Computational Optimization and Applications

, Volume 52, Issue 2, pp 315–344

A preconditioning technique for Schur complement systems arising in stochastic optimization



Deterministic sample average approximations of stochastic programming problems with recourse are suitable for a scenario-based parallelization. In this paper the parallelization is obtained by using an interior-point method and a Schur complement mechanism for the interior-point linear systems. However, the direct linear solves involving the dense Schur complement matrix are expensive, and adversely affect the scalability of this approach. We address this issue by proposing a stochastic preconditioner for the Schur complement matrix and by using Krylov iterative methods for the solution of the dense linear systems. The stochastic preconditioner is built based on a subset of existing scenarios and can be assembled and factorized on a separate process before the computation of the Schur complement matrix finishes on the remaining processes. The expensive factorization of the Schur complement is removed from the parallel execution flow and the scaling of the optimization solver is considerably improved with this approach. The spectral analysis indicates an exponentially fast convergence in probability to 1 of the eigenvalues of the preconditioned matrix with the number of scenarios incorporated in the preconditioner. Numerical experiments performed on the relaxation of a unit commitment problem show good performance, in terms of both the accuracy of the solution and the execution time.


Stochastic programming Saddle-point preconditioning Krylov methods Interior-point method Sample average approximations Parallel computing 


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  1. 1.
    Altman, A., Gondzio, J.: Regularized symmetric indefinite systems in interior-point methods for linear and quadratic optimization. Optim. Methods Softw. 11(1–4), 275–302 (1999) MathSciNetCrossRefGoogle Scholar
  2. 2.
    Benzi, M., Golub, G.H., Liesen, J.: Numerical solution of saddle point problems. Acta Numer. 14, 1–137 (2005) MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Bergamaschi, L., Gondzio, J., Zilli, G.: Preconditioning indefinite systems in interior point methods for optimization. Comput. Optim. Appl. 28(2), 149–171 (2004) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Birge, J.R.: Current trends in stochastic programming computation and applications. Tech. rep., Department of Industrial and Operations Engineering, University of Michigan, Ann Harbour, Michigan (1995) Google Scholar
  5. 5.
    Birge, J.R., Holmes, D.F.: Efficient solution of two stage stochastic linear programs using interior point methods. Comput. Optim. Appl. 1, 245–276 (1992) MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Birge, J.R., Louveaux, F.: Introduction to Stochastic Programming. Springer, New York (1997) MATHGoogle Scholar
  7. 7.
    Birge, J.R., Qi, L.: Computing block-angular Karmarkar projections with applications to stochastic programming. Manag. Sci. 34(12), 1472–1479 (1988) MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Birge, J.R., Chen, X., Qi, L.: A stochastic Newton method for stochastic quadratic programs with recourse. Tech. rep., Applied Mathematics Preprint AM94/9, School of Mathematics, The University of New South Wales (1995) Google Scholar
  9. 9.
    Blackford, L.S., Choi, J., Cleary, A., D’Azevedo, E., Demmel, J., Dhillon, I., Dongarra, J., Hammarling, S., Henry, G., Petitet, A., Stanley, K., Walker, D., Whaley, R.C.: ScaLAPACK Users’ Guide. Society for Industrial and Applied Mathematics, Philadelphia (1997) MATHCrossRefGoogle Scholar
  10. 10.
    Bunch, J.R., Kaufman, L.: Some stable methods for calculating inertia and solving symmetric linear systems. Math. Comput. 31(137), 163–179 (1977) MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Bunch, J.R., Parlett, B.N.: Direct methods for solving symmetric indefinite systems of linear equations. SIAM J. Numer. Anal. 8(4), 639–655 (1971) MathSciNetCrossRefGoogle Scholar
  12. 12.
    Cafieri, S., D’Apuzzo, M., Marino, M., Mucherino, A., Toraldo, G.: Interior-point solver for large-scale quadratic programming problems with bound constraints. J. Optim. Theory Appl. 129, 55–75 (2006) MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Castro, J.: A specialized interior-point algorithm for multicommodity network flows. SIAM J. Optim. 10(3), 852–877 (2000) MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Constantinescu, E.M., Zavala, V.M., Rocklin, M., Lee, S., Anitescu, M.: A computational framework for uncertainty quantification and stochastic optimization in unit commitment with wind power generation. IEEE Trans. Power Syst. 26(1), 431–441 (2011) CrossRefGoogle Scholar
  15. 15.
    Czyzyk, J., Mehrotra, S., Wright, S.J.: PCx user guide. Tech. Rep. OTC 96/01, Optimization Technology Center, Argonne National Laboratory and Northwestern University (1996) Google Scholar
  16. 16.
    Dantzig, G.B., Infanger, G.: Large-scale stochastic linear programs—Importance sampling and Benders decomposition. In: Computational and Applied Mathematics, I, pp. 111–120. North-Holland, Amsterdam (1992) Google Scholar
  17. 17.
    D’Apuzzo, M., Simone, V., Serafino, D.: On mutual impact of numerical linear algebra and large-scale optimization with focus on interior point methods. Comput. Optim. Appl. 45, 283–310 (2010) MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Dollar, H.S., Wathen, A.J.: Approximate factorization constraint preconditioners for saddle-point matrices. SIAM J. Sci. Comput. 27(5), 1555–1572 (2006) MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Dollar, H.S., Gould, N.I.M., Schilders, W.H.A., Wathen, A.J.: Implicit-factorization preconditioning and iterative solvers for regularized saddle-point systems. SIAM J. Matrix Anal. Appl. 28(1), 170–189 (2006) MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Ermoliev, Y.M.: Stochastic quasigradient methods. In: Numerical Techniques for Stochastic Optimization. Springer Ser. Comput. Math., vol. 10, pp. 141–185. Springer, Berlin (1988) CrossRefGoogle Scholar
  21. 21.
    Gertz, E.M., Wright, S.J.: Object-oriented software for quadratic programming. ACM Trans. Math. Softw. 29(1), 58–81 (2003) MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Golub, G.H., Van Loan, C.F.: Matrix Computations, 3rd edn. Johns Hopkins University Press, Baltimore (1996) MATHGoogle Scholar
  23. 23.
    Gondzio, J.: HOPDM (version 2.12)—a fast LP solver based on a primal-dual interior point method. Eur. J. Oper. Res. 85, 221–225 (1995) MATHCrossRefGoogle Scholar
  24. 24.
    Gondzio, J., Grothey, A.: Direct solution of linear systems of size 109 arising in optimization with interior point methods. In: PPAM, pp. 513–525 (2005) Google Scholar
  25. 25.
    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) MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Gondzio, J., Grothey, A.: Exploiting structure in parallel implementation of interior point methods for optimization. Comput. Manag. Sci. 6(2), 135–160 (2009) MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Gondzio, J., Makowski, M.: Solving a class of LP problems with a primal–dual logarithmic barrier method. Eur. J. Oper. Res. 80, 184–192 (1995) MATHCrossRefGoogle Scholar
  28. 28.
    Gondzio, J., Sarkissian, R.: Parallel interior point solver for structured linear programs. Math. Program. 96, 561–584 (2000) MathSciNetCrossRefGoogle Scholar
  29. 29.
    Gould, N.I.M., Hribar, M.E., Nocedal, J.: On the solution of equality constrained quadratic programming problems arising in optimization. SIAM J. Sci. Comput. 23(4), 1376–1395 (2001) MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    Greenbaum, A.: Iterative Methods for Solving Linear Systems. SIAM Series on Frontiers in Applied Mathematics, Philadelphia (1997) MATHCrossRefGoogle Scholar
  31. 31.
    Güler, O.: Existence of interior points and interior paths in nonlinear monotone complementarity problems. Math. Oper. Res. 18(1), 128–147 (1993) MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    Higle, J.L., Sen, S.: Stochastic decomposition: an algorithm for two-stage linear programs with recourse. Math. Oper. Res. 16, 650–669 (1991) MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    Keller, C., Gould, N.I.M., Wathen, A.J.: Constraint preconditioning for indefinite linear systems. SIAM J. Matrix Anal. Appl. 21(4), 1300–1317 (2000) MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    King, A.J.: An implementation of the Lagrangian finite-generation method. In: Numerical Techniques for Stochastic Optimization. Springer Ser. Comput. Math., vol. 10, pp. 295–311. Springer, Berlin (1988) CrossRefGoogle Scholar
  35. 35.
    Linderoth, J., Wright, S.J.: Decomposition algorithms for stochastic programming on a computational grid. Comput. Optim. Appl. 24(2–3), 207–250 (2003) MathSciNetMATHCrossRefGoogle Scholar
  36. 36.
    Luks̆an, L., Vlc̆ek, J.: Indefinitely preconditioned inexact Newton method for large sparse equality constrained nonlinear programming problems. Numer. Linear Algebra Appl. 5(3), 219–247 (1998) MathSciNetCrossRefGoogle Scholar
  37. 37.
    Lustig, I.J., Marsten, R.E., Shanno, D.F.: On implementing Mehrotra’s predictor–corrector interior-point method for linear programming. SIAM J. Optim. 2(3), 435–449 (1992) MathSciNetMATHCrossRefGoogle Scholar
  38. 38.
    McKay, M.D., Beckman, R.J., Conover, W.J.: A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics 21, 239–245 (1979) MathSciNetMATHGoogle Scholar
  39. 39.
    Mehrotra, S.: On the implementation of a primal-dual interior point method. SIAM J. Optim. 2(4), 575–601 (1992) MathSciNetMATHCrossRefGoogle Scholar
  40. 40.
    Mehrotra, S., Ozevin, M.G.: Decomposition-based interior point methods for two-stage stochastic semidefinite programming. SIAM J. Optim. 18(1), 206–222 (2007) MathSciNetMATHCrossRefGoogle Scholar
  41. 41.
    Mehrotra, S., Ozevin, M.G.: Decomposition based interior point methods for two-stage stochastic convex quadratic programs with recourse. Oper. Res. 57(4), 964–974 (2009) MathSciNetMATHCrossRefGoogle Scholar
  42. 42.
    Mehrotra, S., Ozevin, M.G.: On the implementation of interior point decomposition algorithms for two-stage stochastic conic programs. SIAM J. Optim. 19(4), 1846–1880 (2009) MathSciNetMATHCrossRefGoogle Scholar
  43. 43.
    Monteiro, R.D.C., Pang, J.S.: Properties of an interior-point mapping for mixed complementarity problems. Math. Oper. Res. 21(3), 629–654 (1996) MathSciNetMATHCrossRefGoogle Scholar
  44. 44.
    Murphy, M.F., Golub, G.H., Wathen, A.J.: A note on preconditioning for indefinite linear systems. SIAM J. Sci. Comput. 21(6), 1969–1972 (2000) MathSciNetMATHCrossRefGoogle Scholar
  45. 45.
    Pflug, G.C., Halada, L.: A note on the recursive and parallel structure of the Birge and Qi factorization for tree structured linear programs. Comput. Optim. Appl. 24(2–3), 251–265 (2003) MathSciNetMATHCrossRefGoogle Scholar
  46. 46.
    Poulson, J., Marker, B., van de Geijn, R.A.: Elemental: A new framework for distributed memory dense matrix computations (flame working note #44). Tech. Rep., Institute for Computational Engineering and Sciences. The University of Texas at Austin (2010) Google Scholar
  47. 47.
    Rockafellar, R.T., Wets, R.J.B.: A Lagrangian finite generation technique for solving linear-quadratic problems in stochastic programming. Math. Program. Stud. 28, 63–93 (1986). Stochastic programming 84. II MathSciNetMATHCrossRefGoogle Scholar
  48. 48.
    Rockafellar, R.T., Wets, R.J.B.: Scenarios and policy aggregation in optimization under uncertainty. Math. Oper. Res. 16(1), 119–147 (1991) MathSciNetMATHCrossRefGoogle Scholar
  49. 49.
    Rosa, C.H., Ruszczyński, A.: On augmented Lagrangian decomposition methods for multistage stochastic programs. Ann. Oper. Res. 64, 289–309 (1996). Stochastic programming, algorithms and models (Lillehammer, 1994) MathSciNetMATHCrossRefGoogle Scholar
  50. 50.
    Ruszczyński, A.: A regularized decomposition method for minimizing a sum of polyhedral functions. Math. Program. 35, 309–333 (1986) MATHCrossRefGoogle Scholar
  51. 51.
    Shapiro, A., Dentcheva, D., Ruszczyński, A.: Lectures on Stochastic Programming: Modeling and Theory. MPS/SIAM Series on Optimization, vol. 9. SIAM, Philadelphia (2009) MATHCrossRefGoogle Scholar
  52. 52.
    Van Slyke, R., Wets, R.J.: L-shaped linear programs with applications to control and stochastic programming. SIAM J. Appl. Math. 17, 638–663 (1969) MathSciNetMATHCrossRefGoogle Scholar
  53. 53.
    Wright, S.J.: Primal-Dual Interior-Point Methods. Society for Industrial and Applied Mathematics, Philadelphia (1997) MATHCrossRefGoogle Scholar
  54. 54.
    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
  55. 55.
    Zavala, V.M., Constantinescu, E.M., Krause, T., Anitescu, M.: On-line economic optimization of energy systems using weather forecast information. J. Process Control 19, 1725–1736 (2009) CrossRefGoogle Scholar
  56. 56.
    Zhang, D., Zhang, Y.: A Mehrotra-type predictor-corrector algorithm with polynomiality and Q-subquadratic convergence. Ann. Oper. Res. 62, 131–150 (1996) MathSciNetMATHCrossRefGoogle Scholar
  57. 57.
    Zhang, Y.: Solving large-scale linear programs by interior-point methods under the Matlab environment. Tech. Rep. TR96-01, University of Maryland Baltimore County (1996) Google Scholar
  58. 58.
    Zhao, G.: A log-barrier method with Benders decomposition for solving two-stage stochastic linear programs. Math. Program. 90(3), 507–536 (2001) MathSciNetMATHCrossRefGoogle Scholar

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© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Mathematics and Computer Science DivisionArgonne National LaboratoryArgonneUSA

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