Journal of Global Optimization

, Volume 2, Issue 1, pp 21–40 | Cite as

The complementary convex structure in global optimization

  • Hoang Tuy


We show the importance of exploiting the complementary convex structure for efficiently solving a wide class of specially structured nonconvex global optimization problems. Roughly speaking, a specific feature of these problems is that their nonconvex nucleus can be transformed into a complementary convex structure which can then be shifted to a subspace of much lower dimension than the original underlying space. This approach leads to quite efficient algorithms for many problems of practical interest, including linear and convex multiplicative programming problems, concave minimization problems with few nonlinear variables, bilevel linear optimization problems, etc...

Key words

Complementary convex structure Generalized Rank k Property global optimization 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Thach, P. T. (1987), D. C. Sets, D. C. Functions and Systems of Equations, Preprint, Institute of Mathematics, Hanoi. To appear in Mathematical Programming.Google Scholar
  2. 2.
    Tuy, H. and R., Horst (1991), The Geometric Complementarity Problem and Transcending Stationarity in Global Optimization in “Applied Geometry and Discrete Mathematics”, DIMACS Series in Discrete Mathematics and Theoretical Computer Science 4, 341–354.Google Scholar
  3. 3.
    Henderson, J. M. and R. E. Quant (1971), Microeconomic Theory, McGraw-Hill.Google Scholar
  4. 4.
    Maling, K., S. H. Mueller, and W. R. Heller (1982), On Finding Most Optimal Rectangular Package Plans, Proceedings of the 19th Design Automation Conference, 663–670.Google Scholar
  5. 5.
    Konno, H. and M., Inori (1988), Bon Portfolio Optimization by Bilinear Fractional Programming, J. of Oper. Res. of Japan 32, 143–158.Google Scholar
  6. 6.
    Paradalos, P. M. (1988), Polynomial Time Algorithms for Some Classes of Constrained Non-Convex Quadratic Problems, Preprint, Computer Science Department, the Pennsylvania State University.Google Scholar
  7. 7.
    Forgo, F. (1975), The Solution of a Special Quadratic Problem, Szigma, 53–59 (in Hungarian).Google Scholar
  8. 8.
    Gabasov, R. and F. M. Kirillova (1980), Linear Programming Methods, Part 3 (Special Problems), Minsk (in Russian).Google Scholar
  9. 9.
    Pardalos, P. M. (1988), On the Global Minimization of the Product of Two Linear Functions over a Polytope, Preprint, Computer Science Department, The Pennsylvania State University.Google Scholar
  10. 10.
    Konno H. and T. Kuno (1989), Linear Multiplicative Programming, IHSS Report 89-13, Institute of Human and Social Sciences, Tokyo Institute of Technology.Google Scholar
  11. 11.
    Tuy, H. and B. T. Tam (1990), An Efficient Solution Method for Rank Two Quasiconcave Minimization Problems, to appear in Optimization.Google Scholar
  12. 12.
    Idrissi, H., P., Loridan, and C., Michelot (1988), Approximation of Solutions for Location Problems, Journal of Optimization Theory and Applications 56, 127–143.Google Scholar
  13. 13.
    Tuy, H. and Faiz A. Al-Khayyal (1991), A Class of Global Optimization Problems Solvable by Sequential Unconstrained Convex Minimization, to appear in C. Floudas and P. Pardalos (eds), Recent Advances in Global Optimization, Princeton University Press.Google Scholar
  14. 14.
    Ben-Ayed, O. and C. E., Blair (1990), Computational Difficulties of Bilevel Linear Programming, Operations Research 38, 556–560.Google Scholar
  15. 15.
    Tuy, H., A. Migdalas, and P. Värbrand (1990), A Global Optimization Approach for the Linear Two Level Program, Preprint, Department of Mathematics, Linköping University (submitted).Google Scholar
  16. 16.
    Tuy, H. (1990), On Polyhedral Annexation Method for Concave Minimization, in Lev, J. Leifman (ed.), Functional Analysis, Optimization, and Mathematical Economics (volume dedicated to the memory of Kantorovich), Oxford University Press, New York, 248–260.Google Scholar
  17. 17.
    Konno, H. (1976), A Cutting Plane Algorithm for Solving Bilinear Programs, Mathematical Programming 11, 14–27.Google Scholar
  18. 18.
    Tuy, H. (1987), Convex Programs with an Additional Reverse Convex Constraint, Journal of Optimization Theory and Applications 52, 463–485.Google Scholar
  19. 19.
    Tuy, H. (1991), Polyhedral Annexation, Dualization, and Dimension Reduction Technique in Global Optimization, Journal of Global Optimization, 1, 229–244.Google Scholar
  20. 20.
    Thach, P. T. (1991), Quasiconjugates of Functions and Duality Correspondence between Quasiconcave Minimization under a Reverse Convex Constraint and Quasi Convex Maximization under a Convex Constraint, Journal of Mathematical Analysis and Applications.Google Scholar
  21. 21.
    Thach, P. T., and R. E. Burkard (1990), Reverse Convex Programs Dealing with the Product of Two Linear Functions, Preprint, Institute of Mathematics, Graz Technical University.Google Scholar
  22. 22.
    Thach, P. T. and H., Tuy (1990), Dual Solution Methods for Concave Programs and Reverse Convex Programs, Preprint, IHSS, Tokyo, Institute of Technology.Google Scholar
  23. 23.
    Horst, R. and H. Tuy (1990), Global Optimization (Deterministic Approaches), Springer-Verlag.Google Scholar
  24. 24.
    Kuno, T. and H. Konno (1990), Parametric Successive Underestimation Method for Convex Multiplicative Programming Problems, Preprint IHSS, Tokyo Institute of Technology.Google Scholar
  25. 25.
    Konno H. and T. Kuno (1991), Generalized Linear Multiplicative and Fractional Programming, to appear in Annals of Operations Research.Google Scholar
  26. 26.
    Rockafella, R. T. (1970), Convex Analysis, Princeton University Press.Google Scholar
  27. 27.
    Horst, R., N. V., Thoai, and, Vries (1988), On Finding New Vertices and Redundant Constraints in Cutting Plane Algorithms for Global Optimization, Operations Research Letters 7, 85–90.Google Scholar
  28. 28.
    Tuy, H. (1991), Normal Conical Algorithm for Concave Minimization over Polytopes, Mathematical Programming, 51, 229–245.Google Scholar
  29. 29.
    Falk, J. E. (1973), A Linear Max-Min Problem, Mathematical Programming 5, 169–188.Google Scholar
  30. 30.
    Tuy, H. and N. V., Thuong (1988), On the Global Minimization of a Convex Function under General Nonconvex Constraints, Applied Mathematics and Optimization 18, 119–142.Google Scholar
  31. 31.
    Kuno T. and H. Konno (1990), A Parametric Successive Underestimation Method for Convex Programming Problems with an Additional Convex Multiplicative Constraint, Preprint, IHSS, Tokyo Institute of Technology.Google Scholar

Copyright information

© Kluwer Academic Publishers 1992

Authors and Affiliations

  • Hoang Tuy
    • 1
  1. 1.Institute of MathematicsHanoi

Personalised recommendations