Abstract
We present a Branch and Cut algorithm of the software package LaGO to solve nonconvex mixed-integer nonlinear programs (MINLPs). A linear outer approximation is constructed from a convex relaxation of the problem. Since we do not require an algebraic representation of the problem, reformulation techniques for the construction of the convex relaxation cannot be applied, and we are restricted to sampling techniques in case of nonquadratic nonconvex functions. The linear relaxation is further improved by mixed-integer-rounding cuts. Also box reduction techniques are applied to improve efficiency. Numerical results on medium size test problems are presented to show the efficiency of the method.
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References
Adjiman CS and Floudas CA (1997). Rigorous convex underestimators for general twice-differentiable problems. J Global Optim 9: 23–40
Adjiman CS, Dallwig S, Floudas CA and Neumaier A (1998). A global optimization method, αBB, for general twice-differentiable constrained NLPs—I. Theor Adv Comp Chem Eng 22: 1137–1158
Adjiman CS, Androulakis IP and Floudas CA (2000). Global optimization of mixed-integer nonlinear problems. AIChE J 46: 1769–1797
Ahadi-Oskui T (2006) Optimierung des Entwurfs komplexer Energieumwandlungsanlagen. Fortschritts-Berichte VDI, Reihe 6, Nr. 543. VDI-Verlag
Ahadi-Oskui T, Tsatsaronis G (2006) Optimization of the design of a complex energy conversion system using mathematical programming and genetic algorithms. In: Proceedings of IMECE2006
Ahadi-Oskui T, Nowak I, Tsatsaronis G, Vigerske S (2007) Optimizing the design of complex energy conversion systems by Branch and Cut. Preprint 07-11. Department of Mathematics, Humboldt University Berlin, http://www.math.hu-berlin.de/publ/pre/2007/P-07-11.pdf (submitted)
Akrotirianakis IG and Floudas CA (2004). A new class of improved convex underestimators for twice differentiable constrained NLPs. J Global Optim 30(4): 367–390
Bussieck MR, Drud AS and Meeraus A (2003). MINLPLib—a collection of test models for mixed-integer nonlinear programming. INFORMS J Comput 15(1): 114–119
Duran MA and Grossmann IE (1986). An outer-approximation algorithm for a class of mixed-integer nonlinear programs. Math Prog 36: 307–339
Fletcher R and Leyffer S (1994). Solving mixed integer nonlinear programs by outer approximation. Math Program 66(3(A)): 327–349
Floudas CA, Akrotirianakis IG, Caratzoulas C, Meyer CA and Kallrath J (2005). Global optimization in the 21st century: advances and challenges. Comput Chem Eng 29(6): 1185–1202
GAMS Development Corp. (2003) GAMS—the solver manuals
Gonçalves JPM, Ladanyi L (2005) An implementation of a separation procedure for mixed integer rounding inequalities. Research Report RC23686, IBM Research Division, August
ILOG, Inc. CPLEX. http://www.ilog.com/products/cple.
Lougee-Heimer R (2003) The common optimization interface for operations research. IBM J Res Dev 47(1):57–66. http://www.coin-or.org
Marchand H and Wolsey LA (2001). Aggregation and mixed integer rounding to solve MIPs. Oper Res 49(3): 363–371
Meyer CA and Floudas CA (2005). Convex underestimation of twice continuously differentiable functions by piecewise quadratic perturbation: Spline αBB. J Global Optim 29(6): 1185–1202
Nemhauser GL and Wolsey LA (1988). Integer and combinatorial optimization. Wiley-Interscience, New York
Neumaier A (2004) Complete search in continuous global optimization and constraint satisfaction. In: Acta numerica, vol 13, chap 4. Cambridge University Press, Cambridge, pp 271–370
Nowak I (1999). A new semidefinite programming bound for indefinite quadratic forms over a simplex. J Global Optim 14(4): 357–364
Nowak I (2005a). Lagrangian decomposition of block-separable mixed-integer all-quadratic programs. Math Program 102(2): 295–312
Nowak I (2005b). Relaxation and decomposition methods for mixed integer nonlinear programming. Birkhäuser, Basel
Nowak I, Alperin H, Vigerske S (2003) LaGO—an object oriented library for solving MINLPs. In: Bliek Ch, Jermann Ch, Neumaier A (eds) Global optimization and constraint satisfaction, volume 2861 of Lecture Notes in Computer Science. Springer, Heidelberg, pp 31–43
Nowak I, Vigerske S LaGO—Lagrangian Global Optimizer. https://projects.coin-or.org/LaGO
Sahinidis N, Tawarmalani M (2002) BARON. http://archimedes.scs.uiuc.edu/baron/baron.html
Tawarmalani M and Sahinidis NV (2004). Global optimization of mixed-integer nonlinear programs: a theoretical and computational study. Math Program 99: 563–591
Tawarmalani M and Sahinidis NV (2002). Convexification and global optimization in continuous and mixed-integer nonlinear programming: theory, algorithms, software and applications. Kluwer, Dordrecht
Viswanathan J and Grossmann IE (1990). A combined penalty function and outer-approximation method for MINLP optimization. Comput Chem Eng 14(7): 769–782
Wächter A, Biegler LT (2006) On the implementation of a primal-dual interior point filter line search algorithm for large-scale nonlinear programming. Math Prog 106(1):25–57. http://projects.coin-or.org/Ipopt
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Nowak, I., Vigerske, S. LaGO: a (heuristic) Branch and Cut algorithm for nonconvex MINLPs. cent.eur.j.oper.res. 16, 127–138 (2008). https://doi.org/10.1007/s10100-007-0051-x
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DOI: https://doi.org/10.1007/s10100-007-0051-x