Distributed Nonconvex Multiagent Optimization Over Time-Varying Networks

We study nonconvex distributed optimization in multiagent networks where the communications between nodes is modeled as a time-varying sequence of arbitrary digraphs. We introduce a novel broadcast-based distributed algorithmic framework for the (constrained) minimization of the sum of a smooth (possibly nonconvex and nonseparable) function, i.e., the agents' sum-utility, plus a convex (possibly nonsmooth and nonseparable) regularizer. The latter is usually employed to enforce some structure in the solution, typically sparsity. The proposed method hinges on Successive Convex Approximation (SCA) techniques coupled with i) a tracking mechanism instrumental to locally estimate the gradients of agents' cost functions; and ii) a novel broadcast protocol to disseminate information and distribute the computation among the agents. Asymptotic convergence to stationary solutions is established. A key feature of the proposed algorithm is that it neither requires the double-stochasticity of the consensus matrices (but only column stochasticity) nor the knowledge of the graph sequence to implement. To the best of our knowledge, the proposed framework is the first broadcast-based distributed algorithm for convex and nonconvex constrained optimization over arbitrary, time-varying digraphs. Numerical results show that our algorithm outperforms current schemes on both convex and nonconvex problems.Comment: Copyright 2001 SS&C. Published in the Proceedings of the 50th annual Asilomar conference on signals, systems, and computers, Nov. 6-9, 2016, CA, US

Distributed Big-Data Optimization via Block-Iterative Convexification and Averaging

In this paper, we study distributed big-data nonconvex optimization in multi-agent networks. We consider the (constrained) minimization of the sum of a smooth (possibly) nonconvex function, i.e., the agents' sum-utility, plus a convex (possibly) nonsmooth regularizer. Our interest is in big-data problems wherein there is a large number of variables to optimize. If treated by means of standard distributed optimization algorithms, these large-scale problems may be intractable, due to the prohibitive local computation and communication burden at each node. We propose a novel distributed solution method whereby at each iteration agents optimize and then communicate (in an uncoordinated fashion) only a subset of their decision variables. To deal with non-convexity of the cost function, the novel scheme hinges on Successive Convex Approximation (SCA) techniques coupled with i) a tracking mechanism instrumental to locally estimate gradient averages; and ii) a novel block-wise consensus-based protocol to perform local block-averaging operations and gradient tacking. Asymptotic convergence to stationary solutions of the nonconvex problem is established. Finally, numerical results show the effectiveness of the proposed algorithm and highlight how the block dimension impacts on the communication overhead and practical convergence speed

A Stochastic Gradient Method with Mesh Refinement for PDE Constrained Optimization under Uncertainty

Models incorporating uncertain inputs, such as random forces or material parameters, have been of increasing interest in PDE-constrained optimization. In this paper, we focus on the efficient numerical minimization of a convex and smooth tracking-type functional subject to a linear partial differential equation with random coefficients and box constraints. The approach we take is based on stochastic approximation where, in place of a true gradient, a stochastic gradient is chosen using one sample from a known probability distribution. Feasibility is maintained by performing a projection at each iteration. In the application of this method to PDE-constrained optimization under uncertainty, new challenges arise. We observe the discretization error made by approximating the stochastic gradient using finite elements. Analyzing the interplay between PDE discretization and stochastic error, we develop a mesh refinement strategy coupled with decreasing step sizes. Additionally, we develop a mesh refinement strategy for the modified algorithm using iterate averaging and larger step sizes. The effectiveness of the approach is demonstrated numerically for different random field choices