Distributed Smooth Convex Optimization With Coupled Constraints

This note develops a distributed algorithm to solve a convex optimization problem with coupled constraints. Both coupled equality and inequality constraints are considered, where functions in the equality constraints are affine and functions in the inequality constraints are convex. Different from primal-dual subgradient methods with decreasing stepsizes for nonsmooth optimizations, our algorithm focuses on smooth problems and uses a fixed stepsize to find the exact optimal solution. Convergence analysis is derived with rigorous proofs. Our result is also illustrated by simulations.

[1]  Magnus Egerstedt,et al.  Continuous-time proportional-integral distributed optimisation for networked systems , 2013, 1309.6613.

[2]  Yiguang Hong,et al.  Constrained Consensus Algorithms With Fixed Step Size for Distributed Convex Optimization Over Multiagent Networks , 2017, IEEE Transactions on Automatic Control.

[3]  Han-Fu Chen,et al.  Primal-dual algorithm for distributed constrained optimization , 2015, Syst. Control. Lett..

[4]  Ashish Cherukuri,et al.  Initialization-free distributed coordination for economic dispatch under varying loads and generator commitment , 2014, Autom..

[5]  Shu Liang,et al.  Distributed Nash equilibrium seeking for aggregative games with coupled constraints , 2016, Autom..

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

[7]  Qingshan Liu,et al.  A Collaborative Neurodynamic Approach to Multiple-Objective Distributed Optimization , 2018, IEEE Transactions on Neural Networks and Learning Systems.

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

[9]  Angelia Nedic,et al.  Subgradient Methods for Saddle-Point Problems , 2009, J. Optimization Theory and Applications.

[10]  Ashish Cherukuri,et al.  Distributed Generator Coordination for Initialization and Anytime Optimization in Economic Dispatch , 2015, IEEE Transactions on Control of Network Systems.

[11]  Daniela Pucci de Farias,et al.  Decentralized Resource Allocation in Dynamic Networks of Agents , 2008, SIAM J. Optim..

[12]  Stephen P. Boyd,et al.  Optimal Scaling of a Gradient Method for Distributed Resource Allocation , 2006 .

[13]  Feng Liu,et al.  Initialization-free distributed algorithms for optimal resource allocation with feasibility constraints and application to economic dispatch of power systems , 2015, Autom..

[14]  Shouyang Wang,et al.  Distributed continuous-time approximate projection protocols for shortest distance optimization problems , 2015, Autom..

[15]  Karl Henrik Johansson,et al.  Subgradient methods and consensus algorithms for solving convex optimization problems , 2008, 2008 47th IEEE Conference on Decision and Control.

[16]  Heinz H. Bauschke,et al.  Convex Analysis and Monotone Operator Theory in Hilbert Spaces , 2011, CMS Books in Mathematics.

[17]  Lacra Pavel,et al.  A distributed primal-dual algorithm for computation of generalized Nash equilibria via operator splitting methods , 2017, 2017 IEEE 56th Annual Conference on Decision and Control (CDC).

[18]  Asuman E. Ozdaglar,et al.  Approximate Primal Solutions and Rate Analysis for Dual Subgradient Methods , 2008, SIAM J. Optim..

[19]  Dragana Bajović,et al.  Newton-like Method with Diagonal Correction for Distributed Optimization , 2015, SIAM J. Optim..

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

[21]  Karl Henrik Johansson,et al.  Reaching an Optimal Consensus: Dynamical Systems That Compute Intersections of Convex Sets , 2011, IEEE Transactions on Automatic Control.

[22]  Guang-Hong Yang,et al.  Augmented Lagrange algorithms for distributed optimization over multi-agent networks via edge-based method , 2018, Autom..

[23]  Shu Liang,et al.  Distributed Nonsmooth Optimization With Coupled Inequality Constraints via Modified Lagrangian Function , 2016, IEEE Transactions on Automatic Control.

[24]  Ion Necoara,et al.  On linear convergence of a distributed dual gradient algorithm for linearly constrained separable convex problems , 2014, Autom..

[25]  F. Facchinei,et al.  Finite-Dimensional Variational Inequalities and Complementarity Problems , 2003 .

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

[27]  Fernando Paganini,et al.  Stability of primal-dual gradient dynamics and applications to network optimization , 2010, Autom..

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

[29]  Yiguang Hong,et al.  Distributed Continuous-Time Algorithm for Constrained Convex Optimizations via Nonsmooth Analysis Approach , 2015, IEEE Transactions on Automatic Control.

[30]  Yurii Nesterov,et al.  Introductory Lectures on Convex Optimization - A Basic Course , 2014, Applied Optimization.

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