A semidefinite programming heuristic for quadratic programming problems with complementarity constraints

Computational Optimization and Applications

Volumn 31, Issue 1, 2005, Pages 5-29

  Braun, Stephen ^a   Mitchell, John E ^b  

^a Warren and Selbert   (United States)

^b Rensselaer Polytechnic Institute   (United States)

Author keywords

Complementarity constraints; Quadratic programming; Semidefinite programming

Indexed keywords

ALGORITHMS; CONSTRAINT THEORY; FUNCTION EVALUATION; MATHEMATICAL MODELS; OPTIMIZATION; PROBLEM SOLVING; QUADRATIC PROGRAMMING; VECTORS;

COMPLEMENTARY CONSTRAINTS; LINEAR CONSTRAINTS; LINEAR FUNCTIONS; SEMIDEFINITE PROGRAMMING;

HEURISTIC PROGRAMMING;

EID: 18844379364     PISSN: 09266003     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10589-005-1014-6     Document Type: Article

References (39)

1. E. Balas, S. Ceria, and G. Cornuéjols, "A lift-and-project cutting plane algorithm for mixed 0-1 programs," Mathematical Programming, vol. 58, pp. 295-324, 1993.
2. S.E. Braun, "Solving a quadratic programming problem subject to orthogonality constraints," PhD thesis, Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY, Dec. 2001.
3. A. Charnes and W.W. Cooper, "Programming with linear fractional functionals," Naval Research Logistics Quarterly, vol. 9, pp. 181-186, 1962.
4. J.W. Demmel, Applied Numerical Linear Algebra, SIAM: Philadelphia, PA, 1997.
5. M.C. Ferris, S.P. Dirkse, and A. Meeraus, "Mathematical programs with equilibrium constraints: Automatic reformulation and solution via constrained optimization," Technical Report 00-09, Computer Sciences Department, University of Wisconsin, Madison, WI, July 2002.
6. M.C. Ferris and J.S. Pang, "Engineering and economic applications of complementarity problems," SIAM Review, vol. 39, no. 4, pp. 669-713, 1997.
7. R. Fletcher and S. Leyffer, "Numerical experience with solving MPECs as NLPs," Technical Report NA 210, Department of Mathematics and Computer Science, University of Dundee, Dundee DDl 4HN, UK, Aug. 2002.
8. R. Fletcher, S. Leyffer, D. Ralph, and S. Scholtes, "Local convergence of SQP methods for mathematical programs with equilibrium constraints," Technical Report NA 209, Department of Mathematics and Computer Science, University of Dundee, Dundee DD14HN, UK, May 2002.
9. M. Fukushima, Z.-Q. Luo, and J.S. Pang, "A globally convergent sequential quadratic programming algorithm for mathematical programs with linear complementarity constraints," Computational Optimization and Applications, vol. 10, pp. 5-34, 1998.
10. M. Fukushima and P. Tseng, "An implementable active-set algorithm for computing a B-stationary point of the mathematical program with linear complementarity constraints," Technical Report, Department of Mathematics, University of Washington, Seattle, WA, Oct. 1999.
11. G.H. Golub and C.F. Van Loan, Matrix Computations, Johns Hopkins University Press: Baltimore, 1983.
12. C. Helmberg, Semidefinite Programming Homepage, 1996. http://www-user.tu-chemnitz.de/~helmberg/semidef.html.
13. C. Helmberg, "Fixing variables in semidetinite relaxations," SIAM Journal on Matrix Analysis and Applications, vol. 21, no. 3, pp. 952-969, 2000.
14. C. Helmberg, "SBmethod: A C++ implementation of the spectral bundle method," Technical Report 00-35, TU Berlin, Konrad-Zuse-Zentrum, Berlin, Oct. 2000.
15. C. Helmberg, "Semidefinite programming for combinatorial optimization," Technical Report ZR-00-34, TU Berlin, Konrad-Zuse-Zentrum, Berlin, Habilitationsschrift, Oct. 2000.
16. C. Helmberg, "A cutting plane algorithm for large scale Semidefinite relaxations," Technical Report 01-26, TU Berlin, Konrad-Zuse-Zentrum, Berlin, Oct. 2001.
17. C. Helmberg and F. Rendl, "Solving quadratic (0, 1)-problems by Semidefinite programs and cutting planes," Mathematical Programming, vol. 82, pp. 291-315, 1998.
18. C. Helmberg and F. Rendl, "A spectral bundle method for Semidefinite programming," SIAM Journal on Optimization, vol. 10, no. 3, pp. 673-696, 2000.
19. R.A. Horn and C. Johnson, Matrix Analysis, Cambridge University Press: Cambridge, 1985.
20. K. Krishnan, "Linear programming approaches to Semidefinite programming problems," PhD thesis, Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY 12180, July 2002.
21. K. Krishnan and J.E. Mitchell, "A unifying framework for several cutting plane methods for Semidefinite programming," Technical Report, Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY 12180, Nov. 2002. Revised: Dec. 15, 2003.
22. S. Leyffer, "The penalty interior point method fails to converge for mathematical programs with equilibrium constraints," Technical Report NA 208, Department of Mathematics and Computer Science, University of Dundee, Dundee DD1 4HN, UK, Feb. 2002.
23. L. Lovász and A. Schrijver, "Cones of matrices and set-functions and 0-1 optimization," SIAM Journal on Optimization, vol. 1, no. 2, pp. 166-190, 1991.
24. Z.-Q. Luo, J.-S. Pang, and D. Ralph, Mathematical Programs with Equilibrium Constraints, Cambridge University Press: Cambridge, 1996.
25. H. Markowitz, "Portfolio selection," Journal of Finance, vol. 7, pp. 77-91, 1952.
26. H. Markowitz, Portfolio Selection: Efficient Diversification of Investments, 2nd edition, Black-well, New York, 1991.
27. J.E. Mitchell, "Restarting after branching in the SDP approach to MAX-CUT and similar combinatorial optimization problems," Journal of Combinatorial Optimization, vol. 5, no. 2, pp. 151-166, 2001.
28. J.E. Mitchell and S. Braun, "Rebalancing an investment portfolio in the presence of transaction costs," Technical Report, Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY 12180, Nov. 2002.
29. J.V. Outrata and J. Zowe, "A numerical approach to optimization problems with variational inequality constraints," Mathematical Programming, vol. 68, no. 1, pp. 105-130, 1995.
30. H. Scheel and S. Scholtes, "Mathematical programs with complementarity constraints: Stationarity, optimality, and sensitivity," Mathematics of Operations Research, vol. 25, no. 1, pp. 1-22, 2000.
31. S. Scholtes, "Active set methods for inverse linear complementarity problems," Technical Report, Judge Institute of Management Science, Cambridge University, Cambridge CB2 1AG, Nov. 1999.
32. S. Scholtes, "Convergence properties of a regularisation scheme for mathematical programs with complementarity constraints," SIAM Journal on Optimization, vol. 11, no. 4, pp. 918-936, 2001.
33. S. Scholtes, "Combinatorial structures in nonlinear programming," Technical report, Judge Institute of Management Science, Cambridge University, Cambridge CB2 1AG, April 2002.
34. H.D. Sherali and W.D. Adams, "A reformulation-linearization technique for solving discrete and continuous nonconvex problems," Number 31 in Nonconvex Optimization and its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1998.
35. M.J. Todd, "Semidefinite optimization," Acta Numerica, vol. 10, pp. 515-560, 2001.
36. K.C. Toh, M.J. Todd, and R. Tutuncu, "SDPT3 - A Matlab software package for semidefinite programming," Optimization Methods and Software, vol. 11, pp. 545-581, 1999.
37. K.C. Toh, M.J. Todd, and R. Tutuncu, "SDPT3 - A Matlab software package for semidefinite-quadratic-linear programming," version 3.0. Technical Report, Department of Mathematics, National University of Singapore, Singapore, Aug. 2001.
38. L. Vandenberghe and S. Boyd, "Semidefinite programming," SIAM Review, vol. 38, pp. 49-95, 1996.
39. H. Wolkowicz, R. Saigal, and L. Vandenberghe (Eds.), Handbook of Semidefinite Programming: Theory, Algorithms, and Applications, Kluwer Academic Publishers: Dordrecht, The Netherlands, 2000.