An Alternating Trust Region Algorithm for Distributed Linearly Constrained Nonlinear Programs, Application to the Optimal Power Flow Problem

  • Jean-Hubert HoursEmail author
  • Colin N. Jones


A novel trust region method for solving linearly constrained nonlinear programs is presented. The proposed technique is amenable to a distributed implementation, as its salient ingredient is an alternating projected gradient sweep in place of the Cauchy point computation. It is proven that the algorithm yields a sequence that globally converges to a critical point. As a result of some changes to the standard trust region method, namely a proximal regularisation of the trust region subproblem, it is shown that the local convergence rate is linear with an arbitrarily small ratio. Thus, convergence is locally almost superlinear, under standard regularity assumptions. The proposed method is successfully applied to compute local solutions to alternating current optimal power flow problems in transmission and distribution networks. Moreover, the new mechanism for computing a Cauchy point compares favourably against the standard projected search, as for its activity detection properties.


Nonconvex optimisation Distributed optimisation Coordinate gradient descent Trust region methods 

Mathematics Subject Classification

49M27 49M37 65K05 65K10 90C06 90C26 90C30 



The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2014)/ ERC Grant Agreement No. 307608.


  1. 1.
    Necoara, I., Savorgnan, C., Tran Dinh, Q., Suykens, J., Diehl, M.: Distributed nonlinear optimal control using sequential convex programming and smoothing techniques. In: Proceedings of the \(48^{\text{th}}\) Conference on Decision and Control (2009)Google Scholar
  2. 2.
    Kim, B.H., Baldick, R.: Coarse-grained distributed optimal power flow. IEEE Trans. Power Syst. 12(2), 932–939 (1997)Google Scholar
  3. 3.
    Chiang, M., Low, S., Calderbank, A., Doyle, J.: Layering as optimization decomposition: a mathematical theory of network architectures. Proc. IEEE 95(1), 255–312 (2007)CrossRefGoogle Scholar
  4. 4.
    Hours, J.-H., Jones, C.N.: A parametric non-convex decomposition algorithm for real-time and distributed NMPC. IEEE Trans. Autom. Control 61(2), 287–302 (2016)Google Scholar
  5. 5.
    Gan, L., Li, N., Topcu, U., Low, S.H.: Exact convex relaxation of optimal power flow in radial network. IEEE Trans. Autom. Control 60, 72–87 (2014)Google Scholar
  6. 6.
    Zavala, V.M., Laird, C.D., Biegler, L.T.: Interior-point decomposition approaches for parallel solution of large-scale nonlinear parameter estimation problems. Chem. Eng. Sci. 63, 4834–4845 (2008)CrossRefGoogle Scholar
  7. 7.
    Fei, Y., Guodong, R., Wang, B., Wang, W.: Parallel L-BFGS-B algorithm on GPU. Comput. Graph. 40, 1–9 (2014)CrossRefGoogle Scholar
  8. 8.
    Bertsekas, D.P., Tsitsiklis, J.N.: Parallel and Distributed Computation: Numerical Methods. Athena Scientific, Belmont (1997)zbMATHGoogle Scholar
  9. 9.
    Wächter, A., Biegler, L.T.: On the implementation of a primal-dual interior point filter line-search algorithm for large-scale nonlinear programming. Math. Program. 106(1), 25–57 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Tran-Dinh, Q., Savorgnan, C., Diehl, M.: Combining Lagrangian decomposition and excessive gap smoothing technique for solving large-scale separable convex optimization problems. Comput. Optim. Appl. 55(1), 75–111 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Cohen, G.: Auxiliary problem principle and decomposition of optimization problems. J. Optim. Theory Appl. 32(3), 277–305 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hamdi, A., Mishra, S.K.: Decomposition methods based on augmented Lagrangian: a survey. In: Topics in Nonconvex Optimization. Mishra, S.K. (2011)Google Scholar
  13. 13.
    Hours, J.-H., Jones, C.N.: An augmented Lagrangian coordination–decomposition algorithm for solving distributed non-convex programs. In: Proceedings of the 2014 American Control Conference, pp. 4312–4317 (2014)Google Scholar
  14. 14.
    Tran Dinh, Q., Necoara, I., Diehl, M.: A dual decomposition algorithm for separable nonconvex optimization using the penalty framework. In: Proceedings of the \(52^{\text{ nd }}\) Conference on Decision and Control (2013)Google Scholar
  15. 15.
    Conn, A., Gould, N.I.M., Toint, P.L.: A globally convergent augmented Lagrangian algorithm for optimization with general constraints and simple bounds. SIAM J. Numer. Anal. 28, 545–572 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Fernández, D., Solodov, M.V.: Local convergence of exact and inexact augmented Lagrangian methods under the second-order sufficient optimality condition. SIAM J. Optim. 22(2), 384–407 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Bolte, J., Sabach, S., Teboulle, M.: Proximal alternating linearized minimization for nonconvex and nonsmooth problems. Math. Program. 146(1–2), 459–494 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Conn, A.R., Gould, N.I.M., Toint, P.L.: Trust Region Methods. Society for Industrial and Applied Mathematics, Philadelphia (2000)CrossRefzbMATHGoogle Scholar
  19. 19.
    Zavala, V.M., Anitescu, M.: Scalable nonlinear programming via exact differentiable penalty functions and trust-region Newton methods. SIAM J. Optim. 24(1), 528–558 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    D’Azevedo, E., Eijkhout, V., Romine, C.: LAPACK Working Note 56: Reducing communication costs in the conjugate gradient algorithm on distributed memory multiprocessors. Technical report, University of Tennessee, Knoxville, TN (1993)Google Scholar
  21. 21.
    Verschoor, M., Jalba, A.C.: Analysis and performance estimation of the conjugate gradient method on multiple GPUs. Parallel Comput. 38(10–11), 552–575 (2012)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Conn, A.R., Gould, N.I.M., Toint, P.L.: Global convergence of a class of trust region algorithms for optimization with simple bounds. SIAM J. Numer. Anal. 25(2) (1988)Google Scholar
  23. 23.
    Xue, D., Sun, W., Qi, L.: An alternating structured trust-region algorithm for separable optimization problems with nonconvex constraints. Comput. Optim. Appl. 57, 365–386 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (2009)zbMATHGoogle Scholar
  25. 25.
    Moreau, J.: Décomposition orthogonale d’un espace hilbertien selon deux cônes mutuellement polaires. C.R. Acad. Sci. 255, 238–240 (1962)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Burke, J., Moré, J., Toraldo, G.: Convergence properties of trust region methods for linear and convex constraints. Math. Program. 47, 305–336 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Moré, J.J.: Trust regions and projected gradients. In: Lecture Notes in Control and Information Sciences, vol. 113. Springer, Berlin (1988)Google Scholar
  28. 28.
    Steihaug, T.: The conjugate gradient method and trust regions in large scale optimization. SIAM J. Numer. Anal. 20, 626–637 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Nocedal, J., Wright, S.: Numerical Optimization. Springer, New-York (2006)zbMATHGoogle Scholar
  30. 30.
    Yamashita, N.: Sparse quasi-Newton updates with positive definite matrix completion. Math. Program. 115(1), 1–30 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Curtis, F.E., Gould, N.I.M., Jiang, H., Robinson, D.P.: Adaptive augmented Lagrangian methods: algorithms and practical numerical experience. Technical report 14T-006, COR@L Laboratory, Department of ISE, Lehigh University (2014. To appear in Optimization Methods and Software).
  32. 32.
    Lam, A.Y.S., Zhang, B., Tse, D.N.: Distributed algorithms for optimal power flow. In: Proceedings of the 51st Conference on Decision and Control, pp. 430–437 (2012)Google Scholar
  33. 33.
    Bukhsh, W.A., Grothey, A., McKinnon, K.I.M., Trodden, P.A.: Local solutions of the optimal power flow problem. IEEE Trans. Power Syst. 28(4) (2013)Google Scholar
  34. 34.
    Bertsekas, D.P.: Constrained Optimization and Lagrange Multiplier Methods. Athena Scientific, Belmont (1982)zbMATHGoogle Scholar
  35. 35.
    Zhu, J.: Optimization of Power System Operation. IEEE Press, Piscataway (2009)CrossRefGoogle Scholar
  36. 36.

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Automatic Control LaboratoryEcole Polytechnique Fédérale de LausanneLausanneSwitzerland

Personalised recommendations