Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content
Springer Nature Link
Log in

Cuts for mixed 0-1 conic programming

  • Published:
Mathematical Programming Submit manuscript

Abstract

In this we paper we study techniques for generating valid convex constraints for mixed 0-1 conic programs. We show that many of the techniques developed for generating linear cuts for mixed 0-1 linear programs, such as the Gomory cuts, the lift-and-project cuts, and cuts from other hierarchies of tighter relaxations, extend in a straightforward manner to mixed 0-1 conic programs. We also show that simple extensions of these techniques lead to methods for generating convex quadratic cuts. Gomory cuts for mixed 0-1 conic programs have interesting implications for comparing the semidefinite programming and the linear programming relaxations of combinatorial optimization problems, e.g. we show that all the subtour elimination inequalities for the traveling salesman problem are rank-1 Gomory cuts with respect to a single semidefinite constraint. We also include results from our preliminary computational experiments with these cuts.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alizadeh, F.: Interior point methods in semidefinite programming with applications to combinatorial optimization. SIAM J. Optim. 5 (1), 13–51, (1995)

    Article  MathSciNet  Google Scholar 

  2. Alizadeh, F., Goldfarb, D.: Second-order cone programming. Math. Prog. 95 (1, Ser. B), 3–51 2003

  3. Anjos, M.F., Wolkowicz, H.: Strengthened semidefinite relaxations via a second lifting for the max-cut problem. Disc. Appl. Math. 119 (1-2), 79–106 (2002)

    Google Scholar 

  4. Applegate, D., Bixby, R., Chvátal, V., Cook, W.: On the solution of traveling salesman problems. Documenta Mathematica Journal der Deutschen Mathematiker-Vereinigung, ICM III:645–656, 1998

  5. Balas, E.: Disjunctive programming. Annals of Discrete Mathematics 5, 3–51 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  6. Balas, E., Ceria, S., Cornuéjols., G.: A lift-and-project cutting plane algorithm for mixed 0-1 programs. Mathematical Programming 58, 295–324 (1993)

    Article  MathSciNet  Google Scholar 

  7. Barahona, F., Mahjoub., A.R.: On the cut polytope. Math. Prog. 36, 157–173 (1986)

    MATH  MathSciNet  Google Scholar 

  8. Ben-Tal, A., Margalit, T., Nemirovski, A.: Robust modeling of multi-stage portfolio problems. In: High performance optimization, pp 303–328. Kluwer Acad. Publ., Dordrecht, 2000

  9. Ben-Tal, A., Nemirovski, A.: Robust convex optimization. Math. Oper. Res. 23 (4), 769–805 (1998)

    Article  MathSciNet  Google Scholar 

  10. Benson, H.Y., Vanderbei., R.J.: Solving problems with semidefinite and related constraints using interior-point methods for nonlinear programming. Math. Prog. 95 (2 ser. B), 279–302 (2003)

    Google Scholar 

  11. Bixby, R.E., Ceria, S., McZeal, C.M., Savelsberg, M.W.P.: A mixed integer programming library: MIPLIB3.0. Available at http://www.caam.rice.edu/~bixby/miplib/miplib.html.

  12. Boyd, S., El Ghaoui, L., Feron, E., Balakrishnan, V.: Linear matrix inequalities in system and control theory. SIAM, Philadelphia, PA, 1994

  13. Çezik, M.T.: Semidefinite methods for the traveling salesman and other combinatorial problems. PhD thesis, IEOR Dept., Columbia University, 2000

  14. Çezik, M.T., Iyengar, G.: Quadratic cuts for mixed 0-1 quadratic programs. Technical Report TR-2001-02, CORC, IEOR Dept., Columbia University, 2001. Available at http://www.corc.ieor.columbia.edu/reports/techreports/tr-2001- 02.ps.gz

  15. Cvetković, D., Čangalović, M., Kovačević-Vujčić, V.: Semidefinite programming methods for the symmetric traveling salesman problem. In: IPCO VII (Graz, 1999), pp 126–136. Springer, Berlin, 1999

  16. El Ghaoui, L., Oustry, F., Lebret., H.: Robust solutions to uncertain semidefinite programs. SIAM J. Optim. 9 (1), 33–52 (1998)

    Article  MathSciNet  Google Scholar 

  17. Goemans., M.: Semidefinite programming in combinatorial optimization. Math. Prog. 79, 143–161 (1997)

    Article  Google Scholar 

  18. Goemans, M., Rendl, F.: Combinatorial optimization. In: Handbook of Semidefinite Programming. Kluwer, 2000

  19. Goemans, M., Williamson., D.P.: Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. ACM 42, 1115–1145 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  20. Goldfarb, D.: The simplex method for conic programming. Technical Report TR-2002-05, CORC, IEOR Dept., Columbia University, 2002. Available at http://www.corc.ieor.columbia.edu/reports/techreports/tr-2002-05.ps

  21. Goldfarb, D., Iyengar., G.: Robust portfolio selection problems. Math. Oper. Res. 28 (1), 1–38 (2003)

    Article  MathSciNet  Google Scholar 

  22. Helmberg, C.: A cutting plane algorithm for large scale semidefinite relaxations. In: The sharpest cut, MPS/SIAM Ser. Optim., pages 233–256. SIAM, Philadelphia, PA, 2004

  23. Helmberg, C., Rendl., F.: Solving quadratic (0,1)-problems by semidefinite programs and cutting planes. Math. Prog. 82 (3), 291–315 (1998)

    MathSciNet  Google Scholar 

  24. Karloff., H.: How good is the Goemans-Williamson maxcut algorithm. SIAM J. Comp. 29 (1), 336–350 (1999)

    Google Scholar 

  25. Lasserre., J.B.: Global optimization with polynomials and the methods of moments. SIAM J. Optim. 1, 796–817 (2001)

    Google Scholar 

  26. Laurent., M.: A comparison of Sherali-Adams, Lovász-Schrijver and lasserre relaxations for 0-1 programming. Math. Oper. Res. 28 (3), 470–496 (2003)

    Article  MathSciNet  Google Scholar 

  27. Lawler, E.L., Lenstra, J.K., Rinnooy Kan, A.H.G., Shmoys, D.B. (eds).: The traveling salesman problem. John Wiley & Sons Ltd., Chichester, 1990

  28. Lemke, C., Spielberg., K.: A capital budgeting heuristic algorithm using exchange operations. AIEE Transactions 6, 264–280 (1974)

    Google Scholar 

  29. Lobo, M.S., Fazel, M., Boyd, S.: Portfolio optimization with linear and fixed transaction costs and bounds on risk. Under review in Ann. Oper. Res. Available at http://www.stanford.edu/~boyd/reports/portfolio.pdf.

  30. Lobo, M.S., Vandenberghe, L., Boyd, S., Lebret., H.: Applications of second-order cone programming. Linear Algebra Appl., 284 (1 (3), 193–228 (1998)

    MathSciNet  Google Scholar 

  31. Lovász, L., Schrijver., A.: Cones of matrices and set-functions and 0-1 optimization. SIAM J. Optim. 1 (2), 166–190 (1991)

    Article  Google Scholar 

  32. Mitchell., J.B.: Fixing variables and generating classical cutting planes when using an interior point branch-and-cut method to solve integer programming problems. Eur. J. Oper. Res. 97, 139–148 (1997)

    Article  Google Scholar 

  33. Mitchell., J.B.: Computational experience with an interior point cutting plane algorithm. SIAM J. Optim. 10 (4), 1212–1227 (2000)

    Google Scholar 

  34. Mitchell, J.B., Borchers, B.: Solving linear ordering problems with a combined interior point/simplex cutting plane algorithm. In: High performance optimization, pp 349–366. Kluwer Acad. Publ., Dordrecht, 2000

  35. Mitchell., J.E.: Restarting after branching in the SDP approach to MAX-CUT and similar combinatorial optimization problems. J. Comb. Opt. 5 (2), 151–166 (2001)

    Google Scholar 

  36. Nemhauser, G.L., Wolsey, L.A.: Integer and combinatorial optimization. Wiley-Interscience series in Discrete Mathematics. John Wiley & Sons, Inc., 1999

  37. Nesterov, Y., Nemirovskii, A.: Interior-point polynomial algorithms in convex programming. SIAM, Philadelphia, 1993

  38. Petersen., C.: Computational experience with variants of the Balas algorithm applied to the selection of R&D projects. Mgmt. Sc. 13, 736–750 (1967)

    Article  Google Scholar 

  39. Reinelt., G.: TSPLIB— A traveling salesman problem library. ORSA J. Comp. 3 (4), 376–384 (1991)

    Google Scholar 

  40. Sherali, H.D., Adams., W.P.: A hierarchy of relaxations between the continuous and convex hull representations of zero-one programming problems. SIAM J. Disc. Math. 3, 411–430 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  41. Sherali, H.D., Adams., W.P.: A hierarchy of relaxations for mixed-integer zero-one programming problems. Discrete Appl. Math. 52, 83–106 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  42. Stubbs, R.A., Mehrotra., S.: A branch-and-cut method for 0-1 mixed convex programming. Math. Prog. 86, 515–532 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  43. Stubbs, R.A., Mehrotra., S.: Generating convex polynomial inequalities for mixed 0-1 programs. J. Global Optim. 24 (3), 311–332 (2002)

    Article  MathSciNet  Google Scholar 

  44. Stürm., J.: Using SeDuMi 1. 02, a MATLAB toolbox for optimization over symmetric cones. Optimization Methods and Software 625 (653), 11–12 (1999)

    Google Scholar 

  45. Vandenberghe, L., Boyd, S.: Semidefinite programming. SIAM Review, 1996

Download references

Author information

Authors and Affiliations

  1. GERAD and Départment d’Informatique et de Recherche Opérationnelle, Université de Montréal, Montréal, H3C 3J7, Canada

    M.T. Çezik

  2. IEOR Department, Columbia University New York, New York, 10027, USA

    G. Iyengar

Authors
  1. M.T. Çezik
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. G. Iyengar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to G. Iyengar.

Additional information

Research partially supported by NSF grants CCR-00-09972, DMS-01-04282 and ONR grant N000140310514.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Çezik, M., Iyengar, G. Cuts for mixed 0-1 conic programming. Math. Program. 104, 179–202 (2005). https://doi.org/10.1007/s10107-005-0578-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10107-005-0578-3

Keywords

Search

Navigation