Abstract
The nonmonotone linear complementarity problem (LCP) is formulated as a bilinear program with separable constraints and an objective function that minimizes a natural error residual for the LCP. A linear-programming-based algorithm applied to the bilinear program terminates in a finite number of steps at a solution or stationary point of the problem. The bilinear algorithm solved 80 consecutive cases of the LCP formulation of the knapsack feasibility problem ranging in size between 10 and 3000, with almost constant average number of major iterations equal to four.
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References
K. P. Bennett and O. L. Mangasarian (1993), Bilinear separation of two sets inn-space,Computational Optimization & Applications 2: 207–227.
S.-J. Chung (1989), NP-completeness of the linear complementarity problem,Journal of Optimization Theory and Applications 60: 393–399.
S.-J. Chung and K. G. Murty (1981), Polynomially bounded ellipsoid algorithms for convex quadratic programming, in O. L. Mangasarian, R. R. Meyer, and S. M. Robinson (eds),Nonlinear Programming 4, pages 439–485, New York, Academic Press.
R. W. Cotle, J.-S. Pang, and R. E. Stone (1992).The Linear Complementarity Problem, Academic Press, New York.
J. T. J. Van Eijndhoven (1986), Solving the linear complementarity problem in circuit simulation.SIAM Journal on Control and Optimization,24: 1050–1062.
R. Horst, P. Pardalos, and N. V. Thoai (1994),Introduction to Global Optimization. Kluwer Academic Publishers, Dordrecht, Netherlands.
R. Horst and H. Tuy (1993),Global Optimization, Springer-Verlag, Berlin. (Second, Revised Edition).
Z.-Q. Luo and P. Tseng (1992), Error bound and convergence analysis of matrix splitting algorithms for the affine variational inequality problem.SIAM Journal on Optimization,2: 43–54.
O. L. Mangasarian (1964), Nonlinear programming problems with stochastic objective functions.Management Science,10: 353–359.
O. L. Mangasarian (1979), Simplified characterization of linear complementarity problems solvable as linear programs.Mathematics of Operations Research,4: 268–273.
O. L. Mangasarian and J.-S. Pang (1993), The extended linear complementarity problem.SIAM Journal on Matrix Analysis and Applications 16(2), January 1995.
O. L. Mangasarian and J. Ren (1994), New improved error bounds for the linear complementarity problem,Mathematical Programming 66, 1994, 241–255.
S. Martello and P. Toth (1987), Algorithms for knapsack problems, in S. Martello, G. Laporte, M. Minoux, and C. Ribeiro (eds.),Surveys in Combinatorial Optimization, Annals of Discrete Mathematics 31, North-Holland, Amsterdam.
MathWorks, inc. (1991),PRO-MATLAB for UNIX Computers, The Math Works, Inc., South Natick, MA 01760.
B. A. Murtagh and M. A. Saunders (1983), MINOS 5.0 user's guide. Technical Report SOL 83.20, Stanford University, December 1983. MINOS 5.4 Release Notes, December 1992.
K. G. Murty (1988),Linear Complementarity, Linear and Nonlinear Programming, Helderman-Verlag, Berlin.
J. M. Ortega (1972),Numerical Analysis, A Second Course. Academic Press.
J.-S. Pang (1986), Inexact Newton methods for the nonlinear complementarity problem.Mathematical Programming,36(1): 54–71.
J.-S. Pang, J. C. Trinkle, and G. Lo (1994), A complementarity approach to a quasistatic rigid body motion problem. Department of Mathematical Sciences, The Johns Hopkins University, Baltimore, MD 21218.
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This material is based on research supported by Air Force Office of Scientific Research Grant F49620-94-1-0036 and National Science Foundation Grants CCR-9322479 and CDA-9024618.
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Mangasarian, O.L. The linear complementarity problem as a separable bilinear program. J Glob Optim 6, 153–161 (1995). https://doi.org/10.1007/BF01096765
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DOI: https://doi.org/10.1007/BF01096765