A Decentralized Primal-Dual Method for Constrained Minimization of a Strongly Convex Function

We propose decentralized primal-dual methods for cooperative multi-agent consensus optimization problems over both static and time-varying communication networks, where only local communications are allowed. The objective is to minimize the sum of agent-specific convex functions over conic constraint sets defined by agent-specific nonlinear functions; hence, the optimal consensus decision should lie in the intersection of these private sets. Assuming the sum function is strongly convex, we provide convergence rates for sub-optimality, infeasibility and consensus violation in terms of the number of communications required; examine the effect of underlying network topology on the convergence rates.

[1]  Erfan Yazdandoost Hamedani,et al.  A primal-dual method for conic constrained distributed optimization problems , 2016, NIPS.

[2]  Stephen J. Wright,et al.  Cooperative distributed model predictive control , 2010, Syst. Control. Lett..

[3]  Angelia Nedic,et al.  Distributed Non-Autonomous Power Control through Distributed Convex Optimization , 2009, IEEE INFOCOM 2009.

[4]  Qiong Wu,et al.  Fast distributed optimization over directed graphs , 2016, 2016 American Control Conference (ACC).

[5]  Alejandro Ribeiro,et al.  Consensus in Ad Hoc WSNs With Noisy Links—Part I: Distributed Estimation of Deterministic Signals , 2008, IEEE Transactions on Signal Processing.

[6]  Usman A. Khan,et al.  ADD-OPT: Accelerated Distributed Directed Optimization , 2016, IEEE Transactions on Automatic Control.

[7]  Jorge Cortés,et al.  Distributed subgradient methods for saddle-point problems , 2015, 2015 54th IEEE Conference on Decision and Control (CDC).

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

[9]  Angelia Nedic,et al.  Distributed optimization over time-varying directed graphs , 2013, 52nd IEEE Conference on Decision and Control.

[10]  Martin J. Wainwright,et al.  Dual Averaging for Distributed Optimization: Convergence Analysis and Network Scaling , 2010, IEEE Transactions on Automatic Control.

[11]  Gareth M. James,et al.  Penalized and Constrained Regression , 2013 .

[12]  Michael G. Rabbat,et al.  Consensus-based distributed optimization: Practical issues and applications in large-scale machine learning , 2012, 2012 50th Annual Allerton Conference on Communication, Control, and Computing (Allerton).

[13]  Constantine Caramanis,et al.  Theory and Applications of Robust Optimization , 2010, SIAM Rev..

[14]  Erfan Yazdandoost Hamedani,et al.  Multi-agent constrained optimization of a strongly convex function , 2017, 2017 IEEE Global Conference on Signal and Information Processing (GlobalSIP).

[15]  Erfan Yazdandoost Hamedani,et al.  A Distributed ADMM-like Method for Resource Sharing over Time-Varying Networks , 2016, SIAM J. Optim..

[16]  Antonin Chambolle,et al.  A First-Order Primal-Dual Algorithm for Convex Problems with Applications to Imaging , 2011, Journal of Mathematical Imaging and Vision.

[17]  Tsung-Hui Chang,et al.  A Proximal Dual Consensus ADMM Method for Multi-Agent Constrained Optimization , 2014, IEEE Transactions on Signal Processing.

[18]  Angelia Nedic,et al.  Distributed constrained optimization over noisy networks , 2010, 49th IEEE Conference on Decision and Control (CDC).

[19]  Angelia Nedic,et al.  Stochastic Gradient-Push for Strongly Convex Functions on Time-Varying Directed Graphs , 2014, IEEE Transactions on Automatic Control.

[20]  Torsten Hothorn,et al.  A unified framework of constrained regression , 2014, Stat. Comput..

[21]  Ohad Shamir,et al.  Optimal Distributed Online Prediction , 2011, ICML.

[22]  Laurent El Ghaoui,et al.  Robust Optimization , 2021, ICORES.

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

[24]  Asuman E. Ozdaglar,et al.  A fast distributed proximal-gradient method , 2012, 2012 50th Annual Allerton Conference on Communication, Control, and Computing (Allerton).

[25]  Erfan Yazdandoost Hamedani,et al.  Multi-agent constrained optimization of a strongly convex function over time-varying directed networks , 2017, 2017 55th Annual Allerton Conference on Communication, Control, and Computing (Allerton).

[26]  Convex Optimization in Signal Processing and Communications , 2010 .

[27]  Wotao Yin,et al.  ExtraPush for Convex Smooth Decentralized Optimization over Directed Networks , 2015, ArXiv.

[28]  N. S. Aybat,et al.  A Primal-Dual Algorithm for General Convex-Concave Saddle Point Problems , 2018, 1803.01401.

[29]  Johannes Gehrke,et al.  Gossip-based computation of aggregate information , 2003, 44th Annual IEEE Symposium on Foundations of Computer Science, 2003. Proceedings..

[30]  Yonina C. Eldar,et al.  Convex Optimization in Signal Processing and Communications , 2009 .

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

[32]  Gesualdo Scutari,et al.  NEXT: In-Network Nonconvex Optimization , 2016, IEEE Transactions on Signal and Information Processing over Networks.

[33]  Yangyang Xu,et al.  Lower complexity bounds of first-order methods for convex-concave bilinear saddle-point problems , 2018, Math. Program..

[34]  Niao He,et al.  Mirror Prox algorithm for multi-term composite minimization and semi-separable problems , 2013, Computational Optimization and Applications.

[35]  Stergios I. Roumeliotis,et al.  Multirobot Active Target Tracking With Combinations of Relative Observations , 2011, IEEE Transactions on Robotics.

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

[37]  A. Chambolle,et al.  A remark on accelerated block coordinate descent for computing the proximity operators of a sum of convex functions , 2015 .

[38]  N. S. Aybat,et al.  A distributed ADMM-like method for resource sharing under conic constraints over time-varying networks , 2016 .

[39]  Xiangfeng Wang,et al.  Multi-Agent Distributed Optimization via Inexact Consensus ADMM , 2014, IEEE Transactions on Signal Processing.

[40]  Ali H. Sayed,et al.  Diffusion Adaptation Strategies for Distributed Optimization and Learning Over Networks , 2011, IEEE Transactions on Signal Processing.

[41]  John N. Tsitsiklis,et al.  On distributed averaging algorithms and quantization effects , 2007, 2008 47th IEEE Conference on Decision and Control.

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

[43]  Wenwu Yu,et al.  An Overview of Recent Progress in the Study of Distributed Multi-Agent Coordination , 2012, IEEE Transactions on Industrial Informatics.

[44]  Asuman E. Ozdaglar,et al.  Convergence Rate of Distributed ADMM Over Networks , 2016, IEEE Transactions on Automatic Control.

[45]  Erfan Yazdandoost Hamedani,et al.  A Primal-Dual Algorithm with Line Search for General Convex-Concave Saddle Point Problems , 2020, SIAM J. Optim..

[46]  Wei Shi,et al.  Achieving Geometric Convergence for Distributed Optimization Over Time-Varying Graphs , 2016, SIAM J. Optim..

[47]  C. H. Jeffrey Pang,et al.  Dykstra’s Splitting and an Approximate Proximal Point Algorithm for Minimizing the Sum of Convex Functions , 2017, J. Optim. Theory Appl..

[48]  Mark W. Schmidt,et al.  Convergence Rates of Inexact Proximal-Gradient Methods for Convex Optimization , 2011, NIPS.

[49]  Gesualdo Scutari,et al.  Distributed nonconvex optimization for sparse representation , 2016, 2017 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).

[50]  Shengyuan Xu,et al.  Distributed Primal–Dual Subgradient Method for Multiagent Optimization via Consensus Algorithms , 2011, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics).

[51]  Angelia Nedić,et al.  Distributed Optimization , 2015, Encyclopedia of Systems and Control.

[52]  Antonin Chambolle,et al.  On the ergodic convergence rates of a first-order primal–dual algorithm , 2016, Math. Program..

[53]  Necdet Serhat Aybat,et al.  Decentralized computation of effective resistances and acceleration of consensus algorithms , 2017, 2017 IEEE Global Conference on Signal and Information Processing (GlobalSIP).

[54]  Xiang Li,et al.  Probabilistically Constrained Linear Programs and Risk-Adjusted Controller Design , 2005, SIAM J. Optim..

[55]  Hua Zhou,et al.  Algorithms for Fitting the Constrained Lasso , 2016, Journal of computational and graphical statistics : a joint publication of American Statistical Association, Institute of Mathematical Statistics, Interface Foundation of North America.

[56]  Roberto Tempo,et al.  Distributed Algorithms for Robust Convex Optimization via the Scenario Approach , 2016, IEEE Transactions on Automatic Control.