Journal of Global Optimization

, Volume 2, Issue 1, pp 1–19 | Cite as

A new simplicial cover technique in constrained global optimization

  • R. Horst
  • N. V. Thoai
  • J. De Vries


A simplicial branch and bound-outer approximation technique for solving nonseparable, nonlinearly constrained concave minimization problems is proposed which uses a new simplicial cover rather than classical simplicial partitions. Some geometric properties and convergence results are demonstrated. A report on numerical aspects and experiments is given which shows that the most promising variant of the cover technique can be expected to be more efficient than comparable previous simplicial procedures.

Key words

Constrained global optimization concave minimization simplicial covers branch and bound outer approximation 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Al-Khayyal, F. A. and Falk, J. E. (1983), Jointly Constrained Biconvex Programming, Mathematics of Operations Research 8, 273–286.Google Scholar
  2. Benson, H. P. (1990), Separable Concave Minimization via Partial Outer Approximation and Branch and Bound, Operations Research Letters 9, 389–394.Google Scholar
  3. Benson, H. P. and Erenguc, S. (1988), Using Convex Envelopes to Solve the Interactive Fixed Charge Linear Programming Problem, Journal of Optimization Theory and Applications 59, 223–246.Google Scholar
  4. Benson, H. P. and Horst, R. (1991), A Branch and Bound-Outer Approximation Algorithm for Concave Minimization over a Convex Set, J. Computers and Mathematics with Applications 21, 67–76.Google Scholar
  5. Chen, P. C., Hansen, P., and Jaumard, B. (1991), On-Line and Off-Line Vertex Enumeration by Adjacency Lists, to appear in Operations Research Letters.Google Scholar
  6. Eaves, B. C. and Zangwill, W. I. (1971), Generalized Cutting Plane Algorithms, SIAM Journal on Control 9, 529–542.Google Scholar
  7. Falk, J. E. and Soland, R. M. (1969), An Algorithm for Separable Nonconvex Programming Problems, Management Science 15, 550–569.Google Scholar
  8. Hoffman, K. L. (1981), A Method for Globally Minimizing Concave Functions over Convex Sets, Mathematical Programming 20, 22–32.Google Scholar
  9. Horst, R. (1976), An Algorithm for Nonconvex Programming Problems, Mathematical Programming 10, 312–321.Google Scholar
  10. Horst, R. (1980), A Note on the Convergence of an Algorithm for Noncovex Programming Problems, Mathematical Programming 19, 237–238.Google Scholar
  11. Horst, R. (1986), A General Class of Branch and Bound Methods in Global Optimization with Some New Approaches for Concave Minimization, Journal of Optimization Theory and Applications 51, 271–291.Google Scholar
  12. Horst, R. and Thoai, N. V., (1989), Modification, Implementation and Comparison of Three Algorithms for Globally Solving Linearly Constrained Concave Minimization Problems, Computing 42, 271–289.Google Scholar
  13. Horst, R., Thoai, N. V., and Benson, H. P. (1991), Concave Minimization via Conical Partitions and Polyhedral Outer Approximation, to appear in Mathematical Programming.Google Scholar
  14. Horst, R., Thoai, N. V., and de, Vries, J. (1988), On Finding New Vertices and Redundant Constraints in Cutting Plane Algorithms for Global Optimization, Operations Research Letters 7, 85–90.Google Scholar
  15. Horst, R., Thoai, N. V., and de Vries, J. (1991), On Geometry and Convergence of Simplicial Covers, submitted for publication.Google Scholar
  16. Horst, R., Thoai, N. V., and Tuy, H. (1987), Outer Approximation by Polyhedral Convex Sets, Operations Research Spektrum 9, 153–159.Google Scholar
  17. Horst, R., Thoai, N. V., and Tuy, H. (1989), On an Outer Approximation Concept in Global Optimization, Optimization 20, 255–264.Google Scholar
  18. Horst, R., and Tuy, H. (1987), On the Convergence of Global Methods in Multiextremal Optimization, Journal of Optimization Theory and Applications 54, 253–271.Google Scholar
  19. Horst, R. and Tuy, H. (1990), Global Optimization (Deterministic Approaches), Springer, Berlin.Google Scholar
  20. Pardalos, P. M., and Rosen, J. B. (1987), Constrained Global Optimization: Algorithms and Applications, Lecture Notes in Computer Science 268, Springer-Verlag, Berlin.Google Scholar
  21. Thieu, T. V., Tam, B. T., and Ban, T. V. (1983), An Outer Approximation Method for Globally Minimizing a Concave Function over a Compact Convex Set, Acta Mathematica Vietnamica 8, 21–40.Google Scholar
  22. Thoai, N. V. and de, Vries, J. (1988), Numerical Experiments on Concave Minimization Problems, Proceedings XIII. Symposium on Operations Research, Methods of Operations Research 60, 363–365.Google Scholar
  23. Tuy, H. (1991), Effect of the Subdivision Strategy on Convergence and Efficiency of Some Global Optimization Algorithms. J. of Global Optimization 1, 23–36.Google Scholar
  24. Tuy, H. and Horst, R. (1988), Convergence and Restart in Branch and Bound Algorithms for Global Optimization, Application to Concave Minimization and DC-Optimization Problems, Mathematical Programming 41, 161–183.Google Scholar
  25. Tuy, H., Thieu, T. V., and Thai, N. Q. (1985), A Conical Algorithm for Globally Minimizing a Concave Function Over a Closed Convex Set, Mathematics of Operations Research 10, 498–515.Google Scholar
  26. Veinott, A. F. (1967), Minimum Concave Cost Solution of Leontiev Substitution Models of Multi-Facility Inventory Systems, Operations Research 14, 486–507.Google Scholar

Copyright information

© Kluwer Academic Publishers 1992

Authors and Affiliations

  • R. Horst
    • 1
  • N. V. Thoai
    • 1
  • J. De Vries
    • 1
  1. 1.Fachbereich IV-MathematikUniversity of TrierTrierGermany

Personalised recommendations