Abstract
The general MIP model, discussed in Chap. 2, is reconsidered hereinafter, investigating some possible reformulations, from different points of view (Sect. 3.1). The objective of enucleating implicit implications and introducing valid inequalities, to tighten the model, is examined next (Sect. 3.2).
An erratum to this chapter is available at http://dx.doi.org/10.1007/978-3-319-05005-8_8
An erratum to this chapter can be found at http://dx.doi.org/10.1007/978-3-319-05005-8_8
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- 1.
Note These conditions have been introduced by S. Gliozzi, senior managing consultant at IBM GBS Advanced Analytics and Optimization.
References
Aardal, K., Pochet, Y., Wolsey, L.A.: Capacitated facility location: valid inequalities and facets. Math. Oper. Res. 20, 562–582 (1995)
Andersen, K., Cornuéjols, G., Li, Y.: Reduce-and-split cuts: improving the performance of mixed integer Gomory cuts. Manag. Sci. 51, 1720–1732 (2005)
Andreello, G., Caprara, A., Fischetti, M.: Embedding cuts in a branch and cut framework: a computational study with {0, 1/2}-cuts. INFORMS J. Comput. 19, 229–238 (2007)
Atamtürk, A.: Cover and pack inequalities for (mixed) integer programming. Ann. Oper. Res. 139, 21–38 (2005)
Balas, E., Ceria, S., Cornuéjols, G.: Mixed 0-1 programming by lift-and-project in a branch-and-cut framework. Manag. Sci. 42, 1229–1246 (1996)
Birgin, E.G., Lobato, R.D.: Orthogonal packing of identical rectangles within isotropic convex regions. Comput. Ind. Eng. 59(4), 595–602 (2010)
Birgin, E., Martinez, J., Nishihara, F.H., Ronconi, D.P.: Orthogonal packing of rectangular items within arbitrary convex regions by nonlinear optimization. Comput. Oper. Res. 33(12), 3535–3548 (2006)
Cassioli, A., Locatelli, M.: A heuristic approach for packing identical rectangles in convex regions. Comput. Oper. Res. 38(9), 1342–1350 (2011)
Ceria, S., Cordier, C., Marchand, H., Wolsey, L.A.: Cutting planes for integer programs with general integer variables. Math. Program. 81, 201–214 (1998)
Constantino, M.: Lower bounds in lot-sizing models: a polyhedral study. Math. Oper. Res. 23, 101–118 (1998)
Cordier, C., Marchand, H., Laundry, R., Wolsey, L.A.: A branch-and-cut code for mixed integer programs. Math. Program. 86, 335–354 (2001)
Cornuéjols, G.: Valid inequalities for mixed integer linear programs. Math. Program. 112, 3–44 (2008)
Dash, S., Günlük, O., Lodi, A.: MIR closures of polyhedral sets. Math. Program. 121, 33–60 (2010)
De Farias, I.R., Johnson, E.L., Nemhauser, G.L.: Facets of the complementarity knapsack polytope. Technical Report LEC-98-08, Georgia Institute of Technology, Atlanta (1998)
De Loera, J., Hemmecke, R., Köppe, M.: Global mixed-integer polynomial optimization via summation. In: De Loera, J., Hemmecke, R., Köppe, M. (eds.) Algebraic and Geometric Ideas in the Theory of Discrete Optimization. MPS-SIAM Series on Optimization, pp. 157–177. Society for Industrial and Applied Mathematics, Philadelphia (2012)
Fasano, G.: MIP-based heuristic for non-standard 3D-packing problems. 4OR Q. J. Oper. Res. 6(3), 291–310 (2008)
Hamacher, H.W., Labbé, M., Nickel, S., Sonneborn, T.: Adapting polyhedral properties from facility to hub location problems. Discrete. Appl. Math. 145, 104–116 (2004)
Hanzon, B., Jibetean, D.: Global minimization of a multivariate polynomial using matrix methods. J. Global. Optim. 27, 1–23 (2003)
Jünger, M., Liebling, T.M., Naddef, D., Nemhauser, G.L., Pulleyblank, W.R., et al. (eds.): 50 Years of Integer Programming 1958-2008: From the Early Years to the State-of-the-Art. Springer, Berlin (2009). ISBN 978-3-540-68247-5
Marchand, H., Martin, A., Weismantel, R., Wolsey, L.A.: Cutting planes in integer and mixed integer programming. Technical Report CORE DP9953, Université Catholique de Louvain, Louvain-la-Neuve, Belgium (1999)
Nemhauser, G.L., Wolsey, L.A.: A recursive procedure for generating all cuts for 0-1 mixed integer programs. Math. Program. 46, 379–390 (1990)
Padberg, M.W.: Linear Optimization and Extensions. Springer, Heidelberg (1995)
Padberg, M.W.: Packing small boxes into a big box. Office of Naval Research, N00014-327, New York University (1999)
Padberg, M.W.: Classical cuts for mixed integer programming and branch-and-cut. Math. Meth. Oper. Res. 53, 173–203 (2001)
Padberg, M.W., Rinaldi, G.: A branch and cut algorithm for the resolution of large-scale symmetric traveling salesmen problems. SIAM Rev. 33, 60–100 (1991)
Padberg, M.W., Van Roy, T.J., Wolsey, L.A.: Valid inequalities for fixed charge problems. Oper. Res. 33, 842–861 (1985)
Pintér, J.D.: Global Optimization in Action. Kluwer Academic Publishers, Dordrecht, The Netherlands (1996)
Pintér, J.D.: LGO—a program system for continuous and Lipschitz optimization. In: Bomze, I.M., Csendes, T., Horst, R., Pardalos, P.M. (eds.) Developments in Global Optimization, pp. 183–197. Kluwer Academic Publishers, Dordrecht, The Netherlands (1997)
Pintér, J.D.: Software development for global optimization. In: Pardalos, P.M., Coleman, T.F. (eds.) Global Optimization: Methods and Applications, Fields Institute Communications, vol. 55, pp. 183–204. American Mathematical Society, Providence, RI (2009)
Pochet, Y., Wolsey, L.A.: Polyhedra for lot-sizing with Wagner-Whitin costs. Math. Program. 67, 297–324 (1994)
Schweighofer, M.: Global optimization of polynomials using gradient tentacles and sums of squares. SIAM J. Optim. 17(3), 920–942 (2006)
Suhl, U.H.: Solving large-scale mixed integer programs with fixed charge variables. Math. Program. 32(2), 165–182 (1984)
The MathWorks: MATLAB. The MathWorks, Inc., Natick, MA. www.mathworks.com (2012). Accessed 30 Aug 2013
Van Roy, T.J., Wolsey, L.A.: Solving mixed integer programming problems using automatic reformulation. Oper. Res. 35, 45–57 (1987)
Weismantel, R.: Hilbert bases and the facets of special knapsack polytopes. Math. Oper. Res. 21, 886–904 (1996)
Wolsey, L.A.: Strong formulations for mixed integer programming: a survey. Math. Program. 45, 173–191 (1989)
Wolsey, L.A.: Valid inequalities for mixed integer programs with generalised upper bound constraints. Discrete Appl. Math. 25, 251–261 (1990)
Wolsey, L.A.: Strong formulations for mixed integer programs: valid inequalities and extended formulations. Math. Program. 97(1–2), 423–447 (2003)
Yaman, H.: Polyhedral analysis for the two item uncapacitated lot sizing problem with one way substitution. Discrete Appl. Math. 157, 3133–3151 (2009)
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Fasano, G. (2014). Model Reformulations and Tightening. In: Solving Non-standard Packing Problems by Global Optimization and Heuristics. SpringerBriefs in Optimization. Springer, Cham. https://doi.org/10.1007/978-3-319-05005-8_3
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