Distributed continuous-time gradient-based algorithm for constrained optimization

In this paper, we consider distributed algorithm based on a continuous-time multi-agent system to solve constrained optimization problem. The global optimization objective function is taken as the sum of agents' individual objective functions under a group of convex inequality function constraints. Because the local objective functions cannot be explicitly known by all the agents, the problem has to be solved in a distributed manner with the cooperation between agents. Here we propose a continuous-time distributed gradient dynamics based on the KKT condition and Lagrangian multiplier methods to solve the optimization problem. We show that all the agents asymptotically converge to the same optimal solution with the help of a constructed Lyapunov function and a LaSalle invariance principle of hybrid systems.

[1]  Karl Henrik Johansson,et al.  Dynamical properties of hybrid automata , 2003, IEEE Trans. Autom. Control..

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

[3]  Sonia Martínez,et al.  Distributed convex optimization via continuous-time coordination algorithms with discrete-time communication , 2014, Autom..

[4]  Asuman E. Ozdaglar,et al.  Distributed Subgradient Methods for Multi-Agent Optimization , 2009, IEEE Transactions on Automatic Control.

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

[6]  Xinghu Wang,et al.  Dynamic optimization for multi-agent systems with external disturbances , 2014 .

[7]  Yiguang Hong,et al.  Reaching optimal consensus for multi-agent systems based on approximate projection , 2012, Proceedings of the 10th World Congress on Intelligent Control and Automation.

[8]  Qiang Zhang,et al.  Distributed Parameter Estimation Over Unreliable Networks With Markovian Switching Topologies , 2012, IEEE Transactions on Automatic Control.

[9]  Georgios B. Giannakis,et al.  Distributed Optimal Power Flow for Smart Microgrids , 2012, IEEE Transactions on Smart Grid.

[10]  Karl Henrik Johansson,et al.  Convergence of random sleep algorithms for optimal consensus , 2013, Syst. Control. Lett..

[11]  B. V. Dean,et al.  Studies in Linear and Non-Linear Programming. , 1959 .

[12]  Ali H. Sayed,et al.  Adaptive Penalty-Based Distributed Stochastic Convex Optimization , 2013, IEEE Transactions on Signal Processing.

[13]  Jing Wang,et al.  Control approach to distributed optimization , 2010, 2010 48th Annual Allerton Conference on Communication, Control, and Computing (Allerton).

[14]  Yiguang Hong,et al.  Potential game design for a class of distributed optimisation problems , 2014 .

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

[16]  A. Ruszczynski,et al.  Nonlinear Optimization , 2006 .

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

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

[19]  Yiguang Hong,et al.  An approximate gradient algorithm for constrained distributed convex optimization , 2014, IEEE/CAA Journal of Automatica Sinica.

[20]  Choon Yik Tang,et al.  Zero-gradient-sum algorithms for distributed convex optimization: The continuous-time case , 2011, Proceedings of the 2011 American Control Conference.

[21]  Lacra Pavel,et al.  A continuous-time decentralized optimization scheme with positivity constraints , 2012, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).