Abstract
The Levitin-Poljak gradient-projection method is applied to solve the linear complementarity problem with a nonsymmetric matrixM, which is either a positive-semidefinite matrix or aP-matrix. Further-more, if the quadratic functionx T(Mx + q) is pseudoconvex on the feasible region {x ∈R n |Mx + q ≥ 0,x≥0}, then the gradient-projection method generates a sequence converging to a solution, provided that the problem has a solution. For the case when the matrixM is aP-matrix and the solution is nondegenerate, the gradient-projection method is finite.
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References
Cottle, R. W., andDantzig, G. B.,Complementary Pivot Theory of Mathematical Programming, Linear Algebra and Applications, Vol. 1, pp. 103–125, 1968.
Lemke, C. E.,Bimatrix Equilibrium Points and Mathematical Programming, Management Science, Vol. 11, pp. 681–689, 1965.
Murty, K. G.,Computational Complexity of Complementary Pivot Method, Mathematical Programming Study, Vol. 7, pp. 61–73, 1978.
Bertsekas, D. P.,On the Goldstein-Levitin-Poljak Gradient-Projection Method, IEEE Transactions on Automatic Control, Vol. AC-21, pp. 174–184, 1976.
Levitin, E. S., andPoljak, B. T.,Constrained Minimization Problems, USSR Computational Mathematics and Mathematical Physics, Vol. 6, pp. 1–50, 1966.
Fielder, M., andPtak, V.,On Matrices with Nonpositive Off-Diagonal Elements and Positive Principal Minors, Czechoslovak Mathematics Journal, Vol. 12, pp. 382–400, 1962.
Murty, K. G.,On the Number of Solutions to the Complementarity Problem and Spanning Properties of Complementarity Cones, Linear Algebra and Applications, Vol. 5, pp. 65–108, 1972.
Mangasarian, O. L.,Nonlinear Programming, McGraw-Hill, New York, New York, 1969.
Cheng, Y. C.,Iterative Methods for Solving Linear Complementarity and Linear Programming Problems, University of Wisconsin, Madison, Wisconsin, PhD Thesis, 1981.
Ortega, J. M.,Numerical Analysis: A Second Course, Academic Press, New York, New York, 1972.
Poljak, B. T.,On a Sharp Minimum, University of Wisconsin, Mathematics Research Center, Seminar, 1980.
Mangasarian, O. L., andMeyer, R. R.,Nonlinear Perturbation of Linear Programs, SIAM Journal of Control and Optimization, Vol. 17, pp. 745–752, 1979.
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Communicated by O. L. Mangasarian
This work is based on the author's PhD Dissertation, which was supported by NSF Grant No. MCS-79-01066 at the University of Wisconsin, Madison, Wisconsin.
The author would like to thank Professor O. L. Mangasarian for his guidance of the dissertation.
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Cheng, Y.C. On the gradient-projection method for solving the nonsymmetric linear complementarity problem. J Optim Theory Appl 43, 527–541 (1984). https://doi.org/10.1007/BF00935004
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DOI: https://doi.org/10.1007/BF00935004