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Reformulations in Mathematical Programming: Definitions and Systematics

Published online by Cambridge University Press:  28 January 2009

Leo Liberti
Leo Liberti*
Affiliation:
LIX École Polytechnique, 91128 Palaiseau, France; [email protected]
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Abstract

A reformulation of a mathematical program is a formulation whichshares some properties with, but is in some sense better than, theoriginal program. Reformulations are important with respect to thechoice and efficiency of the solution algorithms; furthermore, it isdesirable that reformulations can be carried outautomatically. Reformulation techniques are widespread in mathematicalprogramming but interestingly they have never been studied under aunified framework. This paper attempts to move some steps in thisdirection. We define a framework for storing and manipulatingmathematical programming formulations and give several fundamentaldefinitions categorizing useful reformulations in essentially fourtypes (opt-reformulations, narrowings, relaxations andapproximations). We establish some theoretical results and givereformulation examples for each type.

Keywords

Reformulationformulationmodellinearizationmathematical program.
Type
Research Article
Information
RAIRO - Operations Research , Volume 43 , Issue 1 , January 2009 , pp. 55 - 85
Copyright
© EDP Sciences, ROADEF, SMAI, 2009

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References

Adams, W.P. and Sherali, H.D., A tight linearization and an algorithm for 0-1 quadratic programming problems. Manage. Sci. 32 (1986) 1274–1290. CrossRef
Adams, W.P. and Sherali, H.D., A hierarchy of relaxations leading to the convex hull representation for general discrete optimization problems. Ann. Oper. Res. 140 (2005) 21–47. CrossRef
Adams, W.P., Forrester, R.J. and Glover, F.W., Comparisons and enhancement strategies for linearizing mixed 0-1 quadratic programs. Discrete Optim. 1 (2004) 99–120. CrossRef
Adjiman, C.S., Dallwig, S., Floudas, C.A. and Neumaier, A., A global optimization method, $\alpha$ BB, for general twice-differentiable constrained NLPs: I. Theoretical advances. Comput. Chem. Eng. 22 (1998) 1137–1158. CrossRef
Adjiman, C.S., Androulakis, I.P. and Floudas, C.A., A global optimization method, $\alpha$ BB, for general twice-differentiable constrained NLPs: II. Implementation and computational results. Comput. Chem. Eng. 22 (1998) 1159–1179. CrossRef
Al-Khayyal, F.A. and Falk, J.E., Jointly constrained biconvex programming. Math. Oper. Res. 8 (1983) 273–286. CrossRef
Alizadeh, F., Interior point methods in semidefinite programming with applications to combinatorial optimization. SIAM J. Optim. 5 (1995) 13–51. CrossRef
Anstreicher, K.M., Recent advances in the solution of quadratic assignment problems. Math. Program. B 97 (2003) 27–42. CrossRef
Audet, C., Hansen, P., Jaumard, B. and Savard, G., Links between linear bilevel and mixed 0-1 programming problems. J. Optim. Theor. Appl. 93 (1997) 273–300. CrossRef
Audet, C., Brimberg, J., Hansen, P., Le Digabel, S. and Mladenović, N., Pooling problem: Alternate formulations and solution methods. Manage. Sci. 50 (2004) 761–776. CrossRef
H.M.T. Ben Amor, J. Desrosiers and A. Frangioni, Stabilization in column generation. Discrete Appl. Math. to appear.
Ben-Ameur, W. and Kerivin, H., Routing of uncertain demands. Optim. Eng. 3 (2005) 283–313. CrossRef
Bertsimas, D. and Sym, M., The price of robustness. Oper. Res. 52 (2004) 35–53. CrossRef
Billionnet, A. and Elloumi, S., Using a mixed-integer quadratic programming solver for the unconstrained quadratic 0-1 problem. Math. Program. 109 (2007) 55–68. CrossRef
A. Billionnet, S. Elloumi and M.-C. Plateau, Improving the performance of standard solvers via a tighter convex reformulation of constrained quadratic 0-1 programs: the QCR method. Discrete Appl. Math., to appear.
Bjorkqvist, J. and Westerlund, T., Automated reformulation of disjunctive constraints in MINLP optimization. Comput. Chem. Eng. 23 (1999) S11–S14. CrossRef
Björk, K.-M., Lindberg, P.O. and Westerlund, T., Some convexifications in global optimization of problems containing signomial terms. Comput. Chem. Eng. 27 (2003) 669–679. CrossRef
A. Brook, D. Kendrick and A. Meeraus, GAMS, a user's guide. ACM SIGNUM Newsletter, 23 (1988) 10–11.
G.B. Dantzig, Linear Programming and Extensions. Princeton University Press, Princeton, NJ (1963).
T. Davidović, L. Liberti, N. Maculan and N. Mladenović, Towards the optimal solution of the multiprocessor scheduling problem with communication delays, in MISTA Proceedings (2007).
L. Di Giacomo, Mathematical programming methods in dynamical nonlinear stochastic Supply Chain management. Ph.D. thesis, DSPSA, Università di Roma “La Sapienza" (2007).
Falk, J.E. and Liu, J., On bilevel programming, part i: general nonlinear cases. Mathem. Program. 70 (1995) 47–72. CrossRef
R. Fletcher and S. Leyffer, User manual for filter. Technical report, University of Dundee, UK (1999).
Fortet, R., Applications de l'algèbre de boole en recherche opérationelle. Revue Française de Recherche Opérationelle 4 (1960) 17–26.
R. Fourer, Personal communication (2004).
R. Fourer and D. Gay, The AMPL Book. Duxbury Press, Pacific Grove (2002).
R. Fourer, J. Ma, K. Martin and W. Sheng, Optimization services 1.0 user manual. Technical report, COIN-OR (2007).
Frangioni, A., On a new class of bilevel programming problems and its use for reformulating mixed-integer problems. Eur. J. Oper. Res. 82 (1995) 615–646. CrossRef
S. Galli, Parsing AMPL internal format for linear and non-linear expressions. Didactical project, DEI, Politecnico di Milano, Italy (2004).
P.E. Gill, User's Guide for SNOPT 5.3. Systems Optimization Laboratory, Department of EESOR, Stanford University, California (1999).
M. Grant, S. Boyd and Y. Ye, Disciplined convex programming, in Liberti and Maculan [56], 155–210.
C. Guéret, C. Prins and M. Sevaux, Applications of optimization with Xpress-MP. Dash optimization. Bilsworth (2000).
S. Gueye and Ph. Michelon, A linearization framework for unconstrained quadratic (0-1) problems. Discrete Appl. Math., to appear.
Gueye, S. and Michelon, P., “miniaturized" linearizations for quadratic 0/1 problems. Ann. Oper. Res. 140 (2005) 235–261. CrossRef
Hägglöf, K., Lindberg, P.O. and Svensson, L., Computing global minima to polynomial optimization problems using Gröbner bases. J. Glob. Optim. 7 (1995) 115–125. CrossRef
P.L. Hammer and S. Rudeanu, Boolean Methods in Operations Research and Related Areas. Springer, Berlin (1968).
Hansen, P., Method of non-linear 0-1 programming. Ann. Discrete Math. 5 (1979) 53–70. CrossRef
P. Hansen and C. Meyer, Improved compact linearizations for the unconstrained quadratic 0-1 minimization problem. Discrete Appl. Math., to appear.
Horst, R., On the convexification of nonlinear programming problems: an applications-oriented approach. Eur. J. Oper. Res. 15 (1984) 382–392. CrossRef
R. Horst and Hoang Tuy, Global Optimization: Deterministic Approaches. Springer-Verlag, Berlin, 3rd ed. (1996).
K.-L. Hsiung, S.-J. Kim and S. Boyd, Tractable approximate robust geometric programming. Optim. Eng. to appear.
ILOG, ILOG CPLEX 10.0 User's Manual. ILOG S.A., Gentilly, France (2005).
Judice, J. and Mitra, G., Reformulation of mathematical programming problems as linear complementarity problems and investigation of their solution methods. J. Optim. Theor. Appl. 57 (1988) 123–149. CrossRef
Kojima, M., Megiddo, N. and An, Y. Ye interior point potential reduction algorithm for the linear complementarity problem. Math. Program. 54 (1992) 267–279. CrossRef
L. Liberti. Comparison of convex relaxations for monomials of odd degree, in Optimization and Optimal Control, edited by I. Tseveendorj, P.M. Pardalos and R. Enkhbat. World Scientific (2003).
Liberti, L., Reduction constraints for the global optimization of NLPs. Int. Trans. Oper. Res. 11 (2004) 34–41.
L. Liberti, Reformulation and Convex Relaxation Techniques for Global Optimization. Ph.D. thesis, Imperial College London, UK, (2004).
L. Liberti, Reformulation and convex relaxation techniques for global optimization. 4OR 2 (2004) 255–258.
Liberti, L., Linearity embedded in nonconvex programs. J. Glob. Optim.n 33 (2005) 157–196. CrossRef
L. Liberti, Writing global optimization software, in Global Optimization: from Theory to Implementation, edited by Liberti and Maculan.Springer, berlin (2006) 211–262.
L. Liberti, Reformulation techniques in mathematical programming, Thèse d'Habilitation à Diriger des Recherches, Université de Paris-Dauphine (2007).
L. Liberti, Compact linearization of binary quadratic problems. 4OR 5 231–245 (2007).
L. Liberti, Automatic generation of symmetry-breaking constraints, in COCOA Proceedings, Lecture Notes in Computer Science, edited by B. Yang, D.-Z. Du and C.A. Wang 5165. Springer, Berlin (2008) 328–338.
Liberti, L. and Pantelides, C.C., Convex envelopes of monomials of odd degree. J. Glob. Optim. 25 (2003) 157–168. CrossRef
Liberti, L. and Pantelides, C.C., An exact reformulation algorithm for large nonconvex NLPs involving bilinear terms. J. Glob. Optim. 36 (2006) 161–189. CrossRef
L. Liberti and N. Maculan, Eds., Global Optimization: from Theory to Implementation. Springer, Berlin (2006).
Liberti, L., Amaldi, E., Maculan, N. and Maffioli, F., Mathematical models and a constructive heuristic for finding minimum fundamental cycle bases. Yugosl. J. Oper. Res. 15 (2005) 15–24. CrossRef
L. Liberti, C. Lavor, M.A.C. Nascimento and N. Maculan, Reformulation in mathematical programming: an application to quantum chemistry. Discrete Appl. Math. to appear.
J.A. de Loera, J. Lee, S. Margulies and S. Onn, Expressing combinatorial optimization problems by systems of polynomial equations and the nullstellensatz, Technical Report RC24276(W0706-020), IBM Corporation (2007).
Lougee-Heimer, R., The common optimization interface for operations research: Promoting open-source software in the operations research community. IBM J. Res. Dev. 47 (2003) 57–66. CrossRef
Mangasarian, O.L., Linear complementarity problems solvable by a single linear program. Math. Program. 10 (1976) 263–270. CrossRef
Mangasarian, O.L., The linear complementarity problem as a separable bilinear program. J. Glob. Optim. 6 (1995) 153–161. CrossRef
Margot, F., Pruning by isomorphism in branch-and-cut. Math. Program. 94 (2002) 71–90. CrossRef
Margot, F., Exploiting orbits in symmetric ilp. Math. Program. B 98 (2003) 3–21. CrossRef
McCormick, G.P., Computability of global solutions to factorable nonconvex programs: Part I – Convex underestimating problems. Math. Program. 10 (1976) 146–175.
Meyer, C.A. and Floudas, C.A., Convex hull of trilinear monomials with mixed sign domains. J. Glob. Optim. 29 (2004) 125–155. CrossRef
Mladenović, N., Plastria, F. and Urošević, D., Reformulation descent applied to circle packing problems. Comput. Oper. Res. 32 (2005) 2419–2434. CrossRef
I. Nowak, Relaxation and Decomposition Methods for Mixed Integer Nonlinear Programming. Birkhäuser, Basel (2005).
C.C. Pantelides, L. Liberti, P. Tsiakis and T. Crombie, Mixed integer linear/nonlinear programming interface specification, Global Cape-Open Deliverable WP2.3-04 (2002).
M.-C. Plateau, Reformulations quadratiques convexes pour la programmation quadratique en variables 0-1. Ph.D. thesis, Conservatoire National d'Arts et Métiers (2006).
J. Puchinger and G.R. Raidl, Relaxation guided variable neighbourhood search, in Proc. of Mini Euro Conference on Variable Neighbourhood Search, Tenerife, Spain (2005).
A. Saxena, V. Goyal and M. Lejeune, MIP reformulations of the probabilistic set covering problem. Technical Report 2007-02-1579, Optimization Online (2007).
Sherali, H.D., Global optimization of nonconvex polynomial programming problems having rational exponents. J. Glob. Optim. 12 (1998) 267–283. CrossRef
H. Sherali, Personal communication (2007).
H.D. Sherali and W.P. Adams, A Reformulation-Linearization Technique for Solving Discrete and Continuous Nonconvex Problems. Kluwer Academic Publishers, Dodrecht (1999).
Sherali, H.D. and Adams, W.P., A hierarchy of relaxations and convex hull characterizations for mixed-integer zero-one programming problems. Discrete Appl. Math. 52 (1994) 83–106. CrossRef
Sherali, H.D. and Adams, W.P., A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems. SIAM J. Discrete Math. 3 (1990) 411–430. CrossRef
Sherali, H.D. and Alameddine, A., A new reformulation-linearization technique for bilinear programming problems. J. Global Optimization 2 (1992) 379–410. CrossRef
H. Sherali and L. Liberti, Reformulation-linearization methods for global optimization, in Encyclopedia of Optimization, edited by C. Floudas and P. Pardalos. Springer, New York, to appear.
H.D. Sherali and C. Smith, Improving discrete model representations via symmetry considerations. Manag. Sci. 47 1396–1407.
Sherali, H. and Tuncbilek, C.H., New reformulation linearization/convexification relaxations for univariate and multivariate polynomial programming problems. Oper. Res. Lett. 21 (1997) 1–9. CrossRef
Sherali, H. and Tuncbilek, C.H., A global optimization algorithm for polynomial programming problems using a reformulation-linearization technique. J. Glob. Optim. 2 (1991) 101–112. CrossRef
Sherali, H.D. and Wang, H., Global optimization of nonconvex factorable programming problems. Math. Program. 89 (2001) 459–478. CrossRef
E.M.B. Smith, On the Optimal Design of Continuous Processes. Ph.D. thesis, Imperial College of Science, Technology and Medicine, University of London (1996).
Smith, E.M.B. and Pantelides, C.C., Global optimisation of nonconvex MINLPs. Comput. Chem. Eng. 21 (1997) S791–S796. CrossRef
Smith, E.M.B. and Pantelides, C.C., A symbolic reformulation/spatial branch-and-bound algorithm for the global optimisation of nonconvex MINLPs. Comput. Chem. Eng. 23 (1999) 457–478. CrossRef
INFORMS Computing Society. The mathematical programming glossary. http://glossary.computing.society.informs.org/second.php?page=R.html.
Strekalovsky, A.S., Extremal problems with d.c. constraints. Comput. Mathem. Math. Phys. 41 (2001) 1742–1751.
A.S. Strekalovsky, On global optimality conditions for d.c. programming problems, Technical Paper, Irkutsk State University (1997).
Tardella, F., Existence and sum decomposition of vertex polyhedral convex envelopes. Optim. Lett. 2 (2008) 363–375. CrossRef
Tawarmalani, M. and Sahinidis, N., Convex extensions and envelopes of semi-continuous functions. Math. Program. 93 (2002) 247–263. CrossRef
Tawarmalani, M. and Sahinidis, N.V., Global optimization of mixed integer nonlinear programs: A theoretical and computational study. Math. Program. 99 (2004) 563–591. CrossRef
Todd, M.J., Semidefinite optimization. Acta Numerica 10 (2001) 515–560. CrossRef
H. Tuy, D.C. optimization: Theory, methods and algorithms. in Handbook of Global Optimization , 1, edited by R. Horst and P.M. Pardalos. Kluwer Academic Publishers, Dordrecht (1995) 149–216.
van Roy, T.J. and Wolsey, L.A., Solving mixed integer programming problems using automatic reformulation. Oper. Res. 35 (1987) 45–57. CrossRef
J.C. Vera, J.F. Pena and L.F. Zuluaga, Exploiting equalities in polynomial programming, Technical Report 2006-05-1338, Optimization Online, 2006.
Wang, X. and Change, T.S., A multivariate global optimization using linear bounding functions. J. Glob. Optim. 12 (1998) 383–404. CrossRef
T. Westerlund, Some transformation techniques in global optimization, in Global optimization: from theory to implementation, edited by L. Liberti and N. Maculan. Springer, Berlin (2006) 45–74.
L.A. Wolsey, Integer Programming. Wiley, New York (1998).