Exact Convergence of Gradient-Free Distributed Optimization Method in a Multi-Agent System

In this paper, a gradient-free algorithm is proposed for a set constrained distributed optimization problem in a multi-agent system under a directed communication network. For each agent, a pseudo-gradient is designed locally and utilized instead of the true gradient information to guide the decision variables update. Compared with most gradient-free optimization methods where a doubly-stochastic weighting matrix is usually employed, this algorithm uses a row-stochastic matrix plus a column-stochastic matrix, and is able to achieve exact asymptotic convergence to the optimal solution.

[1]  Martin J. Wainwright,et al.  Randomized Smoothing for Stochastic Optimization , 2011, SIAM J. Optim..

[2]  Sonia Martínez,et al.  On Distributed Convex Optimization Under Inequality and Equality Constraints , 2010, IEEE Transactions on Automatic Control.

[3]  Shengyuan Xu,et al.  Regularized Primal–Dual Subgradient Method for Distributed Constrained Optimization , 2016, IEEE Transactions on Cybernetics.

[4]  Shengyuan Xu,et al.  Zeroth-Order Method for Distributed Optimization With Approximate Projections , 2016, IEEE Transactions on Neural Networks and Learning Systems.

[5]  Usman A. Khan,et al.  On the distributed optimization over directed networks , 2015, Neurocomputing.

[6]  Lihua Xie,et al.  Convergence of Asynchronous Distributed Gradient Methods Over Stochastic Networks , 2018, IEEE Transactions on Automatic Control.

[7]  Yasumasa Fujisaki,et al.  Distributed Multi-Agent Optimization Based on an Exact Penalty Method with Equality and Inequality Constraints , 2016 .

[8]  Guoqiang Hu,et al.  A distributed optimization method with unknown cost function in a multi-agent system via randomized gradient-free method , 2017, 2017 11th Asian Control Conference (ASCC).

[9]  Yurii Nesterov,et al.  Random Gradient-Free Minimization of Convex Functions , 2015, Foundations of Computational Mathematics.

[10]  Shengyuan Xu,et al.  Gradient‐free method for distributed multi‐agent optimization via push‐sum algorithms , 2015 .

[11]  Changzhi Wu,et al.  Gradient-free method for nonsmooth distributed optimization , 2015, J. Glob. Optim..

[12]  Ohad Shamir,et al.  Stochastic Gradient Descent for Non-smooth Optimization: Convergence Results and Optimal Averaging Schemes , 2012, ICML.

[13]  Anna Scaglione,et al.  Distributed Constrained Optimization by Consensus-Based Primal-Dual Perturbation Method , 2013, IEEE Transactions on Automatic Control.

[14]  J. Cortés,et al.  When does a digraph admit a doubly stochastic adjacency matrix? , 2010, Proceedings of the 2010 American Control Conference.

[15]  Asuman E. Ozdaglar,et al.  Constrained Consensus and Optimization in Multi-Agent Networks , 2008, IEEE Transactions on Automatic Control.

[16]  Qing Ling,et al.  EXTRA: An Exact First-Order Algorithm for Decentralized Consensus Optimization , 2014, 1404.6264.

[17]  Martin J. Wainwright,et al.  Optimal Rates for Zero-Order Convex Optimization: The Power of Two Function Evaluations , 2013, IEEE Transactions on Information Theory.

[18]  Jorge Cortés,et al.  Distributed Strategies for Generating Weight-Balanced and Doubly Stochastic Digraphs , 2009, Eur. J. Control.

[19]  Daniel W. C. Ho,et al.  Randomized Gradient-Free Method for Multiagent Optimization Over Time-Varying Networks , 2015, IEEE Transactions on Neural Networks and Learning Systems.

[20]  Chao Gao,et al.  Strong consistency of random gradient‐free algorithms for distributed optimization , 2017 .

[21]  Usman A. Khan,et al.  Distributed Subgradient Projection Algorithm Over Directed Graphs , 2016, IEEE Transactions on Automatic Control.