Advertisement

Journal of Nonlinear Science

, Volume 29, Issue 4, pp 1247–1272 | Cite as

Distributed Coordination for Nonsmooth Convex Optimization via Saddle-Point Dynamics

  • Jorge CortésEmail author
  • Simon K. Niederländer
Article
  • 241 Downloads

Abstract

This paper considers continuous-time coordination algorithms for networks of agents that seek to collectively solve a general class of nonsmooth convex optimization problems with an inherent distributed structure. Our algorithm design builds on the characterization of the solutions of the nonsmooth convex program as saddle points of an augmented Lagrangian. We show that the associated saddle-point dynamics are asymptotically correct but, in general, not distributed because of the presence of a global penalty parameter. This motivates the design of a discontinuous saddle-point-like algorithm that enjoys the same convergence properties and is fully amenable to distributed implementation. Our convergence proofs rely on the identification of a novel global Lyapunov function for saddle-point dynamics. This novelty also allows us to identify mild convexity and regularity conditions on the objective function that guarantee the exponential convergence rate of the proposed algorithms for convex optimization problems subject to equality constraints. Various examples illustrate our discussion.

Keywords

Distributed multi-agent coordination Nonsmooth convex optimization Saddle-point dynamics Continuous-time optimization algorithms 

Mathematics Subject Classification

90C25 49J52 34A60 34D23 68W15 90C35 

References

  1. Arrow, K., Hurwitz, L., Uzawa, H.: Studies in Linear and Non-linear Programming. Stanford University Press, Stanford (1958)zbMATHGoogle Scholar
  2. Aubin, J.P., Cellina, A.: Differential Inclusions, volume 264 of Grundlehren der Mathematischen Wissenschaften. Springer, New York (1984)Google Scholar
  3. Bacciotti, A., Ceragioli, F.: Stability and stabilization of discontinuous systems and nonsmooth Lyapunov functions. ESAIM Control Optim. Calc. Var. 4, 361–376 (1999)MathSciNetzbMATHGoogle Scholar
  4. Bacciotti, A., Rosier, L.: Liapunov Functions and Stability in Control Theory. Communications and Control Engineering, 2nd edn. Springer, New York (2005)zbMATHGoogle Scholar
  5. Bertsekas, D.P.: Necessary and sufficient conditions for a penalty method to be exact. Math. Program. 9(1), 87–99 (1975)MathSciNetzbMATHGoogle Scholar
  6. Bertsekas, D.P.: Nonlinear Programming, 2nd edn. Athena Scientific, Belmont (1999)zbMATHGoogle Scholar
  7. Bertsekas, D.P., Tsitsiklis, J.N.: Parallel and Distributed Computation: Numerical Methods. Athena Scientific, Belmont (1997)zbMATHGoogle Scholar
  8. Bhat, S.P., Bernstein, D.S.: Nontangency-based Lyapunov tests for convergence and stability in systems having a continuum of equilibria. SIAM J. Control Optim. 42(5), 1745–1775 (2003)MathSciNetzbMATHGoogle Scholar
  9. Brogliato, B., Daniilidis, A., Lemaréchal, C., Acary, V.: On the equivalence between complementarity systems, projected systems, and differential inclusions. Syst. Control Lett. 55(1), 45–51 (2006)MathSciNetzbMATHGoogle Scholar
  10. Bullo, F., Cortés, J., Martínez, S.: Distributed Control of Robotic Networks. Applied Mathematics Series. Princeton University Press (2009). Electronically available at http://coordinationbook.info
  11. Carli, R., Notarstefano, G.: Distributed partition-based optimization via dual decomposition. In: IEEE Conference on Decision and Control, pp. 2979–2984. Firenze, Italy (2013)Google Scholar
  12. Chen, J., Lau, V.K.N.: Convergence analysis of saddle point problems in time varying wireless systems—control theoretical approach. IEEE Trans. Signal Process. 60(1), 443–452 (2012)MathSciNetzbMATHGoogle Scholar
  13. Cherukuri, A., Gharesifard, B., Cortés, J.: Saddle-point dynamics: conditions for asymptotic stability of saddle points. SIAM J. Control Optim. 55(1), 486–511 (2017)MathSciNetzbMATHGoogle Scholar
  14. Chiang, M., Low, S.H., Calderbank, A.R., Doyle, J.C.: Layering as optimization decomposition: a mathematical theory of network architectures. Proc. IEEE 95(1), 255–312 (2007)Google Scholar
  15. Clarke, F.H.: Optimization and Nonsmooth Analysis. Canadian Mathematical Society Series of Monographs and Advanced Texts. Wiley, Hoboken (1983)zbMATHGoogle Scholar
  16. Cortés, J.: Discontinuous dynamical systems—a tutorial on solutions, nonsmooth analysis, and stability. IEEE Control Syst. 28(3), 36–73 (2008)MathSciNetzbMATHGoogle Scholar
  17. Dhingra, N.K., Khong, S.Z., Jovanović, M.R.: The proximal augmented Lagrangian method for nonsmooth composite optimization. IEEE Trans. Autom. Control (2018). arXiv:1610.04514
  18. Duchi, J.C., Agarwal, A., Wainwright, M.J.: Dual averaging for distributed optimization: convergence analysis and network scaling. IEEE Trans. Autom. Control 57(3), 592–606 (2012)MathSciNetzbMATHGoogle Scholar
  19. Ercsey-Ravasz, M., Toroczkai, Z.: The chaos within Sudoku. Sci. Rep. 2, 725 (2012)Google Scholar
  20. Feijer, D., Paganini, F.: Stability of primal-dual gradient dynamics and applications to network optimization. Automatica 46, 1974–1981 (2010)MathSciNetzbMATHGoogle Scholar
  21. Forti, M., Nistri, P., Quincampoix, M.: Generalized neural network for nonsmooth nonlinear programming problems. IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 51(9), 1741–1754 (2004)MathSciNetzbMATHGoogle Scholar
  22. Gharesifard, B., Cortés, J.: Distributed convergence to Nash equilibria in two-network zero-sum games. Automatica 49(6), 1683–1692 (2013)MathSciNetzbMATHGoogle Scholar
  23. Gharesifard, B., Cortés, J.: Distributed continuous-time convex optimization on weight-balanced digraphs. IEEE Trans. Autom. Control 59(3), 781–786 (2014)MathSciNetzbMATHGoogle Scholar
  24. Goebel, R.: Stability and robustness for saddle-point dynamics through monotone mappings. Syst. Control Lett. 108, 16–22 (2017)MathSciNetzbMATHGoogle Scholar
  25. Henry, C.: An existence theorem for a class of differential equations with multivalued right-hand side. J. Math. Anal. Appl. 41, 179–186 (1973)MathSciNetzbMATHGoogle Scholar
  26. Hiriart-Urruty, J.-B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms I. Grundlehren Text Editions. Springer, New York (1993)zbMATHGoogle Scholar
  27. Hiriart-Urruty, J.-B., Strodiot, J.-J., Nguyen, V.H.: Generalized Hessian matrix and second-order optimality conditions for problems with \(\cal{C}^{1,1}\) data. Appl. Math. Optim. 11, 43–56 (1984)MathSciNetzbMATHGoogle Scholar
  28. Holding, T., Lestas, I.: On the convergence of saddle points of concave-convex functions, the gradient method and emergence of oscillations. In: IEEE Conference on Decision and Control, pp. 1143–1148. Los Angeles, CA (2014)Google Scholar
  29. Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1985)zbMATHGoogle Scholar
  30. Johansson, B., Rabi, M., Johansson, M.: A randomized incremental subgradient method for distributed optimization in networked systems. SIAM J. Control Optim. 20(3), 1157–1170 (2009)MathSciNetzbMATHGoogle Scholar
  31. Kelly, F.P., Maulloo, A.K., Tan, D.K.H.: Rate control in communication networks: shadow prices, proportional fairness and stability. J. Oper. Res. Soc. 49(3), 237–252 (1998)zbMATHGoogle Scholar
  32. Kose, T.: Solutions of saddle value problems by differential equations. Econometrica 24(1), 59–70 (1956)MathSciNetzbMATHGoogle Scholar
  33. Li, N., Zhao, C., Chen, L.: Connecting automatic generation control and economic dispatch from an optimization view. IEEE Trans. Control Netw. Syst. 3(3), 254–264 (2016)MathSciNetzbMATHGoogle Scholar
  34. Lu, J., Tang, C.Y.: Zero-gradient-sum algorithms for distributed convex optimization: the continuous-time case. IEEE Trans. Autom. Control 57(9), 2348–2354 (2012)MathSciNetzbMATHGoogle Scholar
  35. Mallada, E., Zhao, C., Low, S.H.: Optimal load-side control for frequency regulation in smart grids. IEEE Trans. Autom. Control 62(12), 6294–6309 (2017)MathSciNetzbMATHGoogle Scholar
  36. Mangasarian, O.L.: Sufficiency of exact penalty minimization. SIAM J. Control Optim. 23(1), 30–37 (1985)MathSciNetzbMATHGoogle Scholar
  37. Mateos-Núñez, D., Cortés, J.: Noise-to-state exponentially stable distributed convex optimization on weight-balanced digraphs. SIAM J. Control Optim. 54(1), 266–290 (2016)MathSciNetzbMATHGoogle Scholar
  38. Nagurney, A., Zhang, D.: Projected Dynamical Systems and Variational Inequalities with Applications. International Series in Operations Research and Management Science, vol. 2. Kluwer Academic Publishers, Dordrecht (1996)zbMATHGoogle Scholar
  39. Necoara, I., Nedelcu, V.: Rate analysis of inexact dual first order methods. IEEE Trans. Autom. Control 59(5), 1232–1243 (2014)zbMATHGoogle Scholar
  40. Necoara, I., Suykens, J.: Application of a smoothing technique to decomposition in convex optimization. IEEE Trans. Autom. Control 53(11), 2674–2679 (2008)MathSciNetzbMATHGoogle Scholar
  41. Nedic, A., Ozdaglar, A.: Distributed subgradient methods for multi-agent optimization. IEEE Trans. Autom. Control 54(1), 48–61 (2009)MathSciNetzbMATHGoogle Scholar
  42. Nedic, A., Ozdaglar, A., Parrilo, P.A.: Constrained consensus and optimization in multi-agent networks. IEEE Trans. Autom. Control 55(4), 922–938 (2010)MathSciNetzbMATHGoogle Scholar
  43. Niederländer, S.K., Cortés, J.: Distributed coordination for separable convex optimization with coupling constraints. In: IEEE Conference on Decision and Control, pp. 694–699. Osaka, Japan (2015)Google Scholar
  44. Niederländer, S.K., Allgöwer, F., Cortés, J.: Exponentially fast distributed coordination for nonsmooth convex optimization. In: IEEE Conference on Decision and Control, pp. 1036–1041. Las Vegas, NV (2016)Google Scholar
  45. Pappalardo, M., Passacantando, M.: Stability for equilibrium problems: from variational inequalities to dynamical systems. J. Optim. Theory Appl. 113(3), 567–582 (2002)MathSciNetzbMATHGoogle Scholar
  46. Polyak, B.T.: Iterative methods using Lagrange multipliers for solving extremal problems with constraints of the equation type. USSR Comput. Math. Math. Phys. 10(5), 1098–1106 (1970)MathSciNetzbMATHGoogle Scholar
  47. Ratliff, L.J., Burden, S.A., Sastry, S.S.: On the characterization of local Nash equilibria in continuous games. IEEE Trans. Autom. Control 61(8), 2301–2307 (2016)MathSciNetzbMATHGoogle Scholar
  48. Richert, D., Cortés, J.: Robust distributed linear programming. IEEE Trans. Autom. Control 60(10), 2567–2582 (2015)MathSciNetzbMATHGoogle Scholar
  49. Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)zbMATHGoogle Scholar
  50. Rockafellar, R.T., Wets, R.J.B.: Variational Analysis, Volume of 317 Comprehensive Studies in Mathematics. Springer, New York (1998)Google Scholar
  51. Shi, W., Ling, Q., Wu, G., Yin, W.: EXTRA: an exact first-order algorithm for decentralized consensus optimization. SIAM J. Optim. 25(2), 944–966 (2015)MathSciNetzbMATHGoogle Scholar
  52. Stegink, T., Persis, C.D., van der Schaft, A.J.: A unifying energy-based approach to stability of power grids with market dynamics. IEEE Trans. Autom. Control 62(6), 2612–2622 (2017)MathSciNetzbMATHGoogle Scholar
  53. Wan, P., Lemmon, M.D.: Event-triggered distributed optimization in sensor networks. In: Symposium on Information Processing of Sensor Networks, pp. 49–60., San Francisco, CA (2009)Google Scholar
  54. Wang, J., Elia, N.: A control perspective for centralized and distributed convex optimization. In: IEEE Conference on Decision and Control, pp. 3800–3805. Orlando, Florida (2011)Google Scholar
  55. Zhang, X., Papachristodoulou, A.: A real-time control framework for smart power networks: design methodology and stability. Automatica 58, 43–50 (2015)MathSciNetzbMATHGoogle Scholar
  56. Zhao, C., Topcu, U., Li, N., Low, S.H.: Design and stability of load-side primary frequency control in power systems. IEEE Trans. Autom. Control 59(5), 1177–1189 (2014)MathSciNetzbMATHGoogle Scholar
  57. Zhu, M., Martínez, S.: On distributed convex optimization under inequality and equality constraints. IEEE Trans. Autom. Control 57(1), 151–164 (2012)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mechanical and Aerospace Engineering, Jacobs School of EngineeringUniversity of California, San DiegoLa JollaUSA
  2. 2.Institute for Systems Theory and Automatic ControlUniversity of StuttgartStuttgartGermany

Personalised recommendations