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A Primal-Dual Slack Approach to Warmstarting Interior-Point Methods for Linear Programming

  • Alexander Engau
  • Miguel F. Anjos
  • Anthony Vannelli
Part of the Operations Research/Computer Science Interfaces book series (ORCS, volume 47)

Abstract

Despite the many advantages of interior-point algorithms over active-set methods for linear programming, one of their practical limitations is the remaining challenge to efficiently solve several related problems by an effective warmstarting strategy. Similar to earlier approaches that modify the initial problem by shifting the boundary of its feasible region, the contribution of this paper is a new but relatively simpler scheme which uses a single set of new slacks to relax the nonnegativity constraints of the original primal-dual variables. Preliminary computational results indicate that this simplified approach yields similar improvements over cold starts as achieved by previous methods.

Keywords:

interior-point methods — linear programming — warmstarting 

Notes

Acknowledgments

We thank the three anonymous referees and the ICS09 program committee for their valuable comments and suggestions on an earlier version of this manuscript.

References

  1. Anjos MF, Burer S (2007) On handling free variables in interior-point methods for conic linear optimization. SIAM J Optim 18(4):1310–1325MathSciNetzbMATHCrossRefGoogle Scholar
  2. Benson HY, Shanno DF (2007) An exact primal-dual penalty method approach to warmstarting interior-point methods for linear programming. Comput Optim Appl 38(3):371–399MathSciNetzbMATHCrossRefGoogle Scholar
  3. El-Bakry AS, Tapia RA, Zhang Y (1994) A study of indicators for identifying zero variables in interior-point methods. SIAM Rev 36(1):45–72MathSciNetzbMATHCrossRefGoogle Scholar
  4. Elhedhli S, Goffin JL (2004) The integration of an interior-point cutting plane method within a branch-and-price algorithm. Math Program 100(2, Ser. A):267–294MathSciNetzbMATHCrossRefGoogle Scholar
  5. Facchinei F, Fischer A, Kanzow C (2000) On the identification of zero variables in an interior-point framework. SIAM J Optim 10(4):1058–1078MathSciNetzbMATHCrossRefGoogle Scholar
  6. Freund RM (1991a) A potential-function reduction algorithm for solving a linear program directly from an infeasible “warm start”. Math Programming 52(3, Ser. B):441–466MathSciNetzbMATHCrossRefGoogle Scholar
  7. Freund RM (1991b) Theoretical efficiency of a shifted-barrier-function algorithm for linear programming. Linear Algebra Appl 152:19–41MathSciNetzbMATHCrossRefGoogle Scholar
  8. Freund RM (1996) An infeasible-start algorithm for linear programming whose complexity depends on the distance from the starting point to the optimal solution. Ann Oper Res 62:29–57MathSciNetzbMATHCrossRefGoogle Scholar
  9. Gondzio J (1998) Warm start of the primal-dual method applied in the cutting-plane scheme. Math Programming 83(1, Ser. A):125–143MathSciNetzbMATHCrossRefGoogle Scholar
  10. Gondzio J, Grothey A (2003) Reoptimization with the primal-dual interior point method. SIAM J Optim 13(3):842–864MathSciNetzbMATHCrossRefGoogle Scholar
  11. Gondzio J, Grothey A (2008) A new unblocking technique to warmstart interior point methods based on sensitivity analysis. SIAM J Optim 19(3):1184–1210MathSciNetzbMATHCrossRefGoogle Scholar
  12. Gondzio J, Vial JP (1999) Warm start and ε-subgradients in a cutting plane scheme for block-angular linear programs. Comput Optim Appl 14(1):17–36MathSciNetzbMATHCrossRefGoogle Scholar
  13. John E, Yildirim EA (2008) Implementation of warm-start strategies in interior-point methods for linear programming in fixed dimension. Comput Optim Appl 41(2):151–183MathSciNetzbMATHCrossRefGoogle Scholar
  14. Karmarkar N (1984) A new polynomial-time algorithm for linear programming. Combinatorica 4(4):373–395MathSciNetzbMATHCrossRefGoogle Scholar
  15. Mitchell JE (1994) An interior point column generation method for linear programming using shifted barriers. SIAM J Optim 4(2):423–440MathSciNetzbMATHCrossRefGoogle Scholar
  16. Mitchell JE (2000) Computational experience with an interior point cutting plane algorithm. SIAM J Optim 10(4):1212–1227MathSciNetzbMATHCrossRefGoogle Scholar
  17. Mitchell JE, Borchers B (1996) Solving real-world linear ordering problems using a primal-dual interior point cutting plane method. Ann Oper Res 62:253–276MathSciNetzbMATHCrossRefGoogle Scholar
  18. Mitchell JE, Todd MJ (1992) Solving combinatorial optimization problems using Karmarkar's algorithm. Math Programming 56(3, Ser. A):245–284MathSciNetzbMATHCrossRefGoogle Scholar
  19. Mizuno S (1994) Polynomiality of infeasible-interior-point algorithms for linear programming. Math Programming 67(1, Ser. A):109–119MathSciNetzbMATHCrossRefGoogle Scholar
  20. Oberlin C, Wright SJ (2006) Active set identification in nonlinear programming. SIAM J Optim 17(2):577–605MathSciNetzbMATHCrossRefGoogle Scholar
  21. Polyak R (1992) Modified barrier functions (theory and methods). Math Programming 54(2, Ser. A):177–222MathSciNetzbMATHCrossRefGoogle Scholar
  22. Roos C, Terlaky T, Vial JP (2006) Interior point methods for linear optimization. Springer, New YorkzbMATHGoogle Scholar
  23. Toh K C, To dd MJ, Tütüncü RH (1999) SDPT3 — a MATLAB software package for semidefinite programming, version 1.3. Optim Methods Softw 11/12(1—4):545–581CrossRefGoogle Scholar
  24. Vanderbei RJ (1999) LOQO: an interior point code for quadratic programming. Optim Methods Softw 11/12(1–4):451–484MathSciNetCrossRefGoogle Scholar
  25. Wright SJ (1997) Primal-dual interior-point methods. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PAzbMATHCrossRefGoogle Scholar
  26. Yildirim EA, Wright SJ (2002) Warm-start strategies in interior-point methods for linear programming. SIAM J Optim 12(3):782–810MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Alexander Engau
    • 1
  • Miguel F. Anjos
    • 1
  • Anthony Vannelli
    • 2
  1. 1.University of WaterlooWaterlooCanada
  2. 2.University of GuelphGuelphCanada

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