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

Strong valid inequalities for orthogonal disjunctions and bilinear covering sets

  • Full Length Paper
  • Series B
  • Published:
Mathematical Programming Submit manuscript

Abstract

In this paper, we derive a closed-form characterization of the convex hull of a generic nonlinear set, when this convex hull is completely determined by orthogonal restrictions of the original set. Although the tools used in this construction include disjunctive programming and convex extensions, our characterization does not introduce additional variables. We develop and apply a toolbox of results to check the technical assumptions under which this convexification tool can be employed. We demonstrate its applicability in integer programming by providing an alternate derivation of the split cut for mixed-integer polyhedral sets and finding the convex hull of certain mixed/pure-integer bilinear sets. We then extend the utility of the convexification tool to relaxing nonconvex inequalities, which are not naturally disjunctive, by providing sufficient conditions for establishing the convex extension property over the non-negative orthant. We illustrate the utility of this result by deriving the convex hull of a continuous bilinear covering set over the non-negative orthant. Although we illustrate our results primarily on bilinear covering sets, they also apply to more general polynomial covering sets for which they yield new tight relaxations.

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. Adjiman C.S., Androulakis I.P., Floudas C.A.: A global optimization method, αBB, for general twice-differentiable constrained NLPs—II. Implementation and computational results. Comput. Chem. Eng. 22, 1159–1179 (1998)

    Article  Google Scholar 

  2. Atamtürk A.: Strong formulations of robust mixed 0–1 programming. Math. Program. 108, 235–250 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  3. Balas E.: Intersection cuts—a new type of cutting planes for integer programming. Oper. Res. 19, 19–39 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  4. Balas E.: Disjunctive programming: cutting planes from logical conditions. In: Mangasarian, O.L., Meyer, R.R., Robinson, S.M. (eds) Nonlinear Programming, pp. 279–312. Academic Press, New York (1975)

    Google Scholar 

  5. Balas E.: Disjunctive programming. Ann. Discret. Math. 5, 3–51 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  6. Balas E.: Disjunctive programming and a hierarchy of relaxations for discrete optimization problems. SIAM J. Algebraic Discret. Methods 6, 466–486 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  7. Balas, E.: Disjunctive programming: properties of the convex hull of feasible points. Discret. Appl. Math. 89(1–3), 3–44, original manuscript was published as a technical report in 1974 (1998)

  8. Balas E.: Projection, lifting and extended formulation in integer and combinatorial optimization. Ann. Oper. Res. 140, 125–161 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  9. Balas E., Perregaard M.: A precise correspondence between lift-and-project cuts, simple disjunctive cuts, and mixed-integer gomory cuts for 0–1 programming. Math. Program. 94, 221–245 (2003)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  11. Balas E., Bockmayr A., Pisaruk N., Wolsey L.: On unions and dominants of polytopes. Math. Program. 99, 223–239 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  12. Belotti, P., Lee, J., Liberti, L., Margot, F., Waechter, A.: Branching and bounds tightening techniques for non-convex MINLP. http://www.optimization-online.org/DB_HTML/2008/08/2059.html (2008)

  13. Bliek, C., Jermann, C., Neumaier, A. (eds.): Global Optimization and Constraint Satisfaction, 5th Annual Workshop on Global Constraint Optimization and Constraint Satisfaction, COCOS, Springer (2002)

  14. Ceria S., Soares J.: Convex programming for disjunctive convex optimization. Math. Program. 86, 595–614 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  15. Cook W., Kannan R., Schrijver A.: Chvátal closures for mixed integer programming problems. Math. Program. 47, 155–174 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  16. Cornuéjols G., Lemaréchal C.: A convex-analysis perspective on disjunctive cuts. Math. Program. 106, 567–586 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  17. Falk J.E., Soland R.M.: An algorithm for separable nonconvex programming problems. Manag. Sci. 15, 550–569 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  18. Fukuda K., Liebling T.M., Lütolf C.: Extended convex hull. Comput. Geom. 20, 13–23 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  19. Harjunkoski I., Westerlund T., Porn R., Skrifvars H.: Different transformations for solving non-convex trim-loss problems by MINLP. Eur. J. Oper. Res. 105, 594–603 (1998)

    Article  MATH  Google Scholar 

  20. Horst R., Tuy H.: Global Optimization: Deterministic Approaches, 3rd edn. Springer, Berlin (1996)

    MATH  Google Scholar 

  21. LINDO Systems Inc: LINGO 11.0 optimization modeling software for linear, nonlinear, and integer programming. http://www.lindo.com (2008)

  22. McCormick G.P.: Computability of global solutions to factorable nonconvex programs: Part I—convex underestimating problems. Math. Program. 10, 147–175 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  23. Neumaier A.: Complete search in continuous global optimization and constraint satisfaction. Acta Numerica 13, 271–369 (2004)

    MathSciNet  Google Scholar 

  24. Rockafellar R.T.: Convex Analysis. Princeton University Press, Princeton (1970)

    MATH  Google Scholar 

  25. Sahinidis, N.V., Tawarmalani, M.: BARON. The Optimization Firm, LLC, Urbana-Champaign, IL. http://www.gams.com/dd/docs/solvers/baron.pdf (2005)

  26. Sawaya N.W., Grossmann I.E.: A cutting plane method for solving linear generalized disjunctive programming problems. Comput. Chem. Eng. 29, 1891–1913 (2005)

    Article  Google Scholar 

  27. Sherali H.D., Sen S.: Cuts from combinatorial disjunctions. Oper. Res. 33, 928–933 (1985)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  29. Tawarmalani M., Sahinidis N.V.: Semidefinite relaxations of fractional programs via novel techniques for constructing convex envelopes of nonlinear functions. J. Glob. Optim. 20, 137–158 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  30. Tawarmalani M., Sahinidis N.V.: Convex extensions and envelopes of lower semi-continuous functions. Math. Program. 93, 247–263 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  31. Tawarmalani M., Sahinidis N.V.: Convexification and Global Optimization in Continuous and Mixed-Integer Nonlinear Programming: Theory, Algorithms, Software, and Applications. Kluwer, Dordrecht (2002)

    MATH  Google Scholar 

  32. Tawarmalani M., Sahinidis N.V.: A polyhedral branch-and-cut approach to global optimization. Math. Program. 103, 225–249 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  33. Tawarmalani, M., Richard, J.P.P., Chung, K.: Strong Valid Inequalities for Orthogonal Disjunctions and Polynomial Covering Sets, Technical Report, Krannert School of Management, Purdue University (2008)

  34. Ziegler G.M.: Lectures on Polytopes. Springer, New York (1998)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Krannert School of Management, Purdue University, 100 S. Grant Street, West Lafayette, IN, 47907-2076, USA

    Mohit Tawarmalani

  2. Department of Industrial and Systems Engineering, University of Florida, 303 Weil Hall, Gainesville, FL, 32611-6595, USA

    Jean-Philippe P. Richard & Kwanghun Chung

Authors
  1. Mohit Tawarmalani
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jean-Philippe P. Richard
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Kwanghun Chung
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mohit Tawarmalani.

Additional information

The work was supported in part by NSF grant CMMI 0900065 and 0856605.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Tawarmalani, M., Richard, JP.P. & Chung, K. Strong valid inequalities for orthogonal disjunctions and bilinear covering sets. Math. Program. 124, 481–512 (2010). https://doi.org/10.1007/s10107-010-0374-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10107-010-0374-6

Mathematics Subject Classification (2000)

Search

Navigation