Mixed integer linear programming formulation techniques

SIAM Review

Volumn 57, Issue 1, 2015, Pages 3-57

  Vielma, Juan Pablo ^a  

^a MASSACHUSETTS INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

Disjunctive programming; Mixed integer linear programming

Indexed keywords

COMPUTATIONAL METHODS;

DISJUNCTIVE PROGRAMMING; MIXED INTEGER LINEAR PROGRAMMING; MIXED INTEGER LINEAR PROGRAMMING (MIP); STATE OF THE ART;

INTEGER PROGRAMMING;

EID: 84923104626     PISSN: 00361445     EISSN: None     Source Type: Journal    
DOI: 10.1137/130915303     Document Type: Review

References (171)

1. T. ACHTERBERG, Constraint Integer Programming, Ph.D. thesis, TU Berlin,2007.
2. T. ACHTERBERG, SCIP: Solving constraint integer programs, Math. Program. Comput., 1 (2009), pp. 1-41.
3. W. ADAMS AND R. FORRESTER, A simple recipe for concise mixed 0-1 linearizations, Oper. Res. Lett., 33 (2005), pp. 55-61.
4. W. ADAMS AND R. FORRESTER, Linear forms of nonlinear expressions: New insights on old ideas, Oper. Res. Lett., 35 (2007), pp. 510-518.
5. W. ADAMS, R. FORRESTER, AND F. GLOVER, Comparisons and enhancement strategies for linearizing mixed 0-1 quadratic programs, Discrete Optim., 1 (2004), pp. 99-120.
6. W. ADAMS AND S. HENRY, Base-2 expansions for linearizing products of functions of discrete variables, Oper. Res., 60 (2012), pp. 1477-1490.
7. W. ADAMS AND H. SHERALI, A tight linearization and an algorithm for zero-one quadratic programming problems, Management Sci., 32 (1986), pp. 1274-1290.
8. F. AL-KHAYYAL AND J. FALK, Jointly constrained biconvex programming, Math. Oper. Res., 8 (1983), pp. 273-286.
9. G. ANGULO, S. AHMED, S. S. DEY, AND V. KAIBEL, Forbidden vertices, Math. Oper. Res., to appear; doi:10.1287/moor.2014.0673.
10. E. BALAS, Disjunctive programming, Ann. Discrete Math., 5 (1979), pp. 3-51.
11. E. BALAS, Disjunctive programming and a hierarchy of relaxations for discrete optimization problems, SIAM J. Algebraic Discrete Methods, 6 (1985), pp. 466-486.
12. E. BALAS, On the convex-hull of the union of certain polyhedra, Oper. Res. Lett., 7 (1988), pp. 279-283.
13. E. BALAS, Disjunctive programming: Properties of the convex hull of feasible points, Discrete Appl. Math., 89 (1998), pp. 3-44.
14. K. M. BALL, An elementary introduction to modern convex geometry, in Flavors of Geometry, S. Levy, ed., Math. Sci. Res. Inst. Publ. 31, Cambridge University Press, Cambridge, 1997, pp. 1-58.
15. C. BARNHART, E. JOHNSON, G. NEMHAUSER, M. SAVELSBERGH, AND P. VANCE, Branch-and- price: Column generation for solving huge integer programs, Oper. Res., 46 (1996), pp. 316-329.
16. A. BARVINOK AND E. VEOMETT, The computational complexity of convex bodies, in Surveys on Discrete and Computational Geometry: Twenty Years Later, J. Goodman, J. Pach, and R. Pollack, eds., Contemp. Math. 453, AMS, Providence, RI, 2008, pp. 117-138.
17. E. M. L. BEALE AND J. A. TOMLIN, Special facilities in a general mathematical programming system for non-convex problems using ordered sets of variables, in OR 69: Proceedings of the Fifth International Conference on Operational Research, J. Lawrence, ed., Tavistock Publications, 1970, pp. 447-454.
18. N. BEAUMONT, An algorithm for disjunctive programs, European J. Oper. Res., 48 (1990), pp. 362-371.
19. A. BEN-TAL AND A. NEMIROVSKI, On polyhedral approximations of the second-order cone, Math. Oper. Res., 26 (2001), pp. 193-205.
20. D. BERTSIMAS, S. GUPTA, AND G. LULLI, Dynamic resource allocation: A flexible and tractable modeling framework, European J. Oper. Res., 236 (2014), pp. 14-26.
21. D. BERTSIMAS AND J. TSITSIKLIS, Introduction to Linear Optimization, Athena Scientific, 1997.
22. D. BERTSIMAS AND R. WEISMANTEL, Optimization over Integers, Dynamic Ideas, 2005.
23. R. BIXBY, M. FENELON, Z. GU, E. ROTHBERG, AND R. WUNDERLING, MIP: Theory and practice - closing the gap, in System Modelling and Optimization, 1999, pp. 19-50.
24. R. BIXBY, M. FENELON, Z. GU, E. ROTHBERG, AND R. WUNDERLING, Mixed-integer programming: A progress report, in The Sharpest Cut: The Impact of Manfred Padberg and His Work, SIAM, Philadelphia, PA, 2004, Chap. 18, pp. 309-325.
25. R. BIXBY AND E. ROTHBERG, Progress in computational mixed integer programming: A look back from the other side of the tipping point, Ann. Oper. Res., 149 (2007), pp. 37-41.
26. C. BLAIR, Representation for multiple right-hand sides, Math. Programming, 49 (1990), pp. 1- 5.
27. D. BRICKER, Reformulation of special ordered sets for implicit enumeration algorithms, with applications in nonconvex separable programming, AIIE Trans., 9 (1977), pp. 195-203.
28. M. N. BROADIE, A theorem about antiprisms, Linear Algebra Appl., 66 (1985), pp. 99-111.
29. S. BURER AND A. LETCHFORD, Non-convex mixed-integer nonlinear programming: A survey, Surv. Oper. Res. Manag. Sci., 17 (2012), pp. 97-106.
30. R. D. CARR AND G. LANCIA, Compact vs. exponential-size LP relaxations, Oper. Res. Lett., 30 (2002), pp. 57-65.
31. R. D. CARR AND G. LANCIA, Compact optimization can outperform separation: A case study in structural proteomics, 4OR, 2 (2004), pp. 221-233.
32. R. Carvajal, M. Constantino, M. Goycoolea, J. P. Vielma, and A. Weintraub, Imposing connectivity constraints in forest planning models, Oper. Res., 61 (2013), pp. 824-836.
33. C. CAVALCANTE, C. C. DE SOUZA, M. SAVELSBERGH, Y. WANG, AND L. WOLSEY, Scheduling projects with labor constraints, Discrete Appl. Math., 112 (2001), pp. 27-52.
34. J. COCHRAN, L. A. COX, JR., P. KESKINOCAK, J. KHAROUFEH, AND J. SMITH, Wiley Encyclopedia of Operations Research and Management Science, John Wiley & Sons, 2011.
35. M. CONFORTI, G. CORNUÉJOLS, AND G. ZAMBELLI, Extended formulations in combinatorial optimization, 4OR, 8 (2010), pp. 1-48.
36. M. CONFORTI, G. CORNUÉJOLS, AND G. ZAMBELLI, Integer Programming, Grad. Texts in Math. 271, Springer, 2014.
37. M. CONFORTI AND L. WOLSEY, Compact formulations as a union of polyhedra, Math. Programming, 114 (2008), pp. 277-289.
38. A. CONN AND M. MONGEAU, Discontinuous piecewise linear optimization, Math. Programming, 80 (1998), pp. 315-380.
39. W. COOK, Fifty-plus years of combinatorial integer programming, in 50 Years of Integer Programming 1958-2008: From the Early Years to the State-of-the-Art, Springer-Verlag, New York, 2010, Chap. 12, pp. 387-430.
40. W. COOK, In Pursuit of the Traveling Salesman: Mathematics at the Limits of Computation, Princeton University Press, Princeton, NJ, 2011.
41. D. COPPERSMITH AND J. LEE, Parsimonious binary-encoding in integer programming, Discrete Optim., 2 (2005), pp. 190-200.
42. K. L. CROXTON, B. GENDRON, AND T. L. MAGNANTI, Variable disaggregation in network flow problems with piecewise linear costs, Oper. Res., 55 (2007), pp. 146-157.
43. C. D'AMBROSIO, A. LODI, AND S. MARTELLO, Piecewise linear approximation of functions of two variables in MILP models, Oper. Res. Lett., 38 (2010), pp. 39-46.
44. G. B. DANTZIG, Discrete-variable extremum problems, Oper. Res., 5 (1957), pp. 266-277.
45. G. B. Dantzig, On the significance of solving linear-programming problems with some integer variables, Econometrica, 28 (1960), pp. 30-44.
46. I. R. DE FARIAS, E. KOZYREFF, R. GUPTA, AND M. ZHAO, Branch-and-cut for separable piecewise linear optimization and intersection with semi-continuous constraints, Math. Program. Comput., 5 (2013), pp. 75-112.
47. I. R. DE FARIAS JR., M. ZHAO, AND H. ZHAO, A special ordered set approach for optimizing a discontinuous separable piecewise linear function, Oper. Res. Lett., 36 (2008), pp. 234- 238.
48. J. DESROSIERS AND M. E. LÜBBECKE, Branch-Price-and-Cut Algorithms, in Wiley Encyclopedia of Operations Research and Management Science, John Wiley & Sons, 2011.
49. S. DEY AND D. MORÁN R., Some properties of convex hulls of integer points contained in general convex sets, Math. Programming, 141 (2013), pp. 507-526.
50. B. DILKINA AND C. GOMES, Solving connected subgraph problems in wildlife conservation, in Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems, Lecture Notes in Comput. Sci. 6140, A. Lodi, M. Milano, and P. Toth, eds., Springer, Berlin, Heidelberg, 2010, pp. 102-116.
51. P. DOMSCHKE, B. GEISSLER, O. KOLB, J. LANG, A. MARTIN, AND A. MORSI, Combination of nonlinear and linear optimization of transient gas networks, INFORMS J. Comput., 23 (2011), pp. 605-617.
52. FAIR ISAAC CORPORATION, FICO Xpress Optimization Suite, http://www.fico.com/en/Products/DMTools/Pages/FICO-Xpress-Optimization-Suite.aspx.
53. S. FIORINI, S. MASSAR, S. POKUTTA, H. R. TIWARY, AND R. DE WOLF, Linear vs. semidefinite extended formulations: Exponential separation and strong lower bounds, in Proceedings of the 44th Symposium on Theory of Computing Conference, STOC 2012, New York, 2012, H. J. Karloff and T. Pitassi, eds., ACM, New York, 2012, pp. 95-106.
54. R. FOURER, A simplex algorithm for piecewise-linear programming I: Derivation and proof, Math. Programming, 33 (1985), pp. 204-233.
55. R. FOURER, A simplex algorithm for piecewise-linear programming II: Finiteness, feasibility and degeneracy, Math. Programming, 41 (1988), pp. 281-315.
56. R. FOURER, A simplex algorithm for piecewise-linear programming III: Computational analysis and applications, Math. Programming, 53 (1992), pp. 213-235.
57. B. GEISSLER, A. MARTIN, A. MORSI, AND L. SCHEWE, Using piecewise linear functions for solving MINLPs, in Mixed Integer Nonlinear Programming, J. Lee and S. Leyffer, eds., Math. Appl. 154, Springer, New York, 2012, pp. 287-314.
58. F. GLOVER, Improved linear integer programming formulations of nonlinear integer problems, Management Sci., 22 (1975), pp. 455-460.
59. F. GLOVER, Improved MIP formulation for products of discrete and continuous variables, J. Inf. Optim. Sci., 5 (1984), pp. 69-71.
60. F. GLOVER AND E. WOOLSEY, Converting the 0-1 polynomial programming problem to a 0-1 linear program, Oper. Res., 22 (1974), pp. 180-182.
61. M. X. GOEMANS, Smallest compact formulation for the permutahedron, Math. Program., to appear; doi:10.1007/s10107-014-0757-1.
62. C. GOUNARIS, R. MISENER, AND C. A. FLOUDAS, Computational comparison of piecewise-linear relaxations for pooling problems, Indust. Engrg. Chem. Res., 48 (2009), pp. 5742- 5766.
63. B. GRÜNBAUM, Convex Polytopes, Grad. Texts in Math., Springer, 2003.
64. S. GUEYE AND P. MICHELON, Miniaturized linearizations for quadratic 0/1 problems, Ann. Oper. Res., 140 (2005), pp. 235-261.
65. S. GUEYE AND P. MICHELON, A linearization framework for unconstrained quadratic (0-1) problems, Discrete Appl. Math., 157 (2009), pp. 1255-1266.
66. M. GUIGNARD, C. RYU, AND K. SPIELBERG, Model tightening for integrated timber harvest and transportation planning, European J. Oper. Res., 111 (1998), pp. 448-460.
67. M. GUIGNARD AND K. SPIELBERG, Logical reduction methods in zero-one programming: Minimal preferred variables, Oper. Res., 29 (1981), pp. 49-74.
68. M. GUIGNARD-SPIELBERG AND K. SPIELBERG, Integer Programming: State of the Art and Recent Advances, Ann. Oper. Res. 139, Springer, New York, 2005.
69. GUROBI OPTIMIZATION, The Gurobi Optimizer, available online from http://www.gurobi.com.
70. P. HANSEN, Methods of nonlinear 0-1 programming, Ann. Discrete Math., 5 (1979), pp. 53-70.
71. P. HANSEN AND C. MEYER, Improved compact linearizations for the unconstrained quadratic 0-1 minimization problem, Discrete Appl. Math., 157 (2009), pp. 1267-1290.
72. I. HARJUNKOSKI, R. PÖRN, T. WESTERLUND, AND H. SKRIFVARS, Different strategies for solving bilinear integer non-linear programming problems with convex transformations, Comput. Chem. Engrg., 21 (1997), pp. S487-S492.
73. R. HEMMECKE AND R. WEISMANTEL, Representation of sets of lattice points, SIAM J. Optim., 18 (2007), pp. 133-137.
74. S. M. HENRY, Tight Polyhedral Representations of Discrete Sets using Projections, Simplices, and Base-2 Expansions, Ph.D. thesis, Clemson University, 2011.
75. J. N. HOOKER, Logic-Based Methods for Optimization: Combining Optimization and Constraint Satisfaction, Wiley Series in Discrete Mathematics and Optimization, John Wiley & Sons, 2000.
76. J. N. HOOKER, A principled approach to mixed integer/linear problem formulation, in Operations Research and Cyber-Infrastructure, J. W. Chinneck, B. Kristjansson, and M. J. Saltzman, eds., Oper. Res./Comput. Sci. Interfaces Ser. 47, Springer, New York, 2009, pp. 79-100.
77. J. N. HOOKER, Hybrid modeling, in Hybrid Optimization: The Ten Years of CPAIOR, P. Van Hentenryck and M. Milano, eds., Springer Optim. Appl. 45, Springer, 2011, pp. 11-62.
78. J. N. HOOKER, Integrated Methods for Optimization, 2nd ed., Internat. Ser. Oper. Res. Management Sci. 170, Springer, 2012.
79. J. N. HOOKER AND M. A. OSORIO, Mixed logical-linear programming, Discrete Appl. Math., 96 (1999), pp. 395-442.
80. R. HORST AND H. TUY, Global Optimization: Deterministic Approaches, Springer, 2003.
81. T. IBARAKI, Integer programming formulation of combinatorial optimization problems, Discrete Math., 16 (1976), pp. 39-52.
82. IBM ILOG, CPLEX High-Performance Mathematical Programming Engine, http://www.ibm.com/software/integration/optimization/cplex/.
83. R. G. JEROSLOW, Some basis theorems for integral monoids, Math. Oper. Res., 3 (1978), pp. 145-154.
84. R. G. JEROSLOW, Representations of unbounded optimization problems as integer programs, J. Optim. Theory Appl., 30 (1980), pp. 339-351.
85. R. G. JEROSLOW, Representability in mixed integer programming I: Characterization results, Discrete Appl. Math., 17 (1987), pp. 223-243.
86. R. G. JEROSLOW, Alternative formulations of mixed integer programs, Ann. Oper. Res., 12 (1988), pp. 241-276.
87. R. G. JEROSLOW, A simplification for some disjunctive formulations, European J. Oper. Res., 36 (1988), pp. 116-121.
88. R. G. JEROSLOW, Logic-Based Decision Support: Mixed Integer Model Formulation, Ann. Discrete Math. 40, North-Holland, Amsterdam, 1989.
89. R. G. JEROSLOW, Representability of functions, Discrete Appl. Math., 23 (1989), pp. 125-137.
90. R. G. JEROSLOW AND J. K. LOWE, Modeling with integer variables, Math. Programming Stud., 22 (1984), pp. 167-184.
91. R. G. JEROSLOW AND J. K. LOWE, Experimental results on the new techniques for integer programming formulations, J. Oper. Res. Soc., 36 (1985), pp. 393-403.
92. E. L. JOHNSON, G. L. NEMHAUSER, AND M. W. P. SAVELSBERGH, Progress in linear programming-based algorithms for integer programming: An exposition, INFORMS J. Comput., 12 (2000), pp. 2-23.
93. W. B. JOHNSON AND J. LINDENSTRAUSS, EDS., Handbook of the Geometry of Banach Spaces. Vol. I, North-Holland, Amsterdam, 2001.
94. M. JÜNGER, T. LIEBLING, D. NADDEF, G. NEMHAUSER, W. PULLEYBLANK, G. REINELT, G. RINALDI, AND L. WOLSEY, 50 Years of Integer Programming 1958-2008: From the Early Years to the State-of-the-Art, Springer-Verlag, New York, 2010.
95. V. KAIBEL, Extended formulations in combinatorial optimization, Optima, 85 (2011), pp. 2-7.
96. V. KAIBEL AND A. LOOS, Branched polyhedral systems, in Integer Programming and Combinatorial Optimization, 14th International Conference, IPCO 2010, Lausanne, Switzerland, June 9-11, 2010. Proceedings, F. Eisenbrand and F. B. Shepherd, eds., Lecture Notes in Comput. Sci. 6080, Springer, 2010, pp. 177-190.
97. V. KAIBEL AND K. PASHKOVICH, Constructing extended formulations from reflection relations, in Proceedings of IPCO 2011, O. Günlük and G. J. Woeginger, eds., Lecture Notes in Comput. Sci. 6655, Springer, 2011, pp. 287-300.
98. B. S. KASHIN, The widths of certain finite-dimensional sets and classes of smooth functions, Izv. Akad. Nauk SSSR Ser. Mat., 41 (1977), pp. 334-351 (in Russian).
99. A. KEHA, I. DE FARIAS, AND G. NEMHAUSER, A branch-and-cut algorithm without binary variables for nonconvex piecewise linear optimization, Oper. Res., 54 (2006), pp. 847- 858.
100. O. KETTANI AND M. ORAL, Equivalent formulations of nonlinear integer problems for efficient optimization, Management Sci., 36 (1990), pp. 115-119.
101. S. KÜC¸UKYAVUZ, On mixing sets arising in chance-constrained programming, Math. Programming, 132 (2012), pp. 31-56.
102. A. H. LAND AND A. G. DOIG, An automatic method for solving discrete programming problems, Econometrica, 28 (1960), pp. 497-520.
103. J. LEE, All-different polytopes, J. Comb. Optim., 6 (2002), pp. 335-352.
104. J. LEE, A celebration of 50 years of integer programming, Optima, 76 (2008), pp. 10-14.
105. J. LEE AND F. MARGOT, On a binary-encoded ILP coloring formulation, INFORMS J. Comput., 19 (2007), pp. 406-415.
106. J. LEE AND D. WILSON, Polyhedral methods for piecewise-linear functions I: The lambda method, Discrete Appl. Math., 108 (2001), pp. 269-285.
107. H. LI AND H. LU, Global optimization for generalized geometric programs with mixed free-sign variables, Oper. Res., 57 (2009), pp. 701-713.
108. L. LIBERTI, Reformulations in mathematical programming: Definitions and systematics, RAIRO Oper. Res., 43 (2009), pp. 55-85.
109. L. LIBERTI AND N. MACULAN, Reformulation techniques in mathematical programming, Discrete Appl. Math., 157 (2009), pp. 1165-1166.
110. E. LIN AND D. BRICKER, Connecting special ordered inequalities and transformation and reformulation technique in multiple choice programming, Comput. Oper. Res., 29 (2002), pp. 1441-1446.
111. E. Y. LIN AND D. L. BRICKER, On the calculation of true and pseudo penalties in multiple choice integer programming, European J. Oper. Res., 55 (1991), pp. 228-236.
112. A. LODI, Mixed integer programming computation, in 50 Years of Integer Programming 1958-2008: From the Early Years to the State-of-the-Art, Springer-Verlag, New York, 2010, Chap. 16, pp. 619-645.
113. J. K. LOWE, Modelling with Integer Variables, Ph.D. thesis, Georgia Institute of Technology, 1984.
114. J. LUEDTKE, S. AHMED, AND G. NEMHAUSER, An integer programming approach for linear programs with probabilistic constraints, Math. Programming, 122 (2010), pp. 247-272.
115. R. K. MARTIN, Using separation algorithms to generate mixed integer model reformulations, Oper. Res. Lett., 10 (1991), pp. 119-128.
116. R. K. MARTIN, R. L. RARDIN, AND B. A. CAMPBELL, Polyhedral characterization of discrete dynamic programming, Oper. Res., 38 (1990), pp. 127-138.
117. G. MCCORMICK, Nonlinear Programming: Theory, Algorithms and Applications, John Wiley & Sons, 1983.
118. R. R. MEYER, On the existence of optimal solutions to integer and mixed-integer programming problems, Math. Programming, 7 (1974), pp. 223-235.
119. R. R. MEYER, Integer and mixed-integer programming models: General properties, J. Optim. Theory Appl., 16 (1975), pp. 191-206.
120. R. R. MEYER, Mixed integer minimization models for piecewise-linear functions of a single variable, Discrete Math., 16 (1976), pp. 163-171.
121. R. R. MEYER, A theoretical and computational comparison of equivalent mixed-integer formulations, Naval Res. Logistics, 28 (1981), pp. 115-131.
122. R. R. MEYER, M. V. THAKKAR, AND W. P. HALLMAN, Rational mixed-integer and polyhedral union minimization models, Math. Oper. Res., 5 (1980), pp. 135-146.
123. R. MISENER AND C. A. FLOUDAS, Global optimization of large-scale generalized pooling problems: Quadratically constrained MINLP models, Indust. Engrg. Chem. Res., 49 (2010), pp. 5424-5438.
124. R. MISENER AND C. A. FLOUDAS, Piecewise-linear approximations of multidimensional functions, J. Optim. Theory Appl., 145 (2010), pp. 120-147.
125. R. MISENER AND C. A. FLOUDAS, Global optimization of mixed-integer quadraticallyconstrained quadratic programs (MIQCQP) through piecewise-linear and edge-concave relaxations, Math. Programming, 136 (2012), pp. 155-182.
126. R. MISENER, C. GOUNARIS, AND C. A. FLOUDAS, Mathematical modeling and global optimization of large-scale extended pooling problems with the (EPA) complex emissions constraints, Comput. Chem. Engrg., 34 (2010), pp. 1432-1456.
127. R. MISENER, J. THOMPSON, AND C. A. FLOUDAS, APOGEE: Global optimization of standard, generalized, and extended pooling problems via linear and logarithmic partitioning schemes, Comput. Chem. Engrg., 35 (2011), pp. 876-892.
128. J. E. MITCHELL, Branch and cut, in Wiley Encyclopedia of Operations Research and Management Science, John Wiley & Sons, 2011.
129. R. H. MÖHRING, A. S. SCHULZ, F. STORK, AND M. UETZ, On project scheduling with irregular starting time costs, Oper. Res. Lett., 28 (2001), pp. 149-154.
130. F. M. MULDOON, W. P. ADAMS, AND H. D. SHERALI, Ideal representations of lexicographic orderings and base-2 expansions of integer variables, Oper. Res. Lett., 41 (2013), pp. 32-39.
131. G. L. NEMHAUSER AND L. A. WOLSEY, Integer and Combinatorial Optimization, Wiley-Interscience, 1988.
132. W. OGRYCZAK, A note on modeling multiple choice requirements for simple mixed integer programming solvers, Comput. Oper. Res., 23 (1996), pp. 199-205.
133. M. ORAL AND O. KETTANI, A linearization procedure for quadratic and cubic mixed-integer problems, Oper. Res., 40 (1992), pp. 109-116.
134. J. OWEN AND S. MEHROTRA, On the value of binary expansions for general mixed-integer linear programs, Oper. Res., 50 (2002), pp. 810-819.
135. M. W. PADBERG, The boolean quadric polytope: Some characteristics, facets and relatives, Math. Programming, 45 (1989), pp. 139-172.
136. M. W. PADBERG, Approximating separable nonlinear functions via mixed zero-one programs, Oper. Res. Lett., 27 (2000), pp. 1-5.
137. M. W. PADBERG AND M. P. RIJAL, Location, Scheduling, Design, and Integer Programming, Springer, 1996.
138. Y. POCHET AND L. WOLSEY, Production Planning by Mixed Integer Programming, Springer Ser. Oper. Res. Financ. Eng., Springer, 2006.
139. R. RAMAN AND I. GROSSMANN, Modelling and computational techniques for logic based integer programming, Comput. Chem. Engrg., 18 (1994), pp. 563-578.
140. T. ROTHVOß, The matching polytope has exponential extension complexity, in Symposium on Theory of Computing (STOC 2014), D. B. Shmoys, ed., ACM, New York, 2014, pp. 263-272.
141. D. M. RYAN AND B. A. FOSTER, An integer programming approach to scheduling, in Computer Scheduling of Public Transport Urban Passenger Vehicle and Crew Scheduling, A. Wren, ed., North-Holland, 1981, pp. 269-280.
142. R. SADYKOV AND F. VANDERBECK, Column generation for extended formulations, Electron. Notes Discrete Math., 37 (2011), pp. 357-362.
143. N. SAWAYA AND I. GROSSMANN, A cutting plane method for solving linear generalized disjunctive programming problems, Comput. Chem. Engrg., 29 (2005), pp. 1891-1913.
144. N. SAWAYA AND I. GROSSMANN, A hierarchy of relaxations for linear generalized disjunctive programming, European J. Oper. Res., 216 (2012), pp. 70-82.
145. A. SCHRIJVER, Theory of Linear and Integer Programming, Wiley Series in Discrete Mathematics & Optimization, John Wiley & Sons, 1998.
146. A. SCHRIJVER, Combinatorial Optimization: Polyhedra and Efficiency, Algorithms and Combinatorics, Springer, 2003.
147. H. SHERALI AND W. P. ADAMS, A Reformulation-Linearization Technique for Solving Discrete and Continuous Nonconvex Problems, Kluwer Academic Publishers, 1999.
148. D. SOMMER, Computational Experience with the Ophelie Mixed Integer Code, talk presented at the International TIMS Conference, Houston, 1972.
149. C. C. SOUZA AND L. A. WOLSEY, Scheduling Projects with Labour Constraints, relatório técnico ic-97-22, IC - UNICAMP, 1997.
150. K. SPIELBERG, Minimal Preferred Variable Methods in 0-1 Programming, Tech. Report 320-3013, IBM Philadelphia Scientific Center, 1972.
151. D. STRALBERG, D. L. APPLEGATE, S. J. PHILLIPS, M. P. HERZOG, N. NUR, AND N. WARNOCK, Optimizing wetland restoration and management for avian communities using a mixed integer programming approach, Biological Conservation, 142 (2009), pp. 94-109.
152. S. SZAREK, Convexity, complexity, and high dimensions, in International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 1599-1621.
153. A. TORIELLO AND J. P. VIELMA, Fitting piecewise linear continuous functions, European J. Oper. Res., 219 (2012), pp. 86-95.
154. F. VANDERBECK AND L. WOLSEY, Reformulation and decomposition of integer programs, in 50 Years of Integer Programming 1958-2008: From the Early Years to the State-of-the-Art, Springer-Verlag, New York, 2010, Chap. 16, pp. 431-502.
155. J. P. VIELMA, S. AHMED, AND G. L. NEMHAUSER, Mixed-integer models for nonseparable piecewise-linear optimization: Unifying framework and extensions, Oper. Res., 58 (2010), pp. 303-315.
156. J. P. VIELMA, S. AHMED, AND G. L. NEMHAUSER, A note on "a superior representation method for piecewise linear functions," INFORMS J. Comput., 22 (2010), pp. 493-497.
157. J. P. VIELMA, S. AHMED, AND G. L. NEMHAUSER, Mixed integer linear programming formulations for probabilistic constraints, Oper. Res. Lett., 40 (2012), pp. 153-158.
158. J. P. VIELMA, A. B. KEHA, AND G. L. NEMHAUSER, Nonconvex, lower semicontinuous piecewise linear optimization, Discrete Optim., 5 (2008), pp. 467-488.
159. J. P. VIELMA AND G. L. NEMHAUSER, Modeling disjunctive constraints with a logarithmic number of binary variables and constraints, in Proceedings of IPCO 2008, A. Lodi, A. Panconesi, and G. Rinaldi, eds., Lecture Notes in Comput. Sci. 5035, Springer, 2008, pp. 199-213.
160. J. P. VIELMA AND G. L. NEMHAUSER, Modeling disjunctive constraints with a logarithmic number of binary variables and constraints, Math. Programming, 128 (2011), pp. 49-72.
161. M. V. VYVE AND L. A. WOLSEY, Approximate extended formulations, Math. Programming, 105 (2006), pp. 501-522.
162. L. WATTERS, Reduction of integer polynomial programming problems to zero-one linear programming problems, Oper. Res., 15 (1967), pp. 1171-1174.
163. H. P. WILLIAMS, Experiments in the formulation of integer programming problems, Math. Programming Stud., 2 (1974), pp. 180-197.
164. H. P. WILLIAMS, Logic and Integer Programming, International Series in Operations Research & Management Science, Springer, 2009.
165. D. L. WILSON, Polyhedral Methods for Piecewise-Linear Functions, Ph.D. thesis, University of Kentucky, Lexington, KY, 1998.
166. L. WOLSEY, Integer Programming, Wiley Series in Discrete Mathematics and Optimization, Wiley, 1998.
167. M. YANNAKAKIS, Expressing combinatorial optimization problems by linear programs, J. Comput. System Sci., 43 (1991), pp. 441-466.
168. S. YILDIZ AND J. P. VIELMA, Incremental and encoding formulations for mixed integer programming, Oper. Res. Lett., 41 (2013), pp. 654-658.
169. M. ZHAO AND I. R. DE FARIAS, JR., The piecewise linear optimization polytope: New inequalities and intersection with semi-continuous constraints, Math. Program., 141 (2013), pp. 217-255.
170. G. M. ZIEGLER, Lectures on Polytopes, Springer-Verlag, 1995.
171. ZUSE INSTITUTE BERLIN, SCIP: Solving Constraint Integer Programs, http://scip.zib.de/.