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Mathematical Programming

, Volume 112, Issue 2, pp 275–301 | Cite as

A sparse proximal implementation of the LP dual active set algorithm

  • Timothy A. Davis
  • William W. HagerEmail author
Full Length Paper

Abstract

We present an implementation of the LP Dual Active Set Algorithm (LP DASA) based on a quadratic proximal approximation, a strategy for dropping inactive equations from the constraints, and recently developed algorithms for updating a sparse Cholesky factorization after a low-rank change. Although our main focus is linear programming, the first and second-order proximal techniques that we develop are applicable to general concave–convex Lagrangians and to linear equality and inequality constraints. We use Netlib LP test problems to compare our proximal implementation of LP DASA to Simplex and Barrier algorithms as implemented in CPLEX.

Keywords

Dual active set algorithm Linear programming Simplex method Barrier method Interior point method Equation dropping 

Mathematics Subject Classification (2000)

90C05 90C06 65Y20 

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References

  1. 1.
    Andersen E.D., Andersen K.D. (1995) Presolving in linear programming. Math. Program. 71, 221–245MathSciNetGoogle Scholar
  2. 2.
    Bergounioux M., Kunisch K. (2002) Primal-dual strategy for state-constrained optimal control problem. Comput. Optim. Appl. 22, 193–224zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bixby R.E. (1994) Progress in linear programming. ORSA J. Comput. 6, 15–22zbMATHMathSciNetGoogle Scholar
  4. 4.
    Björck A., Elfving T. (1979) Accelerated projection methods for computing pseudoinverse solutions of systems of linear equations. BIT 19, 145–163zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Clarke F.H. (1975) Generalized gradients and applications. Trans. Am. Math. Soc. 205, 247–262zbMATHCrossRefGoogle Scholar
  6. 6.
    Davis T.A. (2005) Algorithm 849: a concise sparse Cholesky factorization package. ACM Trans. Math. Softw. 31, 587–591CrossRefGoogle Scholar
  7. 7.
    Davis T.A., Gilbert J.R., Larimore S.I., Ng E.G. (2004) Algorithm 836: COLAMD, a column approximate minimum degree ordering algorithm. ACM Trans. Math. Softw. 30, 377–380zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Davis T.A., Gilbert J.R., Larimore S.I., Ng E.G. (2004) A column approximate minimum degree ordering algorithm. ACM Trans. Math. Softw. 30, 353–376zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Davis T.A., Hager W.W. (1999) Modifying a sparse Cholesky factorization. SIAM J. Matrix Anal. Appl. 20, 606–627zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Davis T.A., Hager W.W. (2001) Multiple-rank modifications of a sparse Cholesky factorization. SIAM J. Matrix Anal. Appl. 22, 997–1013zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Davis T.A., Hager W.W. Dual multilevel optimization, to appear in Mathematical Programming University of Florida (2004)Google Scholar
  12. 12.
    Davis T.A., Hager W.W. (2005) Row modifications of a sparse Cholesky factorization. SIAM J. Matrix Anal. Appl. 26, 621–639zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Davis, T.A., Hager, W.W., Chen, Y.C., Rajamanickam, S. CHOLMOD: a sparse Cholesky factorization and modification package, ACM Trans. Math. Softw. (2006) (in preparation)Google Scholar
  14. 14.
    Dongarra J.J., Du Croz J.J., Duff I.S., Hammarling S. (1990) A set of level 3 basic linear algebra subprograms. ACM Trans. Math. Softw. 16, 1–17zbMATHCrossRefGoogle Scholar
  15. 15.
    Ekeland, I., Temam, R. Convex Analysis and Variational Problems. North-Holland, Amsterdam (1976)Google Scholar
  16. 16.
    Gill P.E., Saunders M.A., Shinnerl J.R. (1996) On the stability of cholesky factorization for quasi-definite systems. SIAM J. Matrix Anal. Appl. 17, 35–46zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Golub G.H., Loan C.F.V. (1979) Unsymmetric positive definite linear systems. Linear Algebra Appl. 28, 85–98zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Hager W.W. (1979) Convex control and dual approximations, part I. Control Cybern. 8, 5–22MathSciNetGoogle Scholar
  19. 19.
    Hager, W.W. Inequalities and approximation. In: Coffman, C.V., Fix, G.J. (eds.) Constructive Approaches to Mathematical Models pp. 189–202. (1979)Google Scholar
  20. 20.
    Hager, W.W. The dual active set algorithm. In: Pardalos, P.M. (ed.) Advances in Optimization and Parallel Computing pp. 137–142. North Holland, Amsterdam (1992)Google Scholar
  21. 21.
    Hager W.W. (1998). The LP dual active set algorithm. In: Leone R.D., Murli A., Pardalos P.M., Toraldo G. (eds). High Performance Algorithms and Software in Nonlinear Optimization. Kluwer, Dordrecht, pp. 243–254Google Scholar
  22. 22.
    Hager W.W. (2000) Iterative methods for nearly singular linear systems. SIAM J. Sci. Comput. 22, 747–766zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Hager W.W. (2002) The dual active set algorithm and its application to linear programming. Comput. Optim. Appl. 21, 263–275zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Hager, W.W. The dual active set algorithm and the iterative solution of linear programs. In: Pardalos, P.M., Wolkowicz, H. (eds.) Novel Approaches to Hard Discrete Optimization, vol. 37, pp. 95–107 Fields Institute Communications, (2003)Google Scholar
  25. 25.
    Hager W.W., Hearn D.W. (1993) Application of the dual active set algorithm to quadratic network optimization. Comput. Optim. Appl. 1, 349–373zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Hager W.W., Ianculescu G. (1984) Dual approximations in optimal control. SIAM J. Control Optim. 22, 423–465zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Hager, W.W., Shi, C.-L., Lundin, E.O. Active set strategies in the LP dual active set algorithm, tech. report. University of Florida, http://www.math.ufl.edu/~hager/LPDASA (1996)Google Scholar
  28. 28.
    Karypis G., Kumar V. (1998) A fast and high quality multilevel scheme for partitioning irregular graphs. SIAM J. Sci. Comput. 20, 359–392CrossRefMathSciNetGoogle Scholar
  29. 29.
    Karypis G., Kumar V. (1999) Multilevel k-way partitioning scheme for irregular graphs. J. Parallel Distrib. Comput. 48, 96–129CrossRefGoogle Scholar
  30. 30.
    Karypis G., Kumar V. (1999) Parallel multilevel k-way partitioning scheme for irregular graphs. SIAM Rev. 41, 278–300zbMATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Liu J.W.H. (1991) A generalized envelope method for sparse factorization by rows. ACM Trans. Math. Softw. 17, 112–129zbMATHCrossRefGoogle Scholar
  32. 32.
    Martinet B. (1970) Régularisation d’inéquations variationnelles par approximations successives. Rev. Francaise Inform. Rech. Oper. Ser. R-3 4, 154–158MathSciNetGoogle Scholar
  33. 33.
    Martinet B. (1972) Determination approachée d’un point fixe d’une application pseudo-contractante. Comptes Rendus des Séances de l’Académie des Sciences 274, 163–165zbMATHMathSciNetGoogle Scholar
  34. 34.
    Ng E.G., Peyton B.W. (1993) Block sparse Cholesky algorithms on advanced uniprocessor computers. SIAM J. Sci. Comput. 14, 1034–1056zbMATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Pan P.Q. (2004) A dual projective pivot algorithm for linear programming. Comput. Optim. Appl. 29, 333–346zbMATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Rockafellar R.T. (1976) Monotone operators and the proximal point algorithm. SIAM J. Control 14, 877–898zbMATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    Roos C., Terlaky T., Vial J.-P. (1997) Theory and Algorithms for Linear Optimization: An Interior Point Approach. Wiley, New YorkzbMATHGoogle Scholar
  38. 38.
    Saunders M.A. (1996). Cholesky-based methods for sparse least squares: The benefits of regularization. In: Adams L., Nazareth J.L. (eds). Linear and Nonlinear Conjugate Gradient-Related Methods. SIAM, Philadelpha, pp. 92–100Google Scholar
  39. 39.
    Shih, C.L. Active Set Strategies in Optimization. PhD Thesis, University of Florida, Department of Mathematics (1995)Google Scholar
  40. 40.
    Vanderbei R.J. (1995) Symmetric quasi-definite matrices. SIAM J. Optim. 5, 100–113zbMATHCrossRefMathSciNetGoogle Scholar
  41. 41.
    Volkwein S. (2003) Lagrange-SQP techniques for the control constrained optimal boundary control for Burger’s equation. Comput. Optim. Appl. 26, 253–284zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Department of Computer and Information Science and EngineeringUniversity of FloridaGainesvilleUSA
  2. 2.Department of MathematicsUniversity of FloridaGainesvilleUSA

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