Advertisement

Mathematical Programming

, Volume 59, Issue 1–3, pp 71–85 | Cite as

Restricted simplicial decomposition for convex constrained problems

  • Jose A. Ventura
  • Donald W. Hearn
Article

Abstract

The strategy of Restricted Simplicial Decomposition is extended to convex programs with convex constraints. The resulting algorithm can also be viewed as an extension of the (scaled) Topkis—Veinott method of feasible directions in which the master problem involves optimization over a simplex rather than the usual line search. Global convergence of the method is proven and conditions are given under which the master problem will be solved a finite number of times. Computational testing with dense quadratic problems confirms that the method dramatically improves the Topkis—Veinott algorithm and that it is competitive with the generalized reduced gradient method.

Keywords

Mathematical Method Finite Number Gradient Method Line Search Global Convergence 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. J. Abadie and J. Carpentier, “Generalization of the Wolfe reduced gradient method to the case of nonlinear constraints,” in: R. Fletcher, ed.,Optimization (Academic Press, 1969).Google Scholar
  2. J. Abadie, “Methods du gradient redirect generalise: Le code GRGA,” Note HI 1756/00, Electricite de France (Paris, France, 1975).Google Scholar
  3. D.P. Bertsekas, “on the Goldstein—Levitin—Polyak gradient projection method,”IEEE Transactions on Automatic Control AC-21 (1976) 174–184.Google Scholar
  4. D.P. Bertsekas, “Projected Newton methods for optimization problems with simple constraints,”SIAM Journal of Control and Optimization 20 (1982) 221–246.Google Scholar
  5. A.R. Colville, “A comparative study of nonlinear programming codes,” in: H.W. Kuhn, ed.,Proceedings of the Princeton Symposium on Mathematical Programming (Princeton University Press, Princeton, NJ, 1970).Google Scholar
  6. G.E. Forsythe and C.B. Moler,Computer Solution of Linear Algebraic Systems (Prentice-Hall, Englewood Cliffs, NJ, 1967).Google Scholar
  7. P.E. Gill, W. Murray, M.A. Saunders and M.H. Wright, “User's guide for SOL/NPSOL+: A Fortran package for nonlinear programming,” Technical Report SOL 83-12, Systems Optimization Laboratory, Stanford University (Stanford, CA, 1983).Google Scholar
  8. D.W. Hearn, S. Lawphongpanich and J.A. Ventura, “Restricted simplicial decomposition: computation and extensions,”Mathematical Programming Study 31 (1987) 99–118.Google Scholar
  9. D.W. Hearn, S. Lawphongpanich and J.A. Ventura, “Finiteness in restricted simplicial decomposition,”Operations Research Letters 4 (1985) 125–130.Google Scholar
  10. D.E. Knuth,The Art of Computer Programming, Vol. 3 (Addison-Wesley, Reading, MA, 1973).Google Scholar
  11. O.L. Mangasarian,Nonlinear Programming (McGraw-Hill, New York, 1969).Google Scholar
  12. D.M. Topkis and A.F. Veinott, “On the convergence of some feasible direction algorithms for nonlinear programming,”SIAM Journal of Control 5 (1967) 268–279.Google Scholar
  13. W.I. Zangwill,Nonlinear Programming:a Unified Approach (Prentice-Hall, Englewood Cliffs, NJ, 1969).Google Scholar

Copyright information

© The Mathematical Programming Society, Inc. 1993

Authors and Affiliations

  • Jose A. Ventura
    • 1
  • Donald W. Hearn
    • 2
  1. 1.Department of Industrial and Management Systems EngineeringThe Pennsylvania State UniversityUniversity ParkUSA
  2. 2.Department of Industrial and Systems EngineeringUniversity of FloridaGainesvilleUSA

Personalised recommendations