomposed to enable algorithm customization for various applications. We apply this framework to the well-known distributed gradient descent (DGD) method, and show that the resulting customized algorithms, which we call DGD t , NEAR-DGD t and NEAR-DGD + , compare favorably to their base algorithms, both theoretically and empirically. The proposed NEAR-DGD + algorithm is an exact first-order method where the communication and computation steps are nested, and when the number of communication steps is adaptively increased, the method converges to the optimal solution. We test the performance and illustrate the flexibility of the methods, as well as practical variants, on quadratic functions and classification problems that arise in machine learning, in terms of iterations, gradient evaluations, communications and the proposed cost framework.

Pursuit is a familiar mechanical activity that humans and animals engage in-athletes chasing balls, predators seeking prey and insects manoeuvring in aerial territorial battles. In this paper, we discuss and compare strategies for pursuit, the occurrence in nature of a strategy known as motion camouflage, and some evolutionary arguments to support claims of prevalence of this strategy, as opposed to alternatives. We discuss feedback laws for a pursuer to realize motion camouflage, as well as two alternative strategies. We then set up a discrete-time evolutionary game to model competition among these strategies. This leads to a dynamics in the probability simplex in three dimensions, which captures the mean-field aspects of the evolutionary game. The analysis of this dynamics as an ascent equation solving a linear programming problem is consistent with observed behaviour in Monte Carlo experiments, and lends support to an evolutionary basis for prevalence of motion camouflage.

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A scalable framework is developed to allocate radio resources across a large number of densely deployed small cells with given traffic statistics on a slow timescale. Joint user association and spectrum allocation is first formulated as a convex optimization problem by dividing the spectrum among all possible transmission patterns of active access points (APs). To improve scalability with the number of APs, the problem is reformulated using local patterns of interfering APs. To maintain global consistency among local patterns, inter-cluster interaction is characterized as hyper-edges in a hyper-graph with nodes corresponding to neighborhoods of APs. A scalable solution is obtained by iteratively solving a convex optimization problem for bandwidth allocation with reduced complexity and constructing a global spectrum allocation using hyper-graph coloring. Numerical results demonstrate the proposed solution for a network with 100 APs and several hundred user equipments. For a given quality of service (QoS), the proposed scheme can increase the network capacity several fold compared to assigning each user to the strongest AP with full-spectrum reuse.