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Quadratic regularizations in an interior-point method for primal block-angular problems

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Abstract

One of the most efficient interior-point methods for some classes of primal block-angular problems solves the normal equations by a combination of Cholesky factorizations and preconditioned conjugate gradient for, respectively, the block and linking constraints. Its efficiency depends on the spectral radius—in [0,1)— of a certain matrix in the definition of the preconditioner. Spectral radius close to 1 degrade the performance of the approach. The purpose of this work is twofold. First, to show that a separable quadratic regularization term in the objective reduces the spectral radius, significantly improving the overall performance in some classes of instances. Second, to consider a regularization term which decreases with the barrier function, thus with no need for an extra parameter. Computational experience with some primal block-angular problems confirms the efficiency of the regularized approach. In particular, for some difficult problems, the solution time is reduced by a factor of two to ten by the regularization term, outperforming state-of-the-art commercial solvers.

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References

  1. Ali, A., Kennington, J.L.: Mnetgen program documentation. Technical Report 77003. Department of Industrial Engineering and Operations Research, Southern Methodist University, Dallas (1977)

  2. Altman A., Gondzio J.: Regularized symmetric indefinite systems in interior point methods for linear and quadratic optimization. Optim. Methods Softw. 11, 275–302 (1999)

    Article  MathSciNet  Google Scholar 

  3. Babonneau F., du Merle O., Vial J.-P.: Solving large-scale linear multicommodity flow problems with an active set strategy and proximal-ACCPM. Oper. Res. 54, 184–197 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bellavia S., Gondzio J., Morini B.: Regularization and preconditioning of KKT systems arising in nonnegative least-squares problems. Numer. Linear Algebra Appl. 16, 39–61 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bergamaschi L., Gondzio J., Zilli G.: Preconditioning indefinite systems in interior point methods for optimization. Comput. Optim. Appl. 28, 149–171 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bienstock D.: Potential Function Methods for Approximately Solving Linear Programming Problems. Theory and Practice. Kluwer, Boston (2002)

    MATH  Google Scholar 

  7. Bienstock D., Raskina O.: Asymptotic analysis of the flow deviation method for the maximum concurrent flow problem. Math. Program. 91, 479–492 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bixby R.E.: Solving real-world linear programs: a decade and more of progress. Oper. Res. 50, 3–15 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  9. Carolan W.J., Hill J.E., Kennington J.L., Niemi S., Wichmann S.J.: An empirical evaluation of the KORBX algorithms for military airlift applications. Oper. Res. 38, 240–248 (1990)

    Article  Google Scholar 

  10. Castro J.: A specialized interior-point algorithm for multicommodity network flows. SIAM J. Optim. 10, 852–877 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  11. Castro J.: Solving difficult multicommodity problems through a specialized interior-point algorithm. Ann. Oper. Res. 124, 35–48 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  12. Castro J.: Solving quadratic multicommodity problems through an interior-point algorithm. In: Sachs, E.W., Tichatschke, R. (eds) System Modelling and Optimization XX, pp. 199–212. Kluwer, Boston (2003)

    Google Scholar 

  13. Castro J.: Quadratic interior-point methods in statistical disclosure control. Comput. Manag. Sci. 2, 107–121 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  14. Castro J.: An interior-point approach for primal block-angular problems. Comput. Optim. Appl. 36, 195–219 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  15. Chardaire P., Lisser A.: Simplex and interior point specialized algorithms for solving nonoriented multicommodity flow problems. Oper. Res. 50, 260–276 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  16. Czyzyk J., Mehrotra S., Wagner M., Wright S.J.: PCx: an interior-point code for linear programming. Optim. Methods Softw. 11/12, 97–430 (1999)

    MathSciNet  Google Scholar 

  17. Fiacco, A.V., McCormick, G.P.: Nonlinear programming: Sequential Unconstrained Minimization Techniques. Wiley, New York (1968). Reprinted in Classics In Applied Mathematics series, SIAM, Philadelphia (1990)

  18. Frangioni A., Gallo G.: A bundle type dual-ascent approach to linear multicommodity min cost flow problems. INFORMS J. Comput. 11, 370–393 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  19. Golub G.H., Van Loan C.F.: Matrix Computations, 3rd edn. Johns Hopkins University Press, Baltimore (1996)

    MATH  Google Scholar 

  20. Gondzio J., Sarkissian R.: Parallel interior-point solver for structured linear programs. Math. Program. 96, 561–584 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  21. Güler O., Ye Y.: Convergence behaviour of some interior-point algorithms. Math. Program. 60, 215–228 (1993)

    Article  MATH  Google Scholar 

  22. Nesterov Y.: Introductory Lectures on Convex Optimization: A Basic Course. Kluwer, Boston (2004)

    MATH  Google Scholar 

  23. Nesterov Y., Nemirovskii A.: Interior-Point Polynomial Algorithms in Convex Programming. SIAM, Philadelphia (1994)

    Book  MATH  Google Scholar 

  24. Nesterov Y., Vial J.-P.: Augmented self-concordant barriers and nonlinear optimization problems with finite complexity. Math. Program. 99, 149–174 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  25. Ng E., Peyton B.W.: Block sparse Cholesky algorithms on advanced uniprocessor computers. SIAM J. Sci. Comput. 14, 1034–1056 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  26. Oliveira A.R.L., Sorensen D.C.: A new class of preconditioners for large-scale linear systems from interior point methods for linear programming. Linear Algebra Appl. 394, 1–24 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  27. Ouorou A., Mahey M., Vial J.-P.: A survey of algorithms for convex multicommodity flow problems. Manag. Sci. 46, 126–147 (2000)

    Article  Google Scholar 

  28. Rockafellar R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  29. Saunders, M.A., Tomlin, J.A.: Solving regularized linear programs using barrier methods and KKT systems. Technical Report SOL 96-4. Department of Operations Research, Stanford University (1996)

  30. Setiono R.: Interior proximal point algorithm for linear programs. J. Optim. Theory Appl. 74, 425–444 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  31. Vanderbei R.J.: Symmetric quasidefinite matrices. SIAM J. Optim. 5, 100–113 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  32. Wright S.J.: Primal-Dual Interior-Point Methods. SIAM, Philadelphia (1996)

    Google Scholar 

Download references

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Correspondence to Jordi Castro.

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Castro, J., Cuesta, J. Quadratic regularizations in an interior-point method for primal block-angular problems. Math. Program. 130, 415–445 (2011). https://doi.org/10.1007/s10107-010-0341-2

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