Skip to main content
SpringerLink
Log in

An LP-based successive overrelaxation method for linear complementarity problems

  • Contributed Papers
  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

    We’re sorry, something doesn't seem to be working properly.

    Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Abstract

A sparsity preserving LP-based SOR method for solving classes of linear complementarity problems including the case where the given matrix is positive-semidefinite is proposed. The LP subproblems need be solved only approximately by a SOR method. Heuristic enhancement is discussed. Numerical results for a special class of problems are presented, which show that the heuristic enhancement is very effective and the resulting program can solve problems of more than 100 variables in a few seconds even on a personal computer.

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. Chung, S. J., andMurty, K. G.,Polynomially Bounded Ellipsoid Algorithms for Convex Quadratic Programming, Nonlinear Programming 4, Edited by O. L. Mangasarian, R. R. Meyer, and S. M. Robinson, Academic Press, New York, New York, pp. 439–485, 1981.

    Google Scholar 

  2. Dantzig, G. B., andCottle, R. W.,Positive Semidefinite Programming, Nonlinear Programming, Edited by J. Abadie, North-Holland Publishing Company, Amsterdam, Holland, pp. 55–73, 1967.

    Google Scholar 

  3. Lemke, C. E.,On Complementary Pivot Theory, Mathematics of the Decision Sciences, Part 1, Edited by G. B. Dantzig and A. F. Veinott, American Mathematical Society, Providence, Rhode Island, pp. 95–114, 1968.

    Google Scholar 

  4. Shiau, T. H.,Iterative Linear Programming for Linear Complementarity and Related Problems, University of Wisconsin, Madison, Wisconsin, Computer Sciences Department, PhD Thesis, 1983.

    Google Scholar 

  5. Ahn, B. H.,Solution of Nonsymmetric Linear Complementarity Problems by Iterative Methods, I, Journal of Optimization Theory and Applications, Vol. 33, pp. 175–185, 1981.

    Google Scholar 

  6. Cheng, Y. C.,Iterative Methods for Solving Linear Complementarity and Linear Programming Problems, University of Wisconsin, Madison, Wisconsin, PhD Thesis, 1981.

    Google Scholar 

  7. Cottle, R. W., Golub, G. H., andSacher, R. S.,On the Solution of Large Structured Complementarity Problems, Applied Mathematics and Optimization, Vol. 4, pp. 347–363, 1978.

    Google Scholar 

  8. Cryer, C. W.,The Solution of a Quadratic Programming Problem Using Systematic Overrelaxation, SIAM Journal on Control, Vol. 9, pp. 385–392, 1971.

    Google Scholar 

  9. Mangasarian, O. L.,Solution of Symmetric Linear Complementarity Problems by Iterative Methods, Journal of Optimization Theory and Applications, Vol. 22, pp. 465–485, 1977.

    Google Scholar 

  10. Cottle, R. W., andDuvall, S. G.,A Lagrangian Relaxation Algorithm for the Constrained Matrix Problem, Stanford University, Stanford, California, Systems Optimization Laboratory, Report No. 82-10, 1982.

    Google Scholar 

  11. Han, S. P., andMangasarian, O. L.,A Dual Differentiable Exact Penalty Function, Mathematical Programming, Vol. 25, pp. 293–306, 1983.

    Google Scholar 

  12. Pang, J. S.,On the Convergence of a Basic Iterative Method for the Implicit Complementarity Problem, Journal of Optimization Theory and Applications, Vol. 17, pp. 149–162, 1982.

    Google Scholar 

  13. Frank, M., andWolfe, P.,An Algorithm for Quadratic Programming, Naval Research Logistics Quarterly, Vol. 3, pp. 95–110, 1956.

    Google Scholar 

  14. Mangasarian, O. L., andShiau, T. H.,Error Bounds for Monotone Linear Complementarity Problems, Mathematical Programming, Vol. 36, pp. 81–89, 1986.

    Google Scholar 

  15. Karmarkar, N.,A New Polynomial-Time Algorithm for Linear Programming, Combinatorica, Vol. 4, pp. 373–395, 1984.

    Google Scholar 

  16. Agmon, S.,The Relaxation Method for Linear Inequalities, Canadian Journal of Mathematics, Vol. 6, pp. 382–392, 1954.

    Google Scholar 

  17. Motzkin, T. S., andSchoenberg, I. J.,The Relaxation Method for Linear Inequalities, Canadian Journal of Mathematics, Vol. 6, pp. 393–404, 1954.

    Google Scholar 

  18. Censor, Y., andLent, A.,A Cyclic Subgradient Projections Method for the Convex Feasibility Problem, University of Haifa, Mount Carmel, Haifa, Israel, Department of Mathematics, Technical Report, 1980.

    Google Scholar 

  19. Ortega, J. M.,Numerical Analysis, A Second Course, Academic Press, New York, New York, 1972.

    Google Scholar 

  20. Varga, R. S.,Matrix Iterative Analysis,Prentice-Hall, Englewood Cliffs, New Jersey, 1962.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, University of Missouri-Columbia, Columbia, Missouri

    T. H. Shiau (Assistant Professor)

Authors
  1. T. H. Shiau
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by O. L. Mangasarian

This research was sponsored by the Air Force Office of Scientific Research, Grant No. AFOSR-86-0124. Part of this material is based on work supported by the National Science Foundation under Grant No. MCS-82-00632.

The author is grateful to Dr. R. De Leone for his helpful and constructive comments on this paper.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Shiau, T.H. An LP-based successive overrelaxation method for linear complementarity problems. J Optim Theory Appl 59, 247–259 (1988). https://doi.org/10.1007/BF00938311

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00938311

Key Words

Search

Navigation