On the Vertex Enumeration Problem in Cutting Plane Algorithms of Global Optimization

  • Reiner Horst

Abstract

We consider the following vertex enumeration problem:
  1. (VE)

    Given a hyperplane H and a polytope P with known vertex set V(P) find the vertex set of the polytope \(\bar P = P \cap H\).

     

Keywords

Hull 

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References

  1. Bulatov, V.P.: Embedding Methods in Optimization Problems, Nauka, Novosibirsk 1977 (in Russian).Google Scholar
  2. Bulatov, V.P., and Kasinkaya, L.I.: Some Methods of Concave Minimization on a Convex Polyhedron and their Applications, in: Methods of Optimization and their Applications, Nauka, Novosibirsk 1982, pp. 71–80 (in Russian).Google Scholar
  3. Chen, P.C., Hansen, P., and Jaumard, B.: On-line and Off-line Vertex Enumeration by Adjacency Lists, Discussion Paper, Rutcor and Gerard, 1990.Google Scholar
  4. Dyer, M.E.: The Complexity of Vertex Enumeration Methods, in: Mathematics of Operations Research 8, 1983, pp. 381–402.Google Scholar
  5. Emelichev, V.A., and Kovalev, M.M.: Solving Certain Concave Programming Problems by Successive Approximation, in: Izvestya Akademii Nauk BSSR 6, 1970, pp. 27–34 (in Russian).Google Scholar
  6. Falk, J.E., and Hoffman, K.L.: A Successive Underestimation Method for Concave Minimization Problems, in: Mathematics of Operations Research 1, 1976, pp. 251–259.Google Scholar
  7. Falk, J.E., and Hoffman, K.L.: Concave Minimization via Collapsing Polytopes, in: Operations Research 34, 1986, pp. 919–926.Google Scholar
  8. Gal, T.: Determination of all Neighbours of a Degenerate Point in Polytopes, Working Paper 17b, Fernuniversität Hagen, Hagen 1978.Google Scholar
  9. Gal, T.: Postoptimal Analysis, Parametric Programming, and Related Topics, New York 1979.Google Scholar
  10. Gal, T.: On the Structure of the Set of Bases of a Degenerate Point, in: Journal of Optimization Theory and Applications 45, 1985, pp. 577–589.CrossRefGoogle Scholar
  11. Gal, T.: Shadow Prices and Sensivity Analysis in Linear Programming under Degeneracy - A State-of-the-Art Survey, in: Operations Research Spektrum 8, 1986, pp. 59–77.CrossRefGoogle Scholar
  12. Gal, T., Kruse, H.J., and Zörnig, P.: Survey and Solved and Open Problems in the Degeneracy Phenomenon, in: Mathematical Programming, Series B, 42, 1988, pp. 125–133.CrossRefGoogle Scholar
  13. Geue, F.: Eine Pivotauswahlregel und die durch sie induzierten Teilgraphen des positiven Entartungsgraphen, Working Paper No. 141, Fernuniversität Hagen, Hagen 1989.Google Scholar
  14. Glover, F.: Polyhedral Annexation in Mixed Integer and Combinatorial Programming, in: Mathematical Programming 8, 1975, pp. 161–188.CrossRefGoogle Scholar
  15. Hoffman, K.L.: A Method for Globally Minimizing Concave Functions over Convex Sets, in: Mathematical Programming 20, 1981, pp. 22–32.CrossRefGoogle Scholar
  16. Horst, R.: On the Global Minimization of Concave Functions: Introduction and Survey, in: Operations Research Spektrum 6, 1984, pp. 195–205.CrossRefGoogle Scholar
  17. Horst, R.: Deterministic Global Optimization: Some Recent Advances and New Fields of Application, in: Naval Research Logistics 37, 1990, pp. 433–471.CrossRefGoogle Scholar
  18. Horst, R., Thoai, N.V., and de Vries, J.: On Finding New Vertices and Redundant Constraints in Cutting Plane Algorithms for Global Optimization, in: Operations Research Letters 7, 1988, pp. 85–90.CrossRefGoogle Scholar
  19. Horst, R., Thoai, N.V., and Tuy, H.: Outer Approximation by Polyhedral Convex Sets, in: Operations Research Spektrum 9, 1987, pp. 153–159.CrossRefGoogle Scholar
  20. Horst, R., Thoai, N.V., and Tuy, H.: On an Outer Approximation Concept in Global Optimization, in: Optimization 20, 1989, pp. 255–264.CrossRefGoogle Scholar
  21. Horst, R., and Tuy, H.: Global Optimization - Deterministic Approaches, Berlin et al. 1990.Google Scholar
  22. Istomin, L.A.: A Modification of Tuy’s Method for Minimizing a Concave Function over a Polytope, in: USSR Computational Mathematics and Mathematical Physics 17, 1977, pp. 1582–1592 (in Russian).CrossRefGoogle Scholar
  23. Khang, D.B., and Fujiwara, O.: A New Algorithm to Find all Vertices over a Polytope, in: Operations Research Letters 8, 1989, pp. 261–264.CrossRefGoogle Scholar
  24. Knuth, D.E.: Searching and Sorting, The Art of Computer Programming, Volume 3, Addison- Wesley 1973.Google Scholar
  25. Kruse, H.J.: Degeneracy Graphs and the Neighbourhood Problem. Lecture Notes in Economics and Mathematical Systems No. 260, Berlin et al. 1986.Google Scholar
  26. Matheiss, T.H., and Rubin, D.S.: A Survey and Comparison of Methods for Finding all Vertices of Convex Polyhedral Sets, in: Mathematics of Operations Research 5, 1980, pp. 167–185.CrossRefGoogle Scholar
  27. Thach, P.T., and Tuy, H.: Global Optimization under Lipschitzian Constraints, in: Japan Journal of Applied Mathematics 4, 1987, pp. 205–217.CrossRefGoogle Scholar
  28. Thieu, T.V., Tam, B.T., and Ban, T.V.: An Outer Approximation Method for Globally Minimizing a Concave Function over a Compact Convex Set, in: Acta Mathematica Vietnamica 8, 1983, pp. 21–40.Google Scholar
  29. Thoai, N.V.: A Modified Version of Tuy’s Method for Solving D.C. Programming Problems, in: Optimization 19, 1988, pp. 665–674.CrossRefGoogle Scholar
  30. Tuy, H.: On Outer Approximation Methods for Solving Concave Minimization Problems, in: Acta Mathematica Vietnamica 8, 1983, pp. 3–34.Google Scholar
  31. Tuy, H.: On a Polyhedral Annexation Method for Concave Minimization, Preprint, Institute of Mathematics, Hanoi 1986.Google Scholar
  32. Tuy, H.: Convex Programs with an Additional Reverse Convex Constraint, in: Journal of Optimization Theory and Applications 52, 1987, pp. 463–485.CrossRefGoogle Scholar
  33. Tuy, H.: Polyhedral Underestimation Method for Concave Minimization, Preprint, Institute of Mathematics, Hanoi 1988.Google Scholar
  34. Tuy, H., and Thuong, N.V.: On the Global Minimization of a Convex Function under General Nonconvex Constraints, in: Applied Mathematics and Optimization 18, 1988, pp. 119–142.CrossRefGoogle Scholar
  35. Vaish, H., and Shetty, C.H.: The Bilinear Programming Problem, in: Naval Research Logistics Quarterly 23, 1976, pp. 303–309.CrossRefGoogle Scholar
  36. Zörnig, P.: Theorie der Entartungsgraphen und ihre Anwendung zur Erklärung der Simplexzyklen, Dissertation, Fernuniversität Hagen, Hagen 1989.Google Scholar

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© Springer-Verlag Berlin Heidelberg 1991

Authors and Affiliations

  • Reiner Horst

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