Stability and robustness for saddle-point dynamics through monotone mappings

Abstract Motivated by recent interest in saddle-point dynamics, where, given a convex in x and concave in y function, trajectories follow the steepest descent in x and the steepest ascent in y , and where convergence of trajectories to saddle points is desired, this note revisits the maximal monotone mapping approach to saddle-point dynamics, mildly improves one convergence result, and proposes new results on the robustness of pointwise asymptotic stability of the set of saddle points. The results apply to nonsmooth convex/concave functions and constraints, and – as a special case – to projected saddle-point dynamics.

[1]  L. Hurwicz,et al.  Gradient Methods for Constrained Maxima , 1957 .

[2]  Enrique Mallada,et al.  The Role of Convexity in Saddle-Point Dynamics: Lyapunov Function and Robustness , 2016, IEEE Transactions on Automatic Control.

[3]  B. Garay,et al.  Multivalued perturbations of a saddle dynamics , 2010 .

[4]  Enrique Mallada,et al.  Asymptotic convergence of constrained primal-dual dynamics , 2015, Syst. Control. Lett..

[5]  Ashish Cherukuri,et al.  Asymptotic stability of saddle points under the saddle-point dynamics , 2015, 2015 American Control Conference (ACC).

[6]  Rafal Goebel,et al.  Robustness of stability through necessary and sufficient Lyapunov-like conditions for systems with a continuum of equilibria , 2014, Syst. Control. Lett..

[7]  Yu. S. Ledyaev,et al.  Asymptotic Stability and Smooth Lyapunov Functions , 1998 .

[8]  Sanjay P. Bhat,et al.  Semistability, Finite-Time Stability, Differential Inclusions, and Discontinuous Dynamical Systems Having a Continuum of Equilibria , 2009, IEEE Transactions on Automatic Control.

[9]  Y. Kōmura,et al.  Nonlinear semi-groups in Hilbert space , 1967 .

[10]  Pierre-Antoine Absil,et al.  On the stable equilibrium points of gradient systems , 2006, Syst. Control. Lett..

[11]  Sjur Didrik Flåm,et al.  Approximating saddle points as equilibria of differential inclusions , 1989 .

[12]  Tosio Kato,et al.  Nonlinear semigroups and evolution equations , 1967 .

[13]  Rafal Goebel,et al.  Set-valued Lyapunov functions for difference inclusions , 2011, Autom..

[14]  Vincent Acary,et al.  On the equivalence between complementarity systems, projected systems and differential inclusions , 2006, Syst. Control. Lett..

[15]  Jing Wang,et al.  A control perspective for centralized and distributed convex optimization , 2011, IEEE Conference on Decision and Control and European Control Conference.

[16]  ASHISH CHERUKURI,et al.  Saddle-Point Dynamics: Conditions for Asymptotic Stability of Saddle Points , 2015, SIAM J. Control. Optim..

[17]  V. I. Venets.  Continuous algorithms for solution of convex optimization problems and finding saddle points of contex-coneave functions with the use of projection operations , 1985 .

[18]  A. Teel,et al.  A smooth Lyapunov function from a class- ${\mathcal{KL}}$ estimate involving two positive semidefinite functions , 2000 .

[19]  A. Iusem,et al.  Set-valued mappings and enlargements of monotone operators , 2008 .

[20]  S. Sorin,et al.  Evolution equations for maximal monotone operators: asymptotic analysis in continuous and discrete time , 2009, 0905.1270.

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

[22]  Alejandro Ribeiro,et al.  Distributed continuous-time online optimization using saddle-point methods , 2016, 2016 IEEE 55th Conference on Decision and Control (CDC).

[23]  Maojiao Ye,et al.  Distributed control for economic dispatch via saddle point dynamics and consensus algorithms , 2016, 2016 IEEE 55th Conference on Decision and Control (CDC).

[24]  Dennis S. Bernstein,et al.  Nontangency-Based Lyapunov Tests for Convergence and Stability in Systems Having a Continuum of Equilibria , 2003, SIAM J. Control. Optim..

[25]  T. Kose.  Solutions of Saddle Value Problems by Differential Equations , 1956 .

[26]  Bahman Gharesifard,et al.  Saddle-point dynamics for distributed convex optimization on general directed graphs , 2016, 2016 IEEE 55th Conference on Decision and Control (CDC).

[27]  Z. Chbani,et al.  Existence and asymptotic behaviour for solutions of dynamical equilibrium systems , 2014 .

[28]  C. Henry.  An existence theorem for a class of di erential equations with multival-ued right-hand side , 1973 .

[29]  Ioannis Lestas,et al.  On the convergence to saddle points of concave-convex functions, the gradient method and emergence of oscillations , 2014, 53rd IEEE Conference on Decision and Control.