Obtaining Tighter Relaxations of Mathematical Programs with Complementarity Constraints

Springer Proceedings in Mathematics and Statistics

Volumn 21, Issue , 2012, Pages 1-23

  Mitchell, John E ^a   Pang, Jong Shi ^b   Yu, Bin ^c  

^a Research Rensselaer Polytechnic Institute   (United States)

^b University of Illinois at Urbana Champaign   (United States)

^c RENSSELAER POLYTECHNIC INSTITUTE   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

MATHEMATICAL PROGRAMMING;

COMPLEMENTARITY CONSTRAINT; COMPUTATIONAL RESULTS; CONVEX QUADRATIC CONSTRAINTS; LINEAR CONSTRAINTS; LINEAR PROGRAMS WITH COMPLEMENTARITY CONSTRAINTS; MATHEMATICAL PROGRAMS WITH COMPLEMENTARITY CONSTRAINTS; MODELING PARADIGMS; SEMI-DEFINITE PROGRAMMING;

OPTIMAL SYSTEMS;

EID: 84887606211     PISSN: 21941009     EISSN: 21941017     Source Type: Conference Proceeding    
DOI: 10.1007/978-1-4614-3924-0_1     Document Type: Conference Paper

References (50)

1. Adjiman, C.S., Androulakis, I.P., Floudas, C.A.: A global optimization method, αBB, for general twice-differentiable constrained NLPs-I. Theoretical advances. Comput. Chem. Eng. 22(9), 1137-1158 (1998)
2. Al-Khayyal, F.A., Falk, J.E.: Jointly constrained biconvex programming. Math. Oper. Res. 8(2), 273-286 (1983)
3. Anstreicher, K.M.: Semidefinite programming versus the reformulation-linearization technique for nonconvex quadratically constrained quadratic programming. J. Global Optim. 43(2-3), 471-484 (2009)
4. Audet, C., Haddad, J., Savard, G.: Disjunctive cuts for continuous linear bilevel programming. Optim. Lett. 1(3), 259-267 (2007)
5. Audet, C., Savard, G., Zghal, W.: New branch-and-cut algorithm for bilevel linear programming. J. Optim. Theory Appl. 38(2), 353-370 (2007)
6. Balas, E.: Intersection cuts-a new type of cutting planes for integer programming. Oper. Res. 19, 19-39 (1971)
7. Balas, E.: Disjunctive programming. Ann. Disc. Math. 5, 3-51 (1979)
8. Balas, E.: Disjunctive programming: properties of the convex hull of feasible points. Disc. Appl. Math. 89(1-3), 3-44 (1998)
9. Balas, E., Ceria, S., Cornuéjols, G.: Mixed 0-1 programming by lift-and-project in a branchand-cut framework. Manage. Sci. 42(9), 1229-1246 (1996)
10. Balas, E., Ceria, S., Cornuéjols, G., Natraj, N.: Gomory cuts revisited. Oper. Res. Lett. 19, 1-9 (1996)
11. Balas, E., Perregaard, M.: Lift-and-project for mixed 0-1 programming: recent progress. Disc. Appl. Math. 123(1-3), 129-154 (2002)
12. Balas, E., Perregaard, M.: A precise correspondence between lift-and-project cuts, simple disjunctive cuts, and mixed integer Gomory cuts for 0-1 programming. Math. Program. 94(2-3), 221-245 (2003)
13. Bao, X., Sahinidis, N.V., Tawarmalani, M.: Multiterm polyhedral relaxations for nonconvex, quadratically-constrained quadratic programs. Optim. Methods Softw. 24(4-5), 485-504 (2009)
14. Belotti, P.: Disjunctive cuts for nonconvex MINLPs. Technical report, Lehigh University, Bethlehem (2009)
15. Belotti, P., Lee, J., Liberti, L., Margot, F., Wachter, A.: Branching and bounds tightening techniques for non-convex MINLP. Optim. Methods Softw. 24(4-5), 597-634 (2009)
16. Belotti, P., Liberti, L., Lodi, A., Nannicini, G., Tramontani, A.: Disjunctive inequalities: applications and extensions. In: Cochran, J.J. (ed.) Encyclopedia of Operations Research and Management Science. Wiley (2010)
17. Borchers, B.: CSDP, a C library for semidefinite programming. Optim. Methods Softw. 11, 613-623 (1999)
18. Burer, S., Monteiro, R.D.C.: A nonlinear programming algorithm for solving semidefinite programs via low-rank factorization. Math. Program. 95(2), 329-357 (2003)
19. Burer, S., Saxena, A.: Old wine in a new bottle: the MILP road to MIQCP. Technical report, Department of Management Sciences, University of Iowa, Iowa City (2009)
20. Byrd, R., Nocedal, J., Waltz, R.: KNITRO: an integrated package for nonlinear optimization. In: di Pillo, G., Roma, M. (eds.) Large-Scale Nonlinear Optimization, pp. 35-59. Springer, Berlin (2006)
21. Ceria, S., Pataki, G.: Solving integer and disjunctive programs by lift-and-project. In: Proceedings of the Sixth IPCO Conference. Lecture Notes in Computer Science, vol. 1412, pp. 271-283. Springer, Berlin (1998)
22. de Farias Jr., I.R., Johnson, E.L., Nemhauser, G.L.: Facets of the complementarity knapsack polytope. Math. Oper. Res. 27(1), 210-226 (2002)
23. Fletcher, R., Leyffer, S.: Solving mathematical programs with complementarity constraints as nonlinear programs. Optim. Methods Softw. 18(1), 15-40 (2004)
24. Helmberg, C.: Numerical evaluation of SB method. Math. Program. 95(2), 381-406 (2003)
25. Hobbs, B.F., Metzler, C.B., Pang, J.S.: Strategic gaming analysis for electric power systems: an MPEC approach. IEEE Trans. Power Syst. 15(2), 638-645 (2000)
26. Hu, J., Mitchell, J.E., Pang, J.S.: An LPCC approach to nonconvex quadratic programs. Math. Program. 133(1-2), 243-277 (2012)
27. Hu, J., Mitchell, J.E., Pang, J.S., Bennett, K.P., Kunapuli, G.: On the global solution of linear programs with linear complementarity constraints. SIAM J. Optim. 19(1), 445-471 (2008)
28. Hu, J., Mitchell, J.E., Pang, J.S., Yu, B.: On linear programs with linear complementarity constraints. Technical report, Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy (2009). Accepted for publication in J. Global Optim.
29. Júdice, J.J., Sherali, H.D., Ribeiro, I.M., Faustino, A.M.:A complementarity-based partitioning and disjunctive cut algorithm for mathematical programming problems with equilibrium constraints. J. Global Optim. 36, 89-114 (2006)
30. Kojima, M., Tuncel, L.: Cones of matrices and successive convex relaxations of nonconvex sets. SIAM J. Optim. 10(3), 750-778 (2000)
31. Krishnan, K., Mitchell, J.E.: A unifying framework for several cutting plane methods for semidefinite programming. Optim. Methods Softw. 21(1), 57-74 (2006)
32. Lovász, L., Schrijver, A.: Cones of matrices and set-functions and 0-1 optimization. SIAM J. Optim. 1(2), 166-190 (1991)
33. Luo, Z.-Q., Ma, W.-K., So, A.M.-C., Ye, Y., Zhang, S.: Semidefinite relaxation of quadratic optimization problems. IEEE Signal Process. Mag. 27(3), 20-34 (2010)
34. McCormick, G.P.: Computability of global solutions to factorable nonconvex programs: part I-convex underestimating problems. Math. Program. 10, 147-175 (1976)
35. Mitchell, J.E., Pang, J.S., Yu, B.: Convex quadratic relaxations of nonconvex quadratically constrained quadratic programs. Technical report, Mathematical Sciences, Rensselaer Polytechnic Institute, Troy (2011)
36. Nemhauser, G.L., Wolsey, L.A.: Integer and Combinatorial Optimization. Wiley, New York (1988)
37. Pang, J.S.: Three modeling paradigms in mathematical programming. Math. Program. 125(2), 297-323 (2010)
38. Pang, J.S., Olson, T., Priebe, C.: A likelihodd-MPEC approach to target classification. Math. Program. 96(1), 1-31 (2003)
39. Pang, J.S., Su, C.L.: On estimating pure characteristics models. Technical report, Industrial and Enterprise Systems Engineering, University of Illinois, Urbana-Champaign (in preparation)
40. Perregaard, M.: Generating disjunctive cuts for mixed integer programs. PhD Thesis, Carnegie Mellon University, Graduate School of Industrial Administration, Pittsburgh (2003)
41. Qualizza, A., Belotti, P., Margot, F.: Linear programming relaxations of quadratically constrained quadratic programs. In: IMA Volume Series. Springer (2010). Accepted. Tepper Working Paper 2009-E20 (revised 4/2010)
42. Richard, J.-P.P., Tawarmalani, M.: Lifted inequalities: a framework for generating strong cuts for nonlinear progams. Math. Program. 121(1), 61-104 (2010)
43. Sahinidis, N.: BARON: a general purpose global optimization software package. J. Global Optim. 8, 201-205 (1996)
44. Saxena, A.: Integer programming, a technology. PhD Thesis, Tepper School of Business, Carnegie Mellon University (2007)
45. Saxena, A., Bonami, P., Lee, J.: Convex relaxations of non-convex mixed integer quadratically constrained programs: extended formulations. Math. Program. 124(1-2), 383-411 (2010)
46. Saxena, A., Bonami, P., Lee, J.: Convex relaxations of non-convex mixed integer quadratically constrained programs: projected formulations. Math. Program. 126(2), 281-314 (2011)
47. Sivaramakrishnan, K.K.: A parallel interior point decomposition algorithm for block angular semidefinite programs. Comput. Optim. Appl. 46(1), 1-29 (2010)
48. Sivaramakrishnan, K.K., Mitchell, J.E.: Properties of a cutting plane method for semidefinite programming. Technical report, Mathematical Sciences, Rensselaer Polytechnic Institute, Troy (2011)
49. Stubbs, R.A., Mehrotra, S.: A branch-and-cut method for 0-1 mixed convex programming. Math. Program. 86, 515-532 (1999)
50. Tawarmalani, M., Sahinidis, N.: Convexification and Global Optimization in Continuous and Mixed-Integer Nonlinear Programming: Theory, Algorithms, Software, and Applications. Kluwer, Dordrecht (2002)