On the global solution of linear programs with linear complementarity constraints

SIAM Journal on Optimization

Volumn 19, Issue 1, 2008, Pages 445-471

  Hu, Jing ^a   Mitchell, John E ^a   Pang, Jong Shi ^b   Bennett, Kristin P ^a   Kunapuli, Gautam ^a  

^a Research Rensselaer Polytechnic Institute   (United States)

^b UNIVERSITY OF ILLINOIS AT URBANA CHAMPAIGN   (United States)

Author keywords

Cutting plane methods; Global resolution; Linear programs with linear complimentarity constraints

Indexed keywords

BENDERS DECOMPOSITIONS; COMPUTATIONAL RESULTS; CUTTING-PLANE METHODS; FEASIBILITY PROBLEMS; FINITE TERMINATIONS; GLOBAL RESOLUTION; GLOBAL SOLUTIONS; INTEGER PROGRAM FORMULATIONS; LINEAR COMPLEMENTARITIES; LINEAR PROGRAMS WITH LINEAR COMPLIMENTARITY CONSTRAINTS; MINI MAXES; PROGRAMMING RELAXATIONS; SATISFIABILITY; SPARSIFICATION; VALID INEQUALITIES;

ALGORITHMS; LINEAR PROGRAMMING; SPACECRAFT;

INTEGER PROGRAMMING;

EID: 61349195260     PISSN: 10526234     EISSN: None     Source Type: Journal    
DOI: 10.1137/07068463x     Document Type: Article

References (50)

1. M. ANITESCU, On using the elastic mode in nonlinear programming approaches to mathematical programs with complementarity constraints, SIAM J. Optim., 15 (2005), pp. 1203-1236.
2. M. ANITESCU, Global convergence of an elastic mode approach for a class of mathematical programs with complementarity constraints, SIAM J. Optim., 16 (2005), pp. 120-145.
3. M. ANITESCU, P. TSENG, AND S.J. WRIGHT, Elastic-mode algorithms for mathematical programs with equilibrium constraints: Global convergence and stationarity properties, Math. Program., 110 (2007), pp. 337-371.
4. C. AUDET, P. HANSEN, B. JAMMARD, AND G. SAVARD, Links between linear bilevel and mixed 0-1 programming problems, 3. Optim. Theory Appl., 93 (1997), pp. 273-300.
5. C. AUDET, G. SAVARD, AND W. ZGHAL, New branch-and-cut algorithm for bilevel linear programming, 3. Optim. Theory Appl., 134 (2007), pp. 353-370.
6. K. BENNETT, X. JI, J. HU, G. KUNAPULI, AND J.S. PANG, Model selection via bilevel programming, in Proceedings of the International Joint Conference on Neural Networks (IJCNN'06) Vancouver, B.C., Canada, IEEE Press, Piscataway, NJ, 2006.
7. B. BORCHERS AND J. FURMAN, A two-phase exact algorithm for MAX-SAT and weighted MAX-SAT Problems, 3. Comb. Optim., 2 (1998), pp. 299-306.
8. L. CHEN AND D. GOLDFARB, An Active-Set Method for Mathematical Programs with Linear Complementarity Constraints, SIAM J. Optim., submitted.
9. G. CODATO AND M. FISCHETTI, Combinatorial Benders' cuts for mixed-integer linear programming, Oper. Res., 54 (2006), pp. 758-766.
10. S. DEMPE, Foundations of Bilevel Programming, Kluwer Academic, Dordrecht, 2002.
11. S. DEMPE, Annotated bibliography on bilevel programming and mathematical programs with equilibrium constraints, Optimization, 52 (2003), pp. 33-359.
12. S. DEMPE, V.V. KALASHNIKOV, AND N. K ALASHNYKOVA, Optimality conditions for bilevel programming problems, in Optimization with Multivalued Mappings: Theory, Applications and Algorithms, S. Dempe and N. Kalashnykova, eds., Springer-Verlag, New York, 2006, pp. 11-36.
13. M.L. FLEGEL AND C. KANZOW, On the Guignard constraint qualification for mathematical programs with equilibrium constraints, Optimization, 54 (2005), pp. 517-534.
14. R. FLETCHER AND S. LEYFFER, Solving mathematical program with complementarity constraints as nonlinear programs, Optim. Methods Softw., 19 (2004), pp. 15-40.
15. R. FLETCHER, S. LEYFFER, D. RALPH, AND S. SCHOLTES, Local convergence of SQP methods for mathematical programs with equilibrium constraints, SIAM J. Optim., 17 (2006), pp. 259-286.
16. C.A. FLOUDAS AND V. VISWESWARAN, A global optimization algorithm (GOP) for certain classes of nonconvex NLPs-I. Theory, Comput. Chem. Engrg., 14 (1990), pp. 1397-1417.
17. C.A. FLOUDAS AND V. VISWESWARAN, Primal-relaxed dual global optimization approach, J. Optim. Theory Appl., 78 (1993), pp. 187-225.
18. M. FUKUSHIMA AND J.S. PANG, Some feasibility issues in mathematical programs with equilibrium constraints, SIAM J. Optim., 8 (1998), pp. 673-681.
19. M. FUKUSHIMA AND J.S. PANG, Convergence of a smoothing continuation method for mathematical programs with complementarity constraints, in Ill-Posed Variational Problems and Regularization Techniques, Lecture Notes in Econom. and Math. Systems 477, M. Thera and R. Tichatschke, eds., Springer-Verlag, Berlin/Heidelberg, 1999, pp. 99-110.
20. M. FUKUSHIMA AND P. TSENG, An implementable active-set algorithm for computing a B-stationary point of a mathematical program with linear complementarity constraints, SIAM J. Optim., 12 (2002), pp. 724-739 [with erratum].
21. P. HANSEN, B. JAUMARD, AND G. SAVARD, New branch-and-bound rules for linear bilevel programming, SIAM J. Sci. Comput., 13 (1992), pp. 1194-1217.
22. J.N. HOOKER, Logic-Based Methods for Optimization, Wiley, New York, 2000.
23. J.N. HOOKER, Integrated Methods for Optimization, Springer-Verlag, New York, 2006.
24. J.N. HOOKER AND G. OTTOSSON, Logic-based Benders decomposition, Math. Program., 96 (2003), pp. 33-60.
25. X.M. HU AND D. RALPH, Convergence of a penalty method for mathematical programming with complementarity constraints, J. Optim. Theory Appl., 123 (2004), pp. 365-390.
26. T. IBARAKI, Complementary programming, Oper. Res., 19 (1971), pp. 1523-1529.
27. T. IBARAKI, The use of cuts in complementary programming, Oper. Res., 21 (1973), pp. 353-359.
28. R.G. JEROSLOW, Cutting-planes for complementarity constraints, SIAM J. Control Optim., 16 (1978), pp. 56-62.
29. H. JIANG AND D. RALPH, Smooth SQP methods for mathematical programs with nonlinear complementarity constraints, SIAM J. Optim., 10 (2000), pp. 779-808.
30. H. JIANG AND D. RALPH, Extension of quasi-Newton methods to mathematical programs with complementarity constraints, Comput. Optim. Appl., 25 (2003), pp. 123-150.
31. S. JOY, J.E. MITCHELL, AND B. BORCHERS, A branch-and-cut algorithm for MAX-SAT and weighted MAX-SAT, Optim. Methods Softw., to appear.
32. G. KUNAPULI, K. BENNETT, J. HU, AND J.S. PANG, Classification model selection via bilevel programming, Optim. Methods Softw., to appear.
33. G. KUNAPULI, K. BENNETT, J. HU, AND J.S. PANG, Bilevel Model Selection for Support Vector Machines, manuscript, Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY, 2007.
34. L. LASDON, Optimization Theory of Large Systems, Dover, New York, 2002.
35. S. LEYFFER, Complementarity constraints as nonlinear equations: Theory and numerical experience, in Optimization and Multivalued Mappings, S. Dempe and V. Kalashnikov, eds., Springer-Verlag, New York, 2006, pp. 169-208.
36. S. LEYFFER, G. LOPÉZ-CALVA, AND J. NOCEDAL, Interior methods for mathematical programs with complementarity constraints, SIAM J. Optim., 17 (2006), pp. 52-77.
37. Z.Q. LUO, J.S. PANG, AND D. RALPH, Mathematical Programs with Equilibrium Constraints, Cambridge University Press, Cambridge, 1996.
38. J.V. OUTRATA, Optimality conditions for a class of mathematical programs with equilibrium constraints, Math. Oper. Res., 24 (1999), pp. 627-644.
39. J. OUTRATA, M. KOÇVARA, AND J. ZOWE, Nonsmooth Approach to Optimization Problems with Equilibrium Constraints, Kluwer Academic, Dordrecht, 1998.
40. J.S. PANG AND M. FUKUSHIMA, Complementarity constraint qualifications and simplified B-stationarity conditions for mathematical programs with quilibrium constraints, Comput. Optim. Appl., 13 (1999), pp. 111-136.
41. J.S. PANG AND S. LEYFFER, On the global minimization of the value-at-risk, Optim. Methods Softw., 19 (2004), pp. 611-631.
42. H. SCHEEL AND S. SCHOLTES, Mathematical programs with complementarity constraints: Stationarity, optimality and sensitivity, Math. Oper. Res., 25 (2000), pp. 1-22.
43. S. SCHOLTES, Convergence properties of a regularization scheme for mathematical programs with complementarity constraints, SIAM J. Optim., 11 (2001), pp. 918-936.
44. M. TAWARMALANI AND N.V. SAHINIDIS, Convexification and Global Optimization in Continuous and Mixed Integer Nonlinear Programming, Kluwer Academic, Dordrecht, 2002.
45. V. VISWESWARAN, C.A. FLOUDAS, M.G. IERAPETRITOU, AND E.N. PISTIKOPOULOS, A decomposition based global optimization approach for solving bilevel linear and nonlinear quadratic programs, in State of the Art in Global Optimization: Computational Methods and Applications, Nonconvex Optim. Appl. 7, C.A. Floudas and P.M. Pardalos, eds., Kluwer Academic, Dordrecht, 1996, pp. 139-162.
46. J.J. YE, Optimality conditions for optimization problems with complementarity constraints, SIAM J. Optim., 9 (1999), pp. 374-387.
47. J.J. YE, Constraint qualifications and necessary optimality conditions for optimization problems with variational inequality constraints, SIAM J. Optim., 10 (2000), pp. 943-962.
48. J.J. YE, Necessary and sufficient optimality conditions for mathematical programs with equilibrium constraints, J. Math. Anal. Appl., 30 (2005), pp. 350-369.
49. http://www-neos.mcs.anl.gov/neos/solvers/index.html.
50. http://www.rpi.edu/~mitchj/generators/lpcc/.