Abstract
Preprocessing techniques are extensively used in the linear and integer programming communities as a means to improve model formulation by reducing size and complexity. Adaptations and extensions of these methods for use within the complementarity framework are detailed and shown to be effective on practical models. The preprocessor developed is comprised of two phases. The first recasts a complementarity problem as a variational inequality over a polyhedral set and exploits the uncovered structure to fix variables and remove constraints. The second discovers information about the function and utilizes complementarity theory to eliminate variables. The methodology is successfully employed to preprocess several models.
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References
E. Andersen and K. Andersen. Presolving in linear programming. Mathematical Programming, 71:221–245, 1995.
A. Brearley, G. Mitra, and H. Williams. Analysis of mathematical programming problems prior to applying the simplex algorithm. Mathematical Programming, 8:54–83, 1975.
A. Brooke, D. Kendrick, and A. Meeraus. GAMS: A User’s Guide. The Scientific Press, South San Francisco, CA, 1988.
S. P. Dirkse and M. C. Ferris. MCPLIB: A collection of nonlinear mixed complementarity problems. Optimization Methods and Software, 5:319–345, 1995.
S. P. Dirkse and M. C. Ferris. The PATH solver: A non-monotone stabilization scheme for mixed complementarity problems. Optimization Methods and Software, 5:123–156, 1995.
M. C. Ferris, R. Fourer, and D. M. Gay. Expressing complementarity problems and communicating them to solvers. SIAM Journal on Optimization, 9:991–1009, 1999.
M. C. Ferris, M. P. Mesnier, and J. More. NEOS and Condor: Solving nonlinear optimization problems over the Internet. ACM Transactions on Mathematical Software, forthcoming, 1999.
M. C. Ferris and T. S. Munson. Case studies in complementarity: Improving model formulation. In M. Thera and R. Tichatschke, editors, Ill-Posed Variational Problems and Regularization Techniques, number 477 in Lecture Notes in Economics and Mathematical Systems, pages 79–98. Springer Verlag, Berlin, 1999.
M. C. Ferris and T. S. Munson. Interfaces to PATH 3.0: Design, implementation and usage. Computational Optimization and Applications, 12:207–227, 1999.
M. C. Ferris and T. S. Munson. Complementarity problems in GAMS and the PATH solver. Journal of Economic Dynamics and Control, 24:165–188, 2000.
M. C. Ferris and J. S. Pang. Engineering and economic applications of complementarity problems. SIAM Review, 39:669–713, 1997.
R. Fourer, D. M. Gay, and B. W. Kernighan. AMPL: A Modeling Language for Mathematical Programming. Duxbury Press, 1993.
D. M. Gay. Electronic mail distribution of linear programming test problems. COAL Newsletter, 13:10–12, 1985.
M. S. Gowda. Applications of degree theory to linear complementarity problems. Mathematics of Operations Research, 18:868–879, 1993.
M. S. Gowda. An analysis of zero set and global error bound properties of a piecewise affine function via its recession function. SIAM Journal on Matrix Analysis and Applications, 17:594–609, 1996.
ILOG CPLEX Division, 889 Alder Avenue, Incline Village, Nevada. CPLEX Optimizer, http://www.cplex.com
S. M. Robinson. Normal maps induced by linear transformations. Mathematics of Operations Research, 17:691–714, 1992.
S. M. Robinson. A reduction method for variational inequalities. Mathematical Programming, 80:161–169, 1998.
R. T. Rockafellar. Convex Analysis. Princeton University Press, Princeton, New Jersey, 1970.
M. W. P. Savelsbergh. Preprocessing and probing techniques for mixed integer programming problems. ORSA Journal on Computing, 6:445–454, 1994.
H. Sellami and S. M. Robinson. Implementation of a continuation method for normal maps. Mathematical Programming, pages 563–578, 1997.
J. Tomlin and J. Welch. Finding duplicate rows in a linear programming model. Operations Research Letters, 5(1):7–11, 1986.
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© 2001 Springer Science+Business Media Dordrecht
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Ferris, M.C., Munson, T.S. (2001). Preprocessing Complementarity Problems. In: Ferris, M.C., Mangasarian, O.L., Pang, JS. (eds) Complementarity: Applications, Algorithms and Extensions. Applied Optimization, vol 50. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-3279-5_7
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DOI: https://doi.org/10.1007/978-1-4757-3279-5_7
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