Abstract
We propose a method for finding a global optimal solution of programs with linear complementarity constraints. This problem arises for instance in bilevel programming. The main idea of the method is to generate a sequence of points either ending at a global optimal solution within a finite number of iterations or converging to a global optimal solution. The construction of such sequence is based on branch-and-bound techniques, which have been used successfully in global optimization. Results on a numerical test of the algorithm are reported.
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References
Z. Q. Luo J. S. Pang D. Ralf (1996) Mathematical Programs with Equilibrium Constraints Cambridge University Press Cambridge, UK
M. Fukushima Z. Q. Luo J. S. Pang (1998) ArticleTitleA Global Convergent Sequential Quadratic Programming Algorithm for Mathematical Programs with Linear Complementarity Constraints Computational Optimization and Applications 10 5–34
J. Z. Zhang G. S. Liu (2001) ArticleTitleA New Extreme-Point Algorithm and Its Application in PSQP Algorithms for Solving Mathematical Programs with Linear Complementarity Constraints Journal of Global Optimization 19 345–361
N. V. Thoai (1987) On Canonical DC Programs and Applications, Essays on Nonlinear Analysis and Optimization Problems National Center for Scientific Research Hanoi, Vietnam 88–100
Thoai, N. V., On a Class of Global Optimization Problems, Methods of Operations Research, Edited by P. Kleinschmidt, F. J. Radermacher, W. Schweitzer, and H. Wildemann, Hain Verlag, Frankfurt, Germany, Vol. 58, pp. 115--130, 1989.
R. W. Cottle J. S. Pang R. E. Stone (1992) The Linear Complementarity Problem Academic Press Boston, Massachusetts
M. Kojima N. Megiddo T. Noma A. Yoshise (1991) A Unified Approach to Interior-Point Algorithms for Linear Complementarity Problems Springer Verlag Berlin, Germany
J. Gotoh N. V. Thoai Y. Yamamoto (2003) ArticleTitleGlobal Optimization Method for Solving the Minimum Maximal Flow Problem Optimization Method and Software 18 395–415
N. V. Thoai H. Tuy (1980) ArticleTitleConvergent Algorithms for Minimizing a Concave Function Mathematics of Operations Research 5 556–566
R. Horst P. Pardalos N. V. Thoai (2000) Introduction to Global Optimization EditionNumber2 Kluwer Academic Publishers Dordrecht, Holland
Z. Q. Luo J. S. Pang D. Ralf S. Q. Wu (1996) ArticleTitleExact Penalization and Stationarity Conditions of Mathematical Programs with Equilibrium Constraints Mathematical Programming 75 19–76
J. F. Bard J. E. Falk (1982) ArticleTitleAn Explicit Solution to the Multilevel Programming Problem Computers and Operations Research 9 77–100
J. F. Bard (1988) ArticleTitleConvex Two-Level Optimization Mathematical Programming 40 15–27
K. Shimizu Y. Ishizuka J. F. Bard (1997) Nondifferentiable and Two-Level Mathematical Programming Kluwer Academic Publishers Boston, Massachusetts
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The main part of this article was written during the first author’s stay as Visiting Professor at the Institute of Policy and Planning Sciences, University of Tsukuba, Tsukuba, Japan. The second and the third authors were supported by Grant-in-Aid for Scientific Research C(2) 13650061 of the Ministry of Education, Culture, Sports, Science, and\break Technology of Japan.
The authors thank P. B. Hermanns, Department of Mathematics, University of Trier, for carrying out the numerical test reported in Section 5. The authors also thank the referees and the Associate Editor for comments and suggestions which helped improving the first version of this article.
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Thoai, N.V., Yamamoto, Y. & Yoshise, A. Global Optimization Method for Solving Mathematical Programs with Linear Complementarity Constraints. J Optim Theory Appl 124, 467–490 (2005). https://doi.org/10.1007/s10957-004-0946-9
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DOI: https://doi.org/10.1007/s10957-004-0946-9