Annals of Operations Research

, Volume 14, Issue 1, pp 41–59 | Cite as

Parallel gradient projection successive overrelaxation for symmetric linear complementarity problems and linear programs

  • O. L. Mangasarian
  • R. De Leone
Article

Abstract

A gradient projection successive overrelaxation (GP-SOR) algorithm is proposed for the solution of symmetric linear complementary problems and linear programs. A key distinguishing feature of this algorithm is that when appropriately parallelized, the relaxation factor interval (0, 2) isnot reduced. In a previously proposed parallel SOR scheme, the substantially reduced relaxation interval mandated by the coupling terms of the problem often led to slow convergence. The proposed parallel algorithm solves a general linear program by finding its least 2-norm solution. Efficiency of the algorithm is in the 50 to 100 percent range as demonstrated by computational results on the CRYSTAL token-ring multicomputer and the Sequent Balance 21000 multiprocessor.

Keywords

Parallel algorithms SOR gradient projection linear programming linear complementarity problem 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    D. De Witt, R. Finkel andM. Solomon, The CRYSTAL multicomputer: Design and implementation experience,IEEE Transactions on Software Engineering, SE-13 (1987) 953–966.Google Scholar
  2. [2]
    E.S. Levitin andB.T. Poljak, Constrained minimization methods,USSR Computational Math. Math. Phys. (English translation) 6 (1966) 1–50.CrossRefGoogle Scholar
  3. [3]
    Y.Y. Lin andJ.S. Pang, Iterative methods for large convex quadratic programs: A survey,SIAM J. Control Optim. 25 (1987) 383–411.CrossRefGoogle Scholar
  4. [4]
    O.L. Mangasarian,Nonlinear programming (McGraw-Hill, New York, 1969).Google Scholar
  5. [5]
    O.L. Mangasarian, Solution of symmetric linear complementarity problems by iterative methods,J. Optim. Theory Appl. 22 (1977) 465–485.CrossRefGoogle Scholar
  6. [6]
    O.L. Mangasarian, Normal solution of linear programs,Math. Prog. Study 22 (1984) 206–216.Google Scholar
  7. [7]
    O.L. Mangasarian, Sparsity-preserving SOR algorithms for separable quadratic and linear programs,Computers and Oper. Res. 11 (1984) 105–112.CrossRefGoogle Scholar
  8. [8]
    O.L. Mangasarian andR. De Leong, Error bounds for strongly convex programs and (super)linearly convergent iterative schemes for the least 2-norm solution of linear programs,Appl. Math. Optim. 17 (1988) 1–14.CrossRefGoogle Scholar
  9. [9]
    O.L. Mangasarian andR. De Leone, Parallel sucessive overrelaxation methods for symmetric linear complementarity problems and linear programs,J. Optim. Theory Appl. 54 (1987) 43–466.CrossRefGoogle Scholar
  10. [10]
    O.L. Mangasarian andT.-H. Shiau, Lipschitz continuity of solutions of linear inequalities, programs and complementarity problems,SIAM J. Control Optim. 25 (1987) 583–595.CrossRefGoogle Scholar
  11. [11]
    K.G. Murty, On the number of solutions to the complementarity problem and spanning properties of complementarity cones,Linear Alg. Appl. 5 (1972) 65–108.CrossRefGoogle Scholar
  12. [12]
    J.W. Ortega,Numerical analysis a second course (Academic Press, New York, 1972).Google Scholar

Copyright information

© J.C. Baltzer A.G., Scientific Publishing Company 1988

Authors and Affiliations

  • O. L. Mangasarian
    • 1
  • R. De Leone
    • 1
  1. 1.Computer Sciences DepartmentUniversity of WisconsinMadisonUSA

Personalised recommendations