Abstract
In this chapter we want to demonstrate that in certain cases general mixed integer nonlinear programs (MINLPs) can be solved by just applying purely techniques from the mixed integer linear world. The way to achieve this is to approximate the nonlinearities by piecewise linear functions. The advantage of applying mixed integer lin- ear techniques are that these methods are nowadays very mature, that is, they are fast, robust, and are able to solve problems with up to millions of variables. In addition, these methods have the potential of finding globally optimal solutions or at least to provide solution guarantees. On the other hand, one tends to say at this point “If you have a hammer, everything is a nail.”[15], because one tries to reformulate or to approximate an ac- tual nonlinear problem until one obtains a model that is tractable by the methods one is common with. Besides the fact that this is a very typical approach in mathematics the question stays whether this is a reasonable approach for the solution of MINLPs or whether the nature of the nonlin- earities inherent to the problem gets lost and the solutions obtained from the mixed integer linear problem have no meaning for the MINLP. The purpose of this chapter is to discuss this question. We will see that the truth lies somewhere in between and that there are problems where this is indeed a reasonable way to go and others where it is not.
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References
A. Balakrishnan and S.C. Graves, A composite algorithm for a concave-cost network flow problem, Networks, 19 (1989), pp. 175–202.
J.J. Bartholdi III and P. Goldsman, The vertex-adjacency dual of a triangulated irregular network has a hamiltonian cycle, Operations Research Letters, 32 (2004), pp. 304–308.
E.M.L. Beale and J.J.H. Forrest, Global optimization using special ordered sets, Math. Programming, 10 (1976), pp. 52–69.
E.M.L. Beale and J.A. Tomlin, Special facilitiess in a general mathematical programming system for non-convex problems using ordered sets of variables,in OR 69, J. Lawrence, ed., International Federation of Operational Research Societies, Travistock Publications, 1970, pp. 447–454.
A. Ben-Tal and A. Nemirovski, On polyhedral approximations of the secondorder cone, Math. Oper. Res., 26 (2001), pp. 193–205.
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.
G.B. Dantzig, On the significance of solving linear programming problems with some integer variables, Econometrica, 28 (1960), pp. 30–44.
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.
C.E. Gounaris and C.A. Floudas, Tight convex underestimators for C2- contiuous problems: I. multivariate functions, Journal of Global Optimization, 42 (2008), pp. 69–89.
, Tight convex underestimators for C2-contiuous problems: I. univariate functions, Journal of Global Optimization, 42 (2008), pp. 51–67.
M. Jach, D. Michaels, and R.Weismantel, The convex envelope of (n-1) convex functions, Siam Journal on Optimization, 19 (3) (2008), pp. 1451–1466.
J.L.W.V. Jensen, Sur les fonctions convexes et les in´egalit´es entre les valeurs moyennes, Acta Methematica, 30 (1) (1906), pp. 175 – 193.
A.B. Keha, I.R. de Farias, Jr., and G.L. Nemhauser, Models for representing piecewise linear cost functions, Oper. Res. Lett., 32 (2004), pp. 44–48.
, A branch-and-cut algorithm without binary variables for nonconvex piecewise linear optimization, Oper. Res., 54 (2006), pp. 847–858.
T. Koch, Personal communication, 2008.
A. Krion, Optimierungsmethoden zur Berechnung von Cross-Border-Flow beim Market-Coupling im europ¨aischen Stromhandel, Master’s thesis, Discrete Optimization Group, Department of Mathematics, Technische Universit¨at Darmstadt, Darmstadt, Germany, 2008.
J. Lee and D. Wilson, Polyhedral methods for piecewise-linear functions. I. The lambda method, Discrete Appl. Math., 108 (2001), pp. 269–285.
S. Leyffer, A. Sartenaer, and E. Wanufell, Branch-and-refine for mixed-integer nonconvex global optimization, Tech. Rep. ANL/MCS-P1547-0908,Argonne National Laboratory, Mathematics and Computer Science Division, 2008.
D. Mahlke, A. Martin, and S. Moritz, A mixed integer approach for timedependent gas network optimization, Optimization Methods and Software, 25 (2010), pp. 625 – 644.
C.D. Maranas and C.A. Floudas, Global minimum potential energy conformations of small molecules, Journal of Global Optimization, 4 (1994), pp. 135–170.
H.M. Markowitz and A.S. Manne, On the solution of discrete programming problems, Econometrica, 25 (1957), pp. 84–110.
R.R. Meyer, Mixed integer minimization models for piecewise-linear functions of a single variable, Discrete Mathematics, 16 (1976), pp. 163 – 171.
G. Nemhauser and J.P. Vielma, Modeling disjunctive constraints with a logarithmic number of binary variables and constraints, in Integer Programming and Combinatorial Optimization, Vol. 5035 of Lecture Notes in Computer Science, 2008, pp. 199–213.
M. Padberg, Approximating separable nonlinear functions via mixed zero–one programs, Oper. Res. Lett., 27 (2000), pp. 1–5.
M. Padberg and M.P. Rijal, Location, scheduling, design and integer programming, Kluwer Academic Publishers, Boston, 1996.
R.T. Rockafellar, Convex Analysis, Princeton University Press, 1970
W.D. Smith, A lower bound for the simplexity of the n-cube via hyperbolic volumes,European Journal of Combinatorics, 21 (2000), pp. 131–137.
F. Tardella, On the existence of polyhedral convex envelopes, in Frontiers in global optimization, C. Floudas and P. M. Pardalos, eds., Vol. 74 of Nonconvex Optimization and its Applications, Springer, 2004, pp. 563 – 573.
M.J. Todd, Hamiltonian triangulations of Rn, in Functional Differential Equations and Approximation of Fixed Points, A. Dold and B. Eckmann, eds., Vol. 730/1979 of Lecture Notes in Mathematics, Springer, 1979, pp. 470 – 483.
J.P. Vielma, S. Ahmed, and G. Nemhauser, Mixed-Integer models for nonseparable Piecewise-Linear optimization: Unifying framework and extensions, Operations Research, 58 (2009), pp. 303–315.
J.P. Vielma, A.B. Keha, and G.L. Nemhauser, Nonconvex, lower semicontinuous piecewise linear optimization, Discrete Optim., 5 (2008), pp. 467–488.
D. Wilson, Polyhedral methods for piecewise-linear functions, Ph.D. thesis in Discrete Mathematics, University of Kentucky, 1998.
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Geißler, B., Martin, A., Morsi, A., Schewe, L. (2012). Using Piecewise Linear Functions for Solving MINLPs . In: Lee, J., Leyffer, S. (eds) Mixed Integer Nonlinear Programming. The IMA Volumes in Mathematics and its Applications, vol 154. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1927-3_10
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