Reformulations in mathematical programming: A computational approach

Studies in Computational Intelligence

Volumn 203, Issue , 2009, Pages 153-234

  Liberti, Leo ^a   Cafieri, Sonia ^a   Tarissan, Fabien ^a  

^a ÉCOLE POLYTECHNIQUE   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 65549123975     PISSN: 1860949X     EISSN: None     Source Type: Book Series    
DOI: 10.1007/978-3-642-01085-9_7     Document Type: Article

References (134)

1. Adams, W., Forrester, R., Glover, F.: Comparisons and enhancement strategies for linearizing mixed 0-1 quadratic programs. Discrete Optimization 1, 99-120 (2004)
2. Adams, W., Sherali, H.: A tight linearization and an algorithm for 0-1 quadratic programming problems. Management Science 32(10), 1274-1290 (1986)
3. Adams, W., Sherali, H.: A hierarchy of relaxations leading to the convex hull representation for general discrete optimization problems. Annals of Operations Research 140, 21-47 (2005)
4. Adjiman, C., Dallwig, S., Floudas, C., Neumaier, A.: A global optimization method, aBB, for general twice-differentiable constrained NLPs: I. Theoretical advances. Computers & Chemical Engineering 22(9), 1137-1158 (1998)
5. Adjiman, C.S., Androulakis, I.P., Floudas, C.A.: A global optimization method, aBB, for general twice-differentiable constrained NLPs: II. Implementation and computational results. Computers & Chemical Engineering 22(9), 1159-1179 (1998)
6. Aho, A., Hopcroft, J., Ullman, J.: Data Structures and Algorithms. Addison-Wesley, Reading (1983)
7. Al-Khayyal, F., Falk, J.: Jointly constrained biconvex programming. Mathematics of Operations Research 8(2), 273-286 (1983)
8. Alizadeh, F.: Interior point methods in semidefinite programming with applications to combinatorial optimization. SIAM Journal on Optimization 5(1), 13-51 (1995)
9. Andersen, K., Cornuéjols, G., Li, Y.: Reduce-and-split cuts: Improving the performance of mixed-integer Gomory cuts. Management Science 51(11), 1720-1732 (2005)
10. Androulakis, I.P., Maranas, C.D., Floudas, C.A.: alphaBB: A global optimization method for general constrained nonconvex problems. Journal of Global Optimization 7(4), 337-363 (1995)
11. Anstreicher, K.: SDP versus RLT for nonconvex QCQPs. In: Floudas, C., Pardalos, P. (eds.) Proceedings of Advances in Global Optimization: Methods and Applications, Mykonos, Greece (2007)
12. Applegate, D., Bixby, R., Chvátal, V., Cook, W.: The Travelling Salesman Problem: A Computational Study. Princeton University Press, Princeton (2007)
13. Audet, C., Hansen, P., Jaumard, B., Savard, G.: Links between linear bilevel and mixed 0-1 programming problems. Journal of Optimization Theory and Applications 93(2), 273-300 (1997)
14. Balas, E.: Intersection cuts - a new type of cutting planes for integer programming. Operations Research 19(1), 19-39 (1971)
15. Balas, E.: Projection, lifting and extended formulation in integer and combinatorial optimization. Annals of Operations Research 140, 125-161 (2005)
16. Balas, E., Jeroslow, R.: Canonical cuts on the unit hypercube. SIAM Journal on Applied Mathematics 23(1), 61-69 (1972)
17. Bauer, C., Frink, A., Kreckel, R.: Introduction to the ginac framework for symbolic computation within the C++ programming language. Journal of Symbolic Computation 33(1), 1-12 (2002)
18. Belotti, P., Lee, J., Liberti, L., Margot, F., Wächter, A.: Branching and bound reduction techniques for non-convex MINLP. Optimization Methods and Software (submitted)
19. Björk, K.M., Lindberg, P., Westerlund, T.: Some convexifications in global optimization of problems containing signomial terms. Computers & Chemical Engineering 27, 669-679 (2003)
20. Bjorkqvist, J., Westerlund, T.: Automated reformulation of disjunctive constraints in MINLP optimization. Computers & Chemical Engineering 23, S11-S14 (1999)
21. Boyd, E.: Fenchel cutting planes for integer programs. Operations Research 42(1), 53-64 (1994)
22. Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)
23. Brook, A., Kendrick, D., Meeraus, A.: GAMS, a user's guide. ACM SIGNUM Newsletter 23(3-4), 10-11 (1988)
24. Caporossi, G., Alamargot, D., Chesnet, D.: Using the computer to study the dyamics of the handwriting processes. In: Suzuki, E., Arikawa, S. (eds.) DS 2004. LNCS (LNAI), vol. 3245, pp. 242-254. Springer, Heidelberg (2004)
25. Cornuéjols, G.: Valid inequalities for mixed integer linear programs. Mathematical Programming B 112(1), 3-44 (2008)
26. Cortellessa, V., Marinelli, F., Potena, P.: Automated selection of software components based on cost/reliability tradeoff. In: Gruhn, V., Oquendo, F. (eds.) EWSA 2006. LNCS, vol. 4344, pp. 66-81. Springer, Heidelberg (2006)
27. Dantzig, G.: Linear Programming and Extensions. Princeton University Press, Princeton (1963)
28. Davidović, T., Liberti, L., Maculan, N., Mladenović, N.: Towards the optimal solution of the multiprocessor scheduling problem with communication delays. In: MISTA Proceedings (2007)
29. Dhyani, K.: Personal communication (2007)
30. Di Giacomo, L.: Mathematical programming methods in dynamical nonlinear stochastic supply chain management. Ph.D. thesis, DSPSA, Università di Roma "La Sapienza" (2007)
31. Duran, M., Grossmann, I.: An outer-approximation algorithm for a class of mixed-integer nonlinear programs. Mathematical Programming 36, 307-339 (1986)
32. Falk, J., Liu, J.: On bilevel programming, part I: General nonlinear cases. Mathematical Programming 70, 47-72 (1995)
33. Falk, J., Soland, R.: An algorithm for separable nonconvex programming problems. Management Science 15, 550-569 (1969)
34. Fischer, A.: New constrained optimization reformulation of complementarity problems. Journal of Optimization Theory and Applications 99(2), 481-507 (1998)
35. Fletcher, R., Leyffer, S.: Solving mixed integer nonlinear programs by outer approximation. Mathematical Programming 66, 327-349 (1994)
36. Floudas, C.: Deterministic Global Optimization. Kluwer Academic Publishers, Dordrecht (2000)
37. Fortet, R.: Applications de l'algèbre de Boole en recherche opérationelle. Revue Française de Recherche Opérationelle 4, 17-26 (1960)
38. Fourer, R.: Personal communication (2004)
39. Fourer, R., Gay, D.: The AMPL Book. Duxbury Press, Pacific Grove (2002)
40. Galli, S.: Parsing AMPL internal format for linear and non-linear expressions, B.Sc. dissertation, DEI, Politecnico di Milano, Italy (2004)
41. Gill, P.: User's Guide for SNOPT 5.3. Systems Optimization Laboratory, Department of EESOR, Stanford University, California (1999)
42. Gill, P.: User's guide for SNOPT version 7. In: Systems Optimization Laboratory. Stanford University, California (2006)
43. Gomory, R.: Essentials of an algorithm for integer solutions to linear programs. Bulletin of the American Mathematical Society 64(5), 256 (1958)
44. Grant, M., Boyd, S., Ye, Y.: Disciplined convex programming. In: Liberti and Maculan [79], pp. 155-210
45. Guéret, C., Prins, C., Sevaux, M.: Applications of optimization with Xpress-MP. Dash Optimization, Bilsworth (2000)
46. Hammer, P., Rudeanu, S.: Boolean Methods in Operations Research and Related Areas. Springer, Berlin (1968)
47. Hansen, P.: Method of non-linear 0-1 programming. Annals of Discrete Mathematics 5, 53-70 (1979)
48. Haverly, C.: Studies of the behaviour of recursion for the pooling problem. ACM SIGMAP Bulletin 25, 19-28 (1978)
49. Horst, R.: On the convexification of nonlinear progrAmming problems: an applications-oriented approach. European Journal of Operations Research 15, 382-392 (1984)
50. Horst, R., Tuy, H.: Global Optimization: Deterministic Approaches, 3rd edn. Springer, Berlin (1996)
51. Horst, R., Van Thoai, N.: Duality bound methods in global optimization. In: Audet, C., Hansen, P., Savard, G. (eds.) Essays and Surveys in Global Optimization, pp. 79-105. Springer, Berlin (2005)
52. ILOG: ILOG CPLEX 11.0 User's Manual. ILOG S.A., Gentilly, France (2008)
53. Judice, J., Mitra, G.: Reformulation of mathematical programming problems as linear complementarity problems and investigation of their solution methods. Journal of Optimization Theory and Applications 57(1), 123-149 (1988)
54. Kaibel, V., Pfetsch, M.: Packing and partitioning orbitopes. Mathematical Programming 114(1), 1-36 (2008)
55. Kaltofen, E.: Challenges of symbolic computation: My favorite open problems. Journal of Symbolic Computation 29, 891-919 (2000), citeseer.nj.nec.com/article/kaltofen99challenge.html
56. Kelley, J.: The cutting plane method for solving convex programs. Journal of SIAM VIII(6), 703-712 (1960)
57. Kesavan, P., Allgor, R., Gatzke, E., Barton, P.: Outer-approximation algorithms for nonconvex mixed-integer nonlinear programs. Mathematical Programming 100(3), 517-535 (2004)
58. Kojima, M., Megiddo, N., Ye, Y.: An interior point potential reduction algorithm for the linear complementarity problem. Mathematical Programming 54, 267-279 (1992)
59. Konno, H.: A cutting plane algorithm for solving bilinear programs. Mathematical Programming 11, 14-27 (1976)
60. Kucherenko, S., Belotti, P., Liberti, L., Maculan, N.: New formulations for the kissing number problem. Discrete Applied Mathematics 155(14), 1837-1841 (2007)
61. Kucherenko, S., Sytsko, Y.: Application of deterministic low-discrepancy sequences in global optimization. Computational Optimization and Applications 30(3), 297-318 (2004)
62. Lavor, C., Liberti, L., Maculan, N.: Computational experience with the molecular distance geometry problem. In: Pintér, J. (ed.) Global Optimization: Scientific and Engineering Case Studies, pp. 213-225. Springer, Berlin (2006)
63. Lavor, C., Liberti, L., Maculan, N., Chaer Nascimento, M.: Solving Hartree-Fock systems with global optimization metohds. Europhysics Letters 5(77), 50,006p1-50,006p5 (2007)
64. Letchford, A., Lodi, A.: Strengthening Chvátal-Gomory cuts and Gomory fractional cuts. Operations Research Letters 30, 74-82 (2002)
65. Liberti, L.: Framework for symbolic computation in C++ using n-ary trees. Tech. rep., CPSE, Imperial College London (2001)
66. Liberti, L.: Comparison of convex relaxations for monomials of odd degree. In: Tseveendorj, I., Pardalos, P., Enkhbat, R. (eds.) Optimization and Optimal Control. World Scientific, Singapore (2003)
67. Liberti, L.: Reduction constraints for the global optimization of NLPs. International Transactions in Operational Research 11(1), 34-41 (2004)
68. Liberti, L.: Reformulation and convex relaxation techniques for global optimization. 4OR 2, 255-258 (2004)
69. Liberti, L.: Reformulation and convex relaxation techniques for global optimization. Ph.D. thesis, Imperial College London, UK (2004)
70. Liberti, L.: Linearity embedded in nonconvex programs. Journal of Global Optimization 33(2), 157-196 (2005)
71. Liberti, L.: Writing global optimization software. In: Liberti and Maculan [79], pp. 211-262
72. Liberti, L.: Compact linearization of binary quadratic problems. 4OR 5(3), 231-245 (2007)
73. Liberti, L.: Reformulations in mathematical programming: Definitions. In: Aringhieri, R., Cordone, R., Righini, G. (eds.) Proceedings of the 7th Cologne-Twente Workshop on Graphs and Combinatorial Optimization, pp. 66-70. Università Statale di Milano, Crema (2008)
74. Liberti, L.: Spherical cuts for integer programming problems. International Transactions in Operational Research 15, 283-294 (2008)
75. Liberti, L.: Reformulations in mathematical programming: Definitions and systematics. RAIRO-RO (accepted for publication)
76. Liberti, L., Dražic, M.: Variable neighbourhood search for the global optimization of constrained NLPs. In: Proceedings of GO Workshop, Almeria, Spain (2005)
77. Liberti, L., Lavor, C., Maculan, N.: Double VNS for the molecular distance geometry problem. In: Proc. of Mini Euro Conference on Variable Neighbourhood Search, Tenerife, Spain (2005)
78. Liberti, L., Lavor, C., Nascimento, M.C., Maculan, N.: Reformulation in mathematical programming: An application to quantum chemistry. Discrete Applied Mathematics (accepted for publication)
79. Liberti, L., MaFulan, N. (eds.): Global Optimization: from Theory to Implementation. Springer, Berlin (2006)
80. Liberti, L., Pantelides, C.: Convex envelopes of monomials of odd degree. Journal of Global Optimization 25, 157-168 (2003)
81. Liberti, L., Pantelides, C.: An exact reformulation algorithm for large nonconvex NLPs involving bilinear terms. Journal of Global Optimization 36, 161-189 (2006)
82. Liberti, L., Tsiakis, P., Keeping, B., Pantelides, C.: ooOPS. Centre for Process Systems Engineering, Chemical Engineering Department, Imperial College, London, UK (2001)
83. Lougee-Heimer, R.: The common optimization interface for operations research: Promoting open-source software in the operations research community. IBM Journal of Research and Development 47(1), 57-66 (2003)
84. Maculan, N., Macambira, E., de Souza, C.: Geometrical cuts for 0-1 integer programming. Tech. Rep. IC-02-006, Instituto de Computação, Universidade Estadual de Campinas (2002)
85. Makhorin, A.: GNU Linear Programming Kit. Free Software Foundation (2003), http://www.gnu.org/software/glpk/
86. Mangasarian, O.: Linear complementarity problems solvable by a single linear program. Mathematical Programming 10, 263-270 (1976)
87. Mangasarian, O.: The linear complementarity problem as a separable bilinear program. Journal of Global Optimization 6, 153-161 (1995)
88. Maranas, C.D., Floudas, C.A.: Finding all solutions to nonlinearly constrained systems of equations. Journal of Global Optimization 7(2), 143-182 (1995)
89. Margot, F.: Pruning by isomorphism in branch-and-cut. Mathematical Programming 94, 71-90 (2002)
90. Margot, F.: Exploiting orbits in symmetric ILP. Mathematical Programming B 98, 3-21 (2003)
91. McCormick, G.: Computability of global solutions to factorable nonconvex programs: Part I - Convex underestimating problems. Mathematical Programming 10, 146-175 (1976)
92. Meyer, C., Floudas, C.: Trilinear monomials with mixed sign domains: Facets of the convex and concave envelopes. Journal of Global Optimization 29, 125-155 (2004)
93. Mladenović, N., Plastria, F., Uroěsević, D.: Reformulation descent applied to circle packing problems. Computers and Operations Research 32(9), 2419-2434 (2005)
94. Nemhauser, G., Wolsey, L.: Integer and Combinatorial Optimization. Wiley, New York (1988)
95. Nowak, I.: Relaxation and Decomposition Methods for Mixed Integer Nonlinear Programming. Birkhäuser, Basel (2005)
96. Pantelides, C., Liberti, L., Tsiakis, P., Crombie, T.: Mixed integer linear/nonlinear programming interface specification. Global Cape-Open Deliverable WP2.3-04 (2002)
97. Pardalos, P., Romeijn, H. (eds.): Handbook of Global Optimization, vol. 2. Kluwer Academic Publishers, Dordrecht (2002)
98. Plateau, M.C.: Reformulations quadratiques convexes pour la programmation quadratique en variables 0-1. Ph.D. thesis, Conservatoire National d'Arts et Métiers (2006)
99. Puchinger, J., Raidl, G.: Relaxation guided variable neighbourhood search. In: Proc. of Mini Euro Conference on Variable Neighbourhood Search, Tenerife, Spain (2005)
100. Raghavachari, M.: On connections between zero-one integer programming and concave programming under linear constraints. Operations Research 17(4), 680-684 (1969)
101. van Roy, T., Wolsey, L.: Solving mixed integer programming problems using automatic reformulation. Operations Research 35(1), 45-57 (1987)
102. Ryoo, H., Sahinidis, N.: Global optimization of nonconvex NLPs and MINLPs with applications in process design. Computers & Chemical Engineering 19(5), 551-566 (1995)
103. Sherali, H.: Global optimization of nonconvex polynomial programming problems having rational exponents. Journal of Global Optimization 12, 267-283 (1998)
104. Sherali, H.: Tight relaxations for nonconvex optimization problems using the reformulation-linearization/convexification technique (RLT). In: Pardalos and Romeijn [97], pp. 1-63
105. Sherali, H.: Personal communication (2007)
106. Sherali, H., Adams, W.: A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems. SIAM Journal of Discrete Mathematics 3, 411-430 (1990)
107. Sherali, H., Adams, W.: A Reformulation-Linearization Technique for Solving Discrete and Continuous Nonconvex Problems. Kluwer Academic Publishers, Dodrecht (1999)
108. Sherali, H., Alameddine, A.: A new reformulation-linearization technique for bilinear programming problems. Journal of Global Optimization 2, 379-410 (1992)
109. Sherali, H., Liberti, L.: Reformulation-linearization technique for global optimization. In: Floudas, C., Pardalos, P. (eds.) Encyclopedia of Optimization, 2nd edn., pp. 3263-3268. Springer, New York (2008)
110. Sherali, H., Tuncbilek, C.: New reformulation linearization/ convexification relaxations for univariate and multivariate polynomial programming problems. Operations Research Letters 21, 1-9 (1997)
111. Sherali, H., Wang, H.: Global optimization of nonconvex factorable programming problems. Mathematical Programming 89, 459-478 (2001)
112. Smith, E.: On the optimal design of continuous processes. Ph.D. thesis, Imperial College of Science, Technology and Medicine, University of London (1996)
113. Smith, E., Pantelides, C.: Global optimisation of nonconvex MINLPs. Computers & Chemical Engineering 21, S791-S796 (1997)
114. Smith, E., Pantelides, C.: A symbolic reformulation/spatial branch-and-bound algorithm for the global optimisation of nonconvex MINLPs. Computers & Chemical Engineering 23, 457-478 (1999)
115. Strekalovsky, A.: On global optimality conditions for d.c. programming problems. Technical Paper, Irkutsk State University (1997)
116. Strekalovsky, A.: Extremal problems with d.c. constraints. Computational Mathematics and Mathematical Physics 41(12), 1742-1751 (2001)
117. Stroustrup, B.: The C++ Programming Language, 3rd edn. Addison-Wesley, Reading (1999)
118. Sutou, A., Dai, Y.: Global optimization approach to unequal sphere packing problems in 3d. Journal of Optimization Theory and Applications 114(3), 671-694 (2002)
119. Tardella, F.: Existence and sum decomposition of vertex polyhedral convex envelopes. Tech. rep., Facoltà di Economia e Commercio, Università di Roma "La Sapienza" (2007)
120. Tawarmalani, M., Ahmed, S., Sahinidis, N.: Global optimization of 0-1 hyperbolic programs. Journal of Global Optimization 24, 385-416 (2002)
121. Tawarmalani, M., Sahinidis, N.: Semidefinite relaxations of fractional programming via novel techniques for constructing convex envelopes of nonlinear functions. Journal of Global Optimization 20(2), 137-158 (2001)
122. Tawarmalani, M., Sahinidis, N.: Convex extensions and envelopes of semicontinuous functions. Mathematical Programming 93(2), 247-263 (2002)
123. Tawarmalani, M., Sahinidis, N.: Exact algorithms for global optimization of mixed-integer nonlinear programs. In: Pardalos and Romeijn [97], pp. 65-86
124. Tawarmalani, M., Sahinidis, N.: Global optimization of mixed integer nonlinear programs: A theoretical and computational study. Mathematical Programming 99, 563-591 (2004)
125. Todd, M.: Semidefinite optimization. Acta Numerica 10, 515-560 (2001)
126. Tsiakis, P., Keeping, B.: ooMILP - a C++ callable object-oriented library and the implementation of its parallel version using corba. In: Liberti and Maculan [79], pp. 155-210
127. Tsiakis, P., Keeping, B., Pantelides, C.: ooMILP. Centre for Process Systems Engineering, Chemical Engineering Department, Imperial College, London, UK, 0.7 edn (2000)
128. Tuy, H.: Concave programming under linear constraints. Soviet Mathematics, 1437-1440 (1964)
129. Tuy, H.: D.c. optimization: Theory, methods and algorithms. In: Horst, R., Pardalos, P. (eds.) Handbook of Global Optimization, vol. 1, pp. 149-216. Kluwer Academic Publishers, Dordrecht (1995)
130. Wang, X., Change, T.: A multivariate global optimization using linear bounding functions. Journal of Global Optimization 12, 383-404 (1998)
131. Westerlund, T.: Some transformation techniques in global optimization. In: Liberti and Maculan [79], pp. 45-74
132. Westerlund, T., Skrifvars, H., Harjunkoski, I., Pörn, R.: An extended cutting plane method for a class of non-convex MINLP problems. Computers & Chemical Engineering 22(3), 357-365 (1998)
133. Wolsey, L.: Integer Programming. Wiley, New York (1998)
134. Zamora, J.M., Grossmann, I.E.: A branch and contract algorithm for problems with concave univariate, bilinear and linear fractional terms. Journal of Global Optimization 14, 217-249 (1999)