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Reformulations in Mathematical Programming: A Computational Approach

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Foundations of Computational Intelligence Volume 3

Part of the book series: Studies in Computational Intelligence ((SCI,volume 203))

Abstract

Mathematical programming is a language for describing optimization problems; it is based on parameters, decision variables, objective function(s) subject to various types of constraints. The present treatment is concerned with the case when objective(s) and constraints are algebraic mathematical expressions of the parameters and decision variables, and therefore excludes optimization of black-box functions. A reformulation of a mathematical program P is a mathematical program Q obtained from P via symbolic transformations applied to the sets of variables, objectives and constraints. We present a survey of existing reformulations interpreted along these lines, some example applications, and describe the implementation of a software framework for reformulation and optimization.

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References

  1. Adams, W., Forrester, R., Glover, F.: Comparisons and enhancement strategies for linearizing mixed 0-1 quadratic programs. Discrete Optimization 1, 99–120 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  2. Adams, W., Sherali, H.: A tight linearization and an algorithm for 0-1 quadratic programming problems. Management Science 32(10), 1274–1290 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  4. Adjiman, C., Dallwig, S., Floudas, C., Neumaier, A.: A global optimization method, αBB, for general twice-differentiable constrained NLPs: I. Theoretical advances. Computers & Chemical Engineering 22(9), 1137–1158 (1998)

    Google Scholar 

  5. Adjiman, C.S., Androulakis, I.P., Floudas, C.A.: A global optimization method, αBB, for general twice-differentiable constrained NLPs: II. Implementation and computational results. Computers & Chemical Engineering 22(9), 1159–1179 (1998)

    Google Scholar 

  6. Aho, A., Hopcroft, J., Ullman, J.: Data Structures and Algorithms. Addison-Wesley, Reading (1983)

    MATH  Google Scholar 

  7. Al-Khayyal, F., Falk, J.: Jointly constrained biconvex programming. Mathematics of Operations Research 8(2), 273–286 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  8. Alizadeh, F.: Interior point methods in semidefinite programming with applications to combinatorial optimization. SIAM Journal on Optimization 5(1), 13–51 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Google Scholar 

  12. Applegate, D., Bixby, R., Chvátal, V., Cook, W.: The Travelling Salesman Problem: a Computational Study. Princeton University Press, Princeton (2007)

    Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  14. Balas, E.: Intersection cuts — a new type of cutting planes for integer programming. Operations Research 19(1), 19–39 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  15. Balas, E.: Projection, lifting and extended formulation in integer and combinatorial optimization. Annals of Operations Research 140, 125–161 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  16. Balas, E., Jeroslow, R.: Canonical cuts on the unit hypercube. SIAM Journal on Applied Mathematics 23(1), 61–69 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Google Scholar 

  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)

    Article  Google Scholar 

  20. Bjorkqvist, J., Westerlund, T.: Automated reformulation of disjunctive constraints in MINLP optimization. Computers & Chemical Engineering 23, S11–S14 (1999)

    Google Scholar 

  21. Boyd, E.: Fenchel cutting planes for integer programs. Operations Research 42(1), 53–64 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  22. Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)

    MATH  Google Scholar 

  23. Brook, A., Kendrick, D., Meeraus, A.: GAMS, a user’s guide. ACM SIGNUM Newsletter 23(3-4), 10–11 (1988)

    Article  Google Scholar 

  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)

    Google Scholar 

  25. Cornuéjols, G.: Valid inequalities for mixed integer linear programs. Mathematical Programming B 112(1), 3–44 (2008)

    Article  MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  27. Dantzig, G.: Linear Programming and Extensions. Princeton University Press, Princeton (1963)

    MATH  Google Scholar 

  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)

    Google Scholar 

  29. Dhyani, K.: Personal communication (2007)

    Google Scholar 

  30. Di Giacomo, L.: Mathematical programming methods in dynamical nonlinear stochastic supply chain management. Ph.D. thesis, DSPSA, Università di Roma “La Sapienza” (2007)

    Google Scholar 

  31. Duran, M., Grossmann, I.: An outer-approximation algorithm for a class of mixed-integer nonlinear programs. Mathematical Programming 36, 307–339 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  32. Falk, J., Liu, J.: On bilevel programming, part I: General nonlinear cases. Mathematical Programming 70, 47–72 (1995)

    MathSciNet  Google Scholar 

  33. Falk, J., Soland, R.: An algorithm for separable nonconvex programming problems. Management Science 15, 550–569 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  34. Fischer, A.: New constrained optimization reformulation of complementarity problems. Journal of Optimization Theory and Applications 99(2), 481–507 (1998)

    Article  MathSciNet  Google Scholar 

  35. Fletcher, R., Leyffer, S.: Solving mixed integer nonlinear programs by outer approximation. Mathematical Programming 66, 327–349 (1994)

    Article  MathSciNet  Google Scholar 

  36. Floudas, C.: Deterministic Global Optimization. Kluwer Academic Publishers, Dordrecht (2000)

    Google Scholar 

  37. Fortet, R.: Applications de l’algèbre de Boole en recherche opérationelle. Revue Française de Recherche Opérationelle 4, 17–26 (1960)

    Google Scholar 

  38. Fourer, R.: Personal communication (2004)

    Google Scholar 

  39. Fourer, R., Gay, D.: The AMPL Book. Duxbury Press, Pacific Grove (2002)

    Google Scholar 

  40. Galli, S.: Parsing AMPL internal format for linear and non-linear expressions, B.Sc. dissertation, DEI, Politecnico di Milano, Italy (2004)

    Google Scholar 

  41. Gill, P.: User’s Guide for SNOPT 5.3. Systems Optimization Laboratory, Department of EESOR, Stanford University, California (1999)

    Google Scholar 

  42. Gill, P.: User’s guide for SNOPT version 7. In: Systems Optimization Laboratory. Stanford University, California (2006)

    Google Scholar 

  43. Gomory, R.: Essentials of an algorithm for integer solutions to linear programs. Bulletin of the American Mathematical Society 64(5), 256 (1958)

    Article  MathSciNet  Google Scholar 

  44. Grant, M., Boyd, S., Ye, Y.: Disciplined convex programming. In: Liberti and Maculan [79], pp. 155–210

    Google Scholar 

  45. Guéret, C., Prins, C., Sevaux, M.: Applications of optimization with Xpress-MP. Dash Optimization, Bilsworth (2000)

    Google Scholar 

  46. Hammer, P., Rudeanu, S.: Boolean Methods in Operations Research and Related Areas. Springer, Berlin (1968)

    MATH  Google Scholar 

  47. Hansen, P.: Method of non-linear 0-1 programming. Annals of Discrete Mathematics 5, 53–70 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  48. Haverly, C.: Studies of the behaviour of recursion for the pooling problem. ACM SIGMAP Bulletin 25, 19–28 (1978)

    Article  Google Scholar 

  49. Horst, R.: On the convexification of nonlinear programming problems: an applications-oriented approach. European Journal of Operations Research 15, 382–392 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  50. Horst, R., Tuy, H.: Global Optimization: Deterministic Approaches, 3rd edn. Springer, Berlin (1996)

    MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  52. ILOG: ILOG CPLEX 11.0 User’s Manual. ILOG S.A., Gentilly, France (2008)

    Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  54. Kaibel, V., Pfetsch, M.: Packing and partitioning orbitopes. Mathematical Programming 114(1), 1–36 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  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

    Article  MATH  MathSciNet  Google Scholar 

  56. Kelley, J.: The cutting plane method for solving convex programs. Journal of SIAM VIII(6), 703–712 (1960)

    MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  58. Kojima, M., Megiddo, N., Ye, Y.: An interior point potential reduction algorithm for the linear complementarity problem. Mathematical Programming 54, 267–279 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  59. Konno, H.: A cutting plane algorithm for solving bilinear programs. Mathematical Programming 11, 14–27 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  60. Kucherenko, S., Belotti, P., Liberti, L., Maculan, N.: New formulations for the kissing number problem. Discrete Applied Mathematics 155(14), 1837–1841 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  61. Kucherenko, S., Sytsko, Y.: Application of deterministic low-discrepancy sequences in global optimization. Computational Optimization and Applications 30(3), 297–318 (2004)

    Article  MathSciNet  Google Scholar 

  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)

    Google Scholar 

  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)

    Google Scholar 

  64. Letchford, A., Lodi, A.: Strengthening Chvátal-Gomory cuts and Gomory fractional cuts. Operations Research Letters 30, 74–82 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  65. Liberti, L.: Framework for symbolic computation in C++ using n-ary trees. Tech. rep., CPSE, Imperial College London (2001)

    Google Scholar 

  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)

    Google Scholar 

  67. Liberti, L.: Reduction constraints for the global optimization of NLPs. International Transactions in Operational Research 11(1), 34–41 (2004)

    Article  MathSciNet  Google Scholar 

  68. Liberti, L.: Reformulation and convex relaxation techniques for global optimization. 4OR 2, 255–258 (2004)

    Article  MATH  Google Scholar 

  69. Liberti, L.: Reformulation and convex relaxation techniques for global optimization. Ph.D. thesis, Imperial College London, UK (2004)

    Google Scholar 

  70. Liberti, L.: Linearity embedded in nonconvex programs. Journal of Global Optimization 33(2), 157–196 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  71. Liberti, L.: Writing global optimization software. In: Liberti and Maculan [79], pp. 211–262

    Google Scholar 

  72. Liberti, L.: Compact linearization of binary quadratic problems. 4OR 5(3), 231–245 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Google Scholar 

  74. Liberti, L.: Spherical cuts for integer programming problems. International Transactions in Operational Research 15, 283–294 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  75. Liberti, L.: Reformulations in mathematical programming: Definitions and systematics. RAIRO-RO (accepted for publication)

    Google Scholar 

  76. Liberti, L., Dražic, M.: Variable neighbourhood search for the global optimization of constrained NLPs. In: Proceedings of GO Workshop, Almeria, Spain (2005)

    Google Scholar 

  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)

    Google Scholar 

  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)

    Google Scholar 

  79. Liberti, L., Maculan, N. (eds.): Global Optimization: from Theory to Implementation. Springer, Berlin (2006)

    MATH  Google Scholar 

  80. Liberti, L., Pantelides, C.: Convex envelopes of monomials of odd degree. Journal of Global Optimization 25, 157–168 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  81. Liberti, L., Pantelides, C.: An exact reformulation algorithm for large nonconvex NLPs involving bilinear terms. Journal of Global Optimization 36, 161–189 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  82. Liberti, L., Tsiakis, P., Keeping, B., Pantelides, C.: \(oo\mathcal{OPS}\). Centre for Process Systems Engineering, Chemical Engineering Department, Imperial College, London, UK (2001)

    Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  87. Mangasarian, O.: The linear complementarity problem as a separable bilinear program. Journal of Global Optimization 6, 153–161 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  89. Margot, F.: Pruning by isomorphism in branch-and-cut. Mathematical Programming 94, 71–90 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  90. Margot, F.: Exploiting orbits in symmetric ILP. Mathematical Programming B 98, 3–21 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  91. McCormick, G.: Computability of global solutions to factorable nonconvex programs: Part I — Convex underestimating problems. Mathematical Programming 10, 146–175 (1976)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  93. Mladenović, N., Plastria, F., Urošević, D.: Reformulation descent applied to circle packing problems. Computers and Operations Research 32(9), 2419–2434 (2005)

    Article  MATH  Google Scholar 

  94. Nemhauser, G., Wolsey, L.: Integer and Combinatorial Optimization. Wiley, New York (1988)

    MATH  Google Scholar 

  95. Nowak, I.: Relaxation and Decomposition Methods for Mixed Integer Nonlinear Programming. Birkhäuser, Basel (2005)

    MATH  Google Scholar 

  96. Pantelides, C., Liberti, L., Tsiakis, P., Crombie, T.: Mixed integer linear/nonlinear programming interface specification. Global Cape-Open Deliverable WP2.3-04 (2002)

    Google Scholar 

  97. Pardalos, P., Romeijn, H. (eds.): Handbook of Global Optimization, vol. 2. Kluwer Academic Publishers, Dordrecht (2002)

    MATH  Google Scholar 

  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)

    Google Scholar 

  99. Puchinger, J., Raidl, G.: Relaxation guided variable neighbourhood search. In: Proc. of Mini Euro Conference on Variable Neighbourhood Search, Tenerife, Spain (2005)

    Google Scholar 

  100. Raghavachari, M.: On connections between zero-one integer programming and concave programming under linear constraints. Operations Research 17(4), 680–684 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  101. van Roy, T., Wolsey, L.: Solving mixed integer programming problems using automatic reformulation. Operations Research 35(1), 45–57 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  103. Sherali, H.: Global optimization of nonconvex polynomial programming problems having rational exponents. Journal of Global Optimization 12, 267–283 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  104. Sherali, H.: Tight relaxations for nonconvex optimization problems using the reformulation-linearization/convexification technique (RLT). In: Pardalos and Romeijn [97], pp. 1–63

    Google Scholar 

  105. Sherali, H.: Personal communication (2007)

    Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  107. Sherali, H., Adams, W.: A Reformulation-Linearization Technique for Solving Discrete and Continuous Nonconvex Problems. Kluwer Academic Publishers, Dodrecht (1999)

    MATH  Google Scholar 

  108. Sherali, H., Alameddine, A.: A new reformulation-linearization technique for bilinear programming problems. Journal of Global Optimization 2, 379–410 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Google Scholar 

  110. Sherali, H., Tuncbilek, C.: New reformulation linearization/convexification relaxations for univariate and multivariate polynomial programming problems. Operations Research Letters 21, 1–9 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  111. Sherali, H., Wang, H.: Global optimization of nonconvex factorable programming problems. Mathematical Programming 89, 459–478 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  112. Smith, E.: On the optimal design of continuous processes. Ph.D. thesis, Imperial College of Science, Technology and Medicine, University of London (1996)

    Google Scholar 

  113. Smith, E., Pantelides, C.: Global optimisation of nonconvex MINLPs. Computers & Chemical Engineering 21, S791–S796 (1997)

    Google Scholar 

  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)

    Article  Google Scholar 

  115. Strekalovsky, A.: On global optimality conditions for d.c. programming problems. Technical Paper, Irkutsk State University (1997)

    Google Scholar 

  116. Strekalovsky, A.: Extremal problems with d.c. constraints. Computational Mathematics and Mathematical Physics 41(12), 1742–1751 (2001)

    MathSciNet  Google Scholar 

  117. Stroustrup, B.: The C++ Programming Language, 3rd edn. Addison-Wesley, Reading (1999)

    Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Google Scholar 

  120. Tawarmalani, M., Ahmed, S., Sahinidis, N.: Global optimization of 0-1 hyperbolic programs. Journal of Global Optimization 24, 385–416 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  122. Tawarmalani, M., Sahinidis, N.: Convex extensions and envelopes of semi-continuous functions. Mathematical Programming 93(2), 247–263 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  123. Tawarmalani, M., Sahinidis, N.: Exact algorithms for global optimization of mixed-integer nonlinear programs. In: Pardalos and Romeijn [97], pp. 65–86

    Google Scholar 

  124. Tawarmalani, M., Sahinidis, N.: Global optimization of mixed integer nonlinear programs: A theoretical and computational study. Mathematical Programming 99, 563–591 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  125. Todd, M.: Semidefinite optimization. Acta Numerica 10, 515–560 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  126. Tsiakis, P., Keeping, B.: \(oo\mathcal{MILP}\) – a C++ callable object-oriented library and the implementation of its parallel version using corba. In: Liberti and Maculan [79], pp. 155–210

    Google Scholar 

  127. Tsiakis, P., Keeping, B., Pantelides, C.: \(oo\mathcal{MILP}\). Centre for Process Systems Engineering, Chemical Engineering Department, Imperial College, London, UK, 0.7 edn (2000)

    Google Scholar 

  128. Tuy, H.: Concave programming under linear constraints. Soviet Mathematics, 1437–1440 (1964)

    Google Scholar 

  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)

    Google Scholar 

  130. Wang, X., Change, T.: A multivariate global optimization using linear bounding functions. Journal of Global Optimization 12, 383–404 (1998)

    Article  MATH  Google Scholar 

  131. Westerlund, T.: Some transformation techniques in global optimization. In: Liberti and Maculan [79], pp. 45–74

    Google Scholar 

  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)

    Article  Google Scholar 

  133. Wolsey, L.: Integer Programming. Wiley, New York (1998)

    MATH  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

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  1. LIX, École Polytechnique, Palaiseau, 91128, France

    Leo Liberti, Sonia Cafieri & Fabien Tarissan

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    Ajith Abraham

  2. College of Business Administration Quantitative and Information System Department, Kuwait University, P.O. Box 5486, 13055, Safat, Kuwait

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  3. Université Paris XII Fac. Sciences, LERISS, 61 avenue du Général de Gaulle Building P2 - Room 350, 94010, Créteil, France

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  4. Department of Computer Science, University of Pretoria, 0002, Pretoria, South Africa

    Andries Engelbrecht

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Liberti, L., Cafieri, S., Tarissan, F. (2009). Reformulations in Mathematical Programming: A Computational Approach. In: Abraham, A., Hassanien, AE., Siarry, P., Engelbrecht, A. (eds) Foundations of Computational Intelligence Volume 3. Studies in Computational Intelligence, vol 203. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-01085-9_7

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