Computational Optimization and Applications

, Volume 67, Issue 2, pp 317–360 | Cite as

Solving nearly-separable quadratic optimization problems as nonsmooth equations

Article
  • 167 Downloads

Abstract

An algorithm for solving nearly-separable quadratic optimization problems (QPs) is presented. The approach is based on applying a semismooth Newton method to solve the implicit complementarity problem arising as the first-order stationarity conditions of such a QP. An important feature of the approach is that, as in dual decomposition methods, separability of the dual function of the QP can be exploited in the search direction computation. Global convergence of the method is promoted by enforcing decrease in component(s) of a Fischer–Burmeister formulation of the complementarity conditions, either via a merit function or through a filter mechanism. The results of numerical experiments when solving convex and nonconvex instances are provided to illustrate the efficacy of the method.

Keywords

Quadratic optimization problems Dual decomposition Complementarity problems Semismooth Newton methods Fischer–Burmeister function 

Mathematics Subject Classification

49M05 49M15 49M27 49M29 65K05 65K10 90C20 90C33 

References

  1. 1.
    Bai, L., Raghunathan, A. U.: Semismooth equation approach to network utility maximization (NUM). In American Control Conference (ACC), pp. 4795–4801, (2013)Google Scholar
  2. 2.
    Bertsekas, D.P.: Nonlinear Programming, 2nd edn. Athena Scientific, Belmont (1999)MATHGoogle Scholar
  3. 3.
    Bertsekas, D.P.: Convex Optimization Theory. Athena Scientific, Belmont (2009)MATHGoogle Scholar
  4. 4.
    Birge, J.R., Louveaux, F.V.: Introduction to Stochastic Programming. Springer, New York (1997)MATHGoogle Scholar
  5. 5.
    Bragalli, C., Ambrosio, C.D., Lee, J., Lodi, A., Toth, P.: On the optimal design of water distribution networks: a practical MINLP approach. Opt. Eng. 13(2), 219–246 (2012)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Ralph, D., Dempe, S.: Directional derivatives of the solution of a parametric nonlinear program. Math. Prog. 70, 159–172 (1995)MathSciNetMATHGoogle Scholar
  7. 7.
    Deng, W., Lai, M.-J., Peng, Z., Yin, W.: Parallel Multi-Block ADMM with \(o(1/k)\) Convergence. Technical report, arXiv:1312.3040 (2014)
  8. 8.
    Dinh, Q.T., Necoara, I., Diehl, M.: Path-following gradient-based decomposition algorithms for separable convex optimization. J. Global Optim. 59(1), 59–80 (2014)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Dinh, Q.T., Necoara, I., Savorgnan, C., Diehl, M.: An inexact perturbed path-following method for lagrangian decomposition in large-scale separable convex optimization. SIAM J. Optim. 23(1), 95–125 (2013)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Dinh, Q.T., 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)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Facchinei, F., Fischer, A., Kanzow, C.: Regularity properties of a semismooth reformulation of variational inequalities. SIAM J. Optim. 8, 850–869 (1998)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Facchinei, F., Soares, J.: A new merit function for nonlinear complementarity problems and a related algorithm. SIAM J. Optim. 7, 225–247 (1997)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Fiacco, A.V.: Sensitivity analysis for nonlinear programming using penalty methods. Math. Program. 10, 287–311 (1976)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Fiacco, A.V., McCormick, G.P.: Nonlinear programming: sequential unconstrained minimization techniques. Class. Appl. Math. Soc. Ind. Appl. Math. (1987)Google Scholar
  15. 15.
    Fischer, A.: A special Newton-type optimization method. Optimization 24, 269–284 (1992)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Fletcher, R., Gould, N.I.M., Leyffer, S., Toint, PhL, Wächter, A.: Global convergence of trust-region SQP-filter algorithms for nonlinear programming. SIAM J. Opt. 13, 635–659 (2002)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Fletcher, R., Leyffer, S.: Nonlinear programming without a penalty function. Math. Program. 91, 239–269 (2002)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Fletcher, R., Leyffer, S.: Filter-type algorithms for solving systems of algebraic equations and inequalities. In: High-Performance Algorithms and Software in Nonlinear Optimization, pp. 259–278. Kluwer, Dordrecht, The Netherlands (2003)Google Scholar
  19. 19.
    Fletcher, R., Leyffer, S., Toint, PhL: On the global convergence of a filter-SQP algorithm. SIAM J. Opt. 13, 44–59 (2002)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Frasch, J.V., Sager, S., Diehl, M.: A parallel quadratic programming method for dynamic optimization problems. Math. Prog. Comput. 7(3), 289–329 (2015)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Gondzio, J., Grothey, A.: Parallel interior-point solver for structured quadratic programs: Application to financial planning problems. Ann. Oper. Res. 152(1), 319–339 (2007)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Gould, N.I.M., Leyffer, S., Toint, PhL: A multidimensional filter algorithm for nonlinear equations and nonlinear least squares. SIAM J. Opt. 15, 17–38 (2004)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Jiang, H., Qi, L.: A new nonsmooth equations approach to nonlinear complementarity problems. SIAM J Control Opt. 35, 178–193 (1997)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Kanzow, C., Petra, S.: Projected filter trust region methods for a semismooth least squares formulation of mixed complementarity problems. Opt. Methods Softw. 22(5), 713–735 (2007)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Qi, L., Sun, J.: A nonsmooth version of Newton’s method. Math. Progr. 58, 353–368 (1993)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Laird, C.D., Biegler, L.T.: Large-scale nonlinear programming for multi-scenario optimization. In: Bock, H.G., Kostina, E., Phu, H.X., Ranacher, R. (eds.) Modeling, simulation and optimization of complex processes, pp. 323–336. Springer, Berlin (2008)CrossRefGoogle Scholar
  27. 27.
    De Luca, T., Facchinei, F., Kanzow, C.: A semismooth equation approach to the solution of nonlinear complementarity problems. Math. Progr. 75, 407–439 (1996)MathSciNetMATHGoogle Scholar
  28. 28.
    Mas-Colell, A., Whinston, M.D., Green, J.R.: Microeconomic Theory. Oxford University Press, Oxford (1991)MATHGoogle Scholar
  29. 29.
    Munson, T.S., Facchinei, F., Ferris, M., Fischer, A., Kanzow, C.: The Semismooth algorithm for large scale complementarity problems. J. Comput. 13, 294–311 (2001)MathSciNetMATHGoogle Scholar
  30. 30.
    Necoara, I., Patrascu, A.: Iteration complexity analysis of dual first-order methods for conic convex programming. Opt. Methods Softw. 31(3), 645–678 (2016)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Necoara, I., Suykens, J.A.K.: Application of a smoothing technique to decomposition in convex optimization. IEEE Trans. Autom. Control 53(11), 2674–2679 (2008)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Necoara, I., Suykens, J.A.K.: Application of a smoothing technique to decomposition in convex optimization. IEEE Trans. Autom. Control 53(11), 2674–2679 (2008)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Necoara, I., Suykens, J.A.K.: Interior-point lagrangian decomposition method for separable convex optimization. J. Optim. Theory Appl. 143(3), 567–588 (2009)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Nesterov, Y.: Introductory Lectures on Convex Optimization: A Basic Course. Kluwer, Boston (2004)CrossRefMATHGoogle Scholar
  35. 35.
    Nesterov, Y.: Excessive gap technique in nonsmooth convex minimization. SIAM J Opt. 16(1), 235–249 (2005)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Nesterov, Y.: Smooth minimization of non-smooth functions. Math. Progr. (A) 103(1), 127–152 (2005)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Nesterov, Y., Nemirovskii, A.: Interior Point Polynomial Algorithms in Convex Programming. Society for Industrial and Applied Mathematics (SIAM Studies in Applied Mathematics), Philadelphia (1994)CrossRefMATHGoogle Scholar
  38. 38.
    Neumaier, A.: MINQ—General Definite and Bound Constrained Indefinite Quadratic Programming (1998)Google Scholar
  39. 39.
    Nie, Pu-yan, Lai, Ming yong, Zhu, Shu jin, Zhang, Pei ai: A line search filter approach for the system of nonlinear equations. Comput. Math. Appl. 55(9), 2134–2141 (2008)MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Qi, L.: Convergence analysis of some algorithms for solving nonsmooth equations. Math. Oper. Res. 51, 101–131 (1991)Google Scholar
  41. 41.
    Mifflin, R.: Semismooth and semiconvex functions in constrained optimization. SIAM J. Control Opt. 15(6), 959–972 (1977)MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    Cottle, R.W., Pang, J.-S., Stone, R.E.: The Linear Complementarity Problem. Classics in Applied Mathematics. SIAM, Philadelphia (2009)CrossRefGoogle Scholar
  43. 43.
    Raghunathan, A.U.: Global optimization of nonlinear network design. SIAM J. Optim. 23(1), 268–295 (2013)MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Raghunathan, A. U., Curtis, F. E., Takaguchi, Y., Hashimoto, H.: Accelerating convergence to competitive equilibrium in electricity markets. In: IEEE Power & Energy Society General Meeting, PESGM2016-000221 (2016)Google Scholar
  45. 45.
    Rockafellar, R.T.: Monotone Operators and the Proximal Point Algorithm. SIAM J. Control Optim. 14(5), 877–898 (1976)MathSciNetCrossRefMATHGoogle Scholar
  46. 46.
    Shor, N.Z.: Minimization Methods for Non-differentiable Functions. Springer, New York (1985)CrossRefMATHGoogle Scholar
  47. 47.
    De Luca, T., Facchinei, F., Kanzow, C.: A theoretical and numerical comparison of some semismooth algorithms for complementarity problems. Comput. Opt. Appl. 16(2), 173–205 (2000)MathSciNetCrossRefMATHGoogle Scholar
  48. 48.
    Ulbrich, M., Ulbrich, S., Vicente, L.N.: A globally convergent primal-dual interior-point filter method for nonlinear programming. Math. Progr. 100(2), 379–410 (2004)MathSciNetCrossRefMATHGoogle Scholar
  49. 49.
    Wächter, A., Biegler, L.T.: On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Math. Program. 106(1), 25–57 (2006)MathSciNetCrossRefMATHGoogle Scholar
  50. 50.
    Wang, G., Negrete-Pincetic, M., Kowli, A., Shafieepoorfard, E., Meyn, S., Shanbhag, U.V.: Dynamic competitive equilibria in electricity markets. In: Chakrabortty, Aranya, Ili, Marija D. (eds.) Control and Optimization Methods for Electric Smart Grids, volume 3 of Power Electronics and Power Systems, pp. 35–62. Springer, New York (2012)CrossRefGoogle Scholar
  51. 51.
    Wood, A.J., Wollenberg, B.F.: Power Generation Operation and Control, 2nd edn. Wiley, Hoboken (1996)Google Scholar
  52. 52.
    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(19), 4834–4845 (2008)CrossRefGoogle Scholar
  53. 53.
    Zimmerman, R.D., Murillo-Sanchez, C.E., Thomas, R.J.: MATPOWER: steady-state operations, planning and analysis tools for power systems research and education. IEEE Trans. Power Syst. 26(1), 12–19 (2011)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of Industrial and Systems EngineeringLehigh UniversityBethlehemUSA
  2. 2.Mitsubishi Electric Research LaboratoriesCambridgeUSA

Personalised recommendations