Skip to main content
SpringerLink
Log in

A new necessary and sufficient global optimality condition for canonical DC problems

  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

The paper proposes a new necessary and sufficient global optimality condition for canonical DC optimization problems. We analyze the rationale behind Tuy’s standard global optimality condition for canonical DC problems, which relies on the so-called regularity condition and thus can not deal with the widely existing non-regular instances. Then we show how to modify and generalize the standard condition to a new one that does not need regularity assumption, and prove that this new condition is equivalent to other known global optimality conditions. Finally, we show that the cutting plane method, when associated with the new optimality condition, could solve the non-regular canonical DC problems, which significantly enlarges the application of existing cutting plane (outer approximation) algorithms.

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

Access this article

Log in via an institution

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. Bazaraa M.S., Sherali H.D.: On the use of exact and heuristic cutting plane methods for the quadratic assignment problem. J. Opl. Res. Soc. 33, 991–1003 (1982)

    Google Scholar 

  2. Ben Saad S., Jacobsen S.: A level set algorithm for a class of reverse convex programs. Ann. Oper. Res. 25, 19–42 (1990)

    Article  Google Scholar 

  3. Ben Saad S., Jacobsen S.: A new cutting plane algorithm for a class of reverse convex 0-1 integer programs. In: Floudas, C.A., Pardalos, P.M. (eds.) Recent Advances in Global Optimization, Princeton Series in Computer Science, pp. 152–164. Princeton University Press, Princeton (1991)

    Google Scholar 

  4. Ben Saad S., Jacobsen S.: Comments on a reverse convex programming algorithm. J. Global Optim. 5, 95–96 (1994)

    Article  Google Scholar 

  5. Bienstock D., Zuckerberg M.: Subset algebra lift operators for 0-1 integer programming. SIAM J. Optim. 15(1), 63–95 (2005)

    Article  Google Scholar 

  6. Bigi G., Frangioni A., Zhang Q.: Approximate optimality conditions and stopping criteria in canonical DC programming. Optim. Methods Softw. 25, 19–27 (2010)

    Article  Google Scholar 

  7. Bigi G., Frangioni A., Zhang Q.: Outer approximation algorithms for canonical DC problems. J. Global Optim. 46, 163–189 (2010)

    Article  Google Scholar 

  8. Burkard R., Cela E., Pardalos P., Pitsoulis L.: The quadratic assignment problem. In: Pardalos, P.M., Resende, M. (eds.) Handbook of Combinatorial Optimization, pp. 241–337. Kluwer, Dordrecht (1998)

    Google Scholar 

  9. Diaby M.: Implicit enumeration for the pure integer 0/1 minimax programming problem. Oper. Res. 41, 1172–1176 (1993)

    Article  Google Scholar 

  10. Dur M., Horst R., Locatelli M.: Necessary and sufficient global optimality conditions for convex maximization revisited. J. Math. Anal. Appl. 217, 637–649 (1998)

    Article  Google Scholar 

  11. Fulop J.A.: Finite cutting plane method for solving linear programs with an additional reverse constraint. Eur. J. Oper. Res. 44, 395–409 (1990)

    Article  Google Scholar 

  12. Grippo L., Sciandrone M.: On the convergence of the block nonlinear gauss-seidel method under convex constraints. Oper. Res. Lett. 26, 127–136 (2000)

    Article  Google Scholar 

  13. Hiriart-Urruty J.: When is a point satisfying \({\nabla f(x) = 0}\) a global minimum of f?.  Am. Math. Monthly 93, 556–558 (1986)

    Article  Google Scholar 

  14. Hiriart-Urruty J.: Conditions for global optimality. In: Horst, R., Pardalos, P.M. (eds.) Handbook of Global Optimization, Kluwer, Dordrechtr (1995)

    Google Scholar 

  15. Hiriart-Urruty J., Ledyaev Y.S.: A note on the characterization of the global maxima of a (tangentially) convex function over a convex set. J. Convex Anal. 3, 55–61 (1996)

    Google Scholar 

  16. Horst R., Pardalos P., Thoai N.: Introduction to Global Optimization, 2nd edn. Kluwer, Dordrecht (2000)

    Google Scholar 

  17. Horst, R., Thoai, N.: DC programming. In: Floudas C, Pardalos P (eds) Encyclopedia of Optimization. Nonconvex Optimizational Applications. Kluwer, Dordrecht, Holland, vol. 53, pp. 375–378 (2001)

  18. Horst R., Tuy H.: Global Optimization. Springer, Berlin (1990)

    Google Scholar 

  19. Nghia M., Hieu N.: A method for solving reverse convex programming problems. Acta Math. Vietnam. 11, 241–252 (1986)

    Google Scholar 

  20. Rikun A.D.: A convex envelope formula for multilinear functions. J. Global Optim. 10, 425–437 (1997)

    Article  Google Scholar 

  21. Singer I.: A Fenchel-Rockafellar type duality theorem for maximization. Bull. Austral. Math. Soc. 20, 193–198 (1979)

    Article  Google Scholar 

  22. Singer I.: Some further duality theorem for optimization problems with reverse convex constraint sets. J. Math. Anal. Appl. 171, 205–219 (1992)

    Article  Google Scholar 

  23. Strekalovsky A.: On the global extremum problem. Soviet Dokl 292, 1062–1066 (1987)

    Google Scholar 

  24. Strekalovsky A.: On problems of global extremum in nonconvex extremal problems. Izv. Vozou. Mat. 8, 74–80 (1990)

    Google Scholar 

  25. Strekalovsky A.: On conditions for global extremum in a nonconvex minimization problem. Izv. Vozou. Mat. 2, 94–96 (1991)

    Google Scholar 

  26. Strekalovsky A., Tsevendorj I.: Testing the \({\mathbb{R}}\)-strategy for a reverse convex problem. J. Global Optim. 13, 61–74 (1998)

    Article  Google Scholar 

  27. Tao P., El Bernoussi S.: Numerical methods for solving a class of global nonconvex optimization problems. Int. Ser. Numer. Math. 87, 97–132 (1989)

    Google Scholar 

  28. Thach P.: Convex programs with several additional reverse convex constraints. Acta Math. Vietnam. 10, 35–57 (1985)

    Google Scholar 

  29. Thoai N.: A modified version of Tuy’s method for solving d.c. programming problems. Optimization 19, 665–674 (1988)

    Article  Google Scholar 

  30. Thoai N.: On Tikhonov’s reciprocity principle and optimality conditions in DC optimization. J. Math. Anal. Appl. 225, 673–678 (1998)

    Article  Google Scholar 

  31. Toland J.F.: Duality in nonconvex optimization. J. Math. Anal. Appl. 66, 399–415 (1978)

    Article  Google Scholar 

  32. Toland J. F.: A duality principle for nonconvex optimization and the calculus of variations. Arch. Rational Mech. Anal. 71, 41–61 (1979)

    Article  Google Scholar 

  33. Tuan H.: Remarks on an algorithm for reverse convex programs. J. Global Optim. 16, 295–297 (2000)

    Article  Google Scholar 

  34. Tuy H.: A general deterministic approach to global optimization via d.c. programming. In: Hiriart-Urruty, J. (Ed.) FERMAT days 85: mathematics for optimization. North-Holland Mathematical Studies 129, pp. 273–303. North-Holland Publishing Co., Toulouse (1986)

    Chapter  Google Scholar 

  35. Tuy H.: Convex programs with an additional reverse convex constraint. J. Optim. Theory Appl. 52, 463–486 (1987)

    Article  Google Scholar 

  36. Tuy H.: Global minimization of a difference of two convex functions. Math. Program. Study 30, 150–182 (1987)

    Article  Google Scholar 

  37. Tuy H.: Canonical DC programming problem: outer approximation methods revisited. Oper. Res. Lett. 18, 99–106 (1995)

    Article  Google Scholar 

  38. Tuy H.: D.C.optimization: theory, methods and algorithms. In: Horst, R., Pardalos, P. (eds.) Handbook of Global Optimization, Nonconvex Optimization and Its Applications, 2, pp. 149–216. Kluwer, Dordrecht (1995)

    Google Scholar 

  39. Tuy H.: On global optimality conditions and cutting plane algorithms. J. Optim. Theory Appl. 118, 201–216 (2003)

    Article  Google Scholar 

  40. Zhang, Q.: Outer approximation algorithms for DC programs and beyond. Ph.D. thesis. University of Pisa, Pisa, Italy (2008)

  41. Zhang Q.: Outer approximation algorithms for DC programs and beyond. 4OR-Q J. Oper. Res. 8, 323–326 (2010)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Wuhan University, Wuchang, Luojia Hill, Wuhan, 430072, China

    Qinghua Zhang

Authors
  1. Qinghua Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Qinghua Zhang.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhang, Q. A new necessary and sufficient global optimality condition for canonical DC problems. J Glob Optim 55, 559–577 (2013). https://doi.org/10.1007/s10898-012-9908-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-012-9908-1

Keywords

Search

Navigation