Saddle Point Seeking for Convex Optimization Problems

Abstract In this paper, we consider convex optimization problems with constraints. By combining the idea of a Lie bracket approximation for extremum seeking systems and saddle point algorithms, we propose a feedback which steers a single-integrator system to the set of saddle points of the Lagrangian associated to the convex optimization problem. We prove practical uniform asymptotic stability of the set of saddle points for the extremum seeking system for strictly convex as well as linear programs. Using a numerical example we illustrate how the approach can be used in distributed optimization problems.

[1]  Stefan Ulbrich,et al.  Nichtlineare Optimierung , 2012, Mathematik Kompakt.

[2]  Martin Guay,et al.  Extremum-seeking control of state-constrained nonlinear systems , 2004, Autom..

[3]  Miroslav Krstic,et al.  Nash Equilibrium Seeking in Noncooperative Games , 2012, IEEE Transactions on Automatic Control.

[4]  Nicanor Quijano,et al.  A Shahshahani Gradient based extremum seeking scheme , 2012, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).

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

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

[7]  Dirk Aeyels,et al.  Practical stability and stabilization , 2000, IEEE Trans. Autom. Control..

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

[9]  João M. Lemos,et al.  STOCHASTIC EXTREMUM SEEKING IN THE PRESENCE OF CONSTRAINTS , 2005 .

[10]  Hans-Bernd Dürr,et al.  A smooth vector field for saddle point problems , 2011, IEEE Conference on Decision and Control and European Control Conference.

[11]  Miroslav Krstic,et al.  Extremum seeking for moderately unstable systems and for autonomous vehicle target tracking without position measurements , 2006 .

[12]  Alireza Nazemi.  A dynamic system model for solving convex nonlinear optimization problems , 2012 .

[13]  Kartik B. Ariyur,et al.  Real-Time Optimization by Extremum-Seeking Control , 2003 .

[14]  Hans-Bernd Dürr,et al.  A smooth vector field for quadratic programming , 2012, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).

[15]  W. Neise,et al.  Adaptive flow control using slope seeking , 2006, 2006 14th Mediterranean Conference on Control and Automation.

[16]  Hans-Bernd Dürr,et al.  Feedback design for multi-agent systems: A saddle point approach , 2012, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).

[17]  Denis Dochain,et al.  Adaptive extremum seeking control of continuous stirred tank bioreactors with unknown growth kinetics , 2004, Autom..

[18]  Dimitri P. Bertsekas,et al.  Nonlinear Programming , 1997 .

[19]  I. Mareels,et al.  Extremum seeking from 1922 to 2010 , 2010, Proceedings of the 29th Chinese Control Conference.

[20]  Milos S. Stankovic,et al.  Lie bracket approximation of extremum seeking systems , 2011, Autom..