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Mixed-integer nonlinear optimization*†

Published online by Cambridge University Press:  02 April 2013

Pietro Belotti ,
Christian Kirches ,
Sven Leyffer ,
Jeff Linderoth ,
James Luedtke  and
Ashutosh Mahajan
Pietro Belotti
Affiliation:
Department of Mathematical Sciences, Clemson University, Clemson, SC 29634, USA
Christian Kirches
Affiliation:
Interdisciplinary Center for Scientific Computing, Heidelberg University, 69120 Heidelberg, Germany Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL 60439, USA
Sven Leyffer
Affiliation:
Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL 60439, USA
Jeff Linderoth
Affiliation:
Department of Industrial and Systems Engineering, University of Wisconsin–Madison, Madison, WI 53706, USA
James Luedtke
Affiliation:
Department of Industrial and Systems Engineering, University of Wisconsin–Madison, Madison, WI 53706, USA
Ashutosh Mahajan
Affiliation:
Industrial Engineering and Operations Research, Indian Institute of Technology Bombay, Powai, Mumbai, MH 400076, India
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Abstract

Many optimal decision problems in scientific, engineering, and public sector applications involve both discrete decisions and nonlinear system dynamics that affect the quality of the final design or plan. These decision problems lead to mixed-integer nonlinear programming (MINLP) problems that combine the combinatorial difficulty of optimizing over discrete variable sets with the challenges of handling nonlinear functions. We review models and applications of MINLP, and survey the state of the art in methods for solving this challenging class of problems.

Most solution methods for MINLP apply some form of tree search. We distinguish two broad classes of methods: single-tree and multitree methods. We discuss these two classes of methods first in the case where the underlying problem functions are convex. Classical single-tree methods include nonlinear branch-and-bound and branch-and-cut methods, while classical multitree methods include outer approximation and Benders decomposition. The most efficient class of methods for convex MINLP are hybrid methods that combine the strengths of both classes of classical techniques.

Non-convex MINLPs pose additional challenges, because they contain non-convex functions in the objective function or the constraints; hence even when the integer variables are relaxed to be continuous, the feasible region is generally non-convex, resulting in many local minima. We discuss a range of approaches for tackling this challenging class of problems, including piecewise linear approximations, generic strategies for obtaining convex relaxations for non-convex functions, spatial branch-and-bound methods, and a small sample of techniques that exploit particular types of non-convex structures to obtain improved convex relaxations.

We finish our survey with a brief discussion of three important aspects of MINLP. First, we review heuristic techniques that can obtain good feasible solution in situations where the search-tree has grown too large or we require real-time solutions. Second, we describe an emerging area of mixed-integer optimal control that adds systems of ordinary differential equations to MINLP. Third, we survey the state of the art in software for MINLP.

Type
Research Article
Information
Acta Numerica , Volume 22 , May 2013 , pp. 1 - 131
Copyright
Copyright © Cambridge University Press 2013

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Footnotes

*

Colour online for monochrome figures available at journals.cambridge.org/anu.

†

This work was supported by the Office of Advanced Scientific Computing Research, Office of Science, US Department of Energy, under Contract DE-AC02-06CH11357.

References

REFERENCES

Abhishek, K., Leyffer, S. and Linderoth, J. T. (2010), ‘FilMINT: An outer-approximation-based solver for nonlinear mixed integer programs’, INFORMS J. Comput. 22, 555–567.CrossRefGoogle Scholar
Abichandani, P., Benson, H. Y. and Kam, M. (2008), Multi-vehicle path coordination under communication constraints. In American Control Conference, IEEE Conference Publications, pp. 650–656.Google Scholar
Abramson, M. A. (2004), ‘Mixed variable optimization of a load-bearing thermal insulation system using a filter pattern search algorithm’, Optim. Engng 5, 157–177.CrossRefGoogle Scholar
Abramson, M., Audet, C., Chrissis, J. and Walston, J. (2009), ‘Mesh adaptive direct search algorithms for mixed variable optimization’, Optim. Lett. 3, 35–47.CrossRefGoogle Scholar
Achterberg, T. (2005), SCIP: A framework to integrate constraint and mixed integer programming. ZIB-Report 04-19, Zuse Institut Berlin.Google Scholar
Achterberg, T. and Berthold, T. (2007), ‘Improving the feasibility pump’, Discrete Optim. 4, 77–86.CrossRefGoogle Scholar
Achterberg, T., Koch, T. and Martin, A. (2004), ‘Branching rules revisited’, Oper. Res. Lett. 33, 42–54.CrossRefGoogle Scholar
Adams, W. (2011), Use of Lagrange interpolating polynomials in the RLT. In Wiley Encyclopedia of Operations Research and Management Science.Google Scholar
Adams, W. and Sherali, H. (1986), ‘A tight linearization and an algorithm for zero-one quadratic programming problems’, Management Sci. 32, 1274–1290.CrossRefGoogle Scholar
Adams, W. and Sherali, H. (2005), ‘A hierarchy of relaxations leading to the convex hull representation for general discrete optimization problems’, Ann. Oper. Res. 140, 21–47.CrossRefGoogle Scholar
Adjiman, C. S., Androulakis, I. and Floudas, C. (1998), ‘A global optimization method, αBB, for general twice-differentiable constrained NLPs, II: Implementation and computational results’, Comput. Chem. Engng 22, 1159–1179.CrossRefGoogle Scholar
Akrotirianakis, I., Maros, I. and Rustem, B. (2001), ‘An outer approximation based branch-and-cut algorithm for convex 0–1 MINLP problems’, Optim. Methods Software 16, 21–47.Google Scholar
Al-Khayyal, F. A. and Falk, J. E. (1983), ‘Jointly constrained biconvex programming’, Math. Oper. Res. 8, 273–286.CrossRefGoogle Scholar
Altunay, M., Leyffer, S., Linderoth, J. T. and Xie, Z. (2011), ‘Optimal security response to attacks on open science grids’, Computer Networks 55, 61–73.CrossRefGoogle Scholar
Andersen, E. D. and Andersen, K. D. (1995), ‘Presolving in linear programming’, Math. Program. 71, 221–245.CrossRefGoogle Scholar
Androulakis, I. P., Maranas, C. D. and Floudas, C. A. (1995), ‘αBB: A global optimization method for general constrained nonconvex problems’, J. Global Optim. 7, 337–363.CrossRefGoogle Scholar
Anstreicher, K. (2012), ‘On convex relaxations for quadratically constrained quadratic programming’, Math. Program. 136, 233–251.CrossRefGoogle Scholar
Anstreicher, K. M. (2009), ‘Semidefinite programming versus the reformulation–linearization technique for nonconvex quadratically constrained quadratic programming’, J. Global Optim. 43, 471–484.CrossRefGoogle Scholar
Atamtürk, A. and Narayanan, V. (2010), ‘Conic mixed-integer rounding cuts’, Math. Program. A 122, 1–20.CrossRefGoogle Scholar
Audet, C. and Dennis, J. E. Jr (2000), ‘Pattern search algorithms for mixed variable programming’, SIAM J. Optim. 11, 573–594.CrossRefGoogle Scholar
Bacher, R. (1997), The Optimal Power Flow (OPF) and its solution by the interior point approach. EES-UETP Madrid, short course.Google Scholar
Baes, M., Pia, A. Del, Nesterov, Y., Onn, S. and Weismantel, R. (2012), ‘Minimizing Lipschitz-continuous strongly convex functions over integer points in polytopes’, Math. Program. 134, 305–322.CrossRefGoogle Scholar
Balakrishnan, A. and Graves, S. (1989), ‘A composite algorithm for a concave-cost network flow problem’, Networks 19, 175–202.CrossRefGoogle Scholar
Balaprakash, P., Wild, S. M. and Hovland, P. D. (2011), ‘Can search algorithms save large-scale automatic performance tuning?’, Procedia Comput. Sci. (ICCS 2011) 4, 2136–2145.CrossRefGoogle Scholar
Balas, E., Ceria, S. and Cornuéjols, G. (1993), ‘A lift-and-project cutting plane algorithm for mixed 0–1 programs’, Math. Program. 58, 295–324.CrossRefGoogle Scholar
Balas, E., Ceria, S. and Cornuéjols, G. (1996), ‘Mixed 0–1 programming by lift-and-project in a branch-and-cut framework’, Management Sci. 42, 1229–1246.CrossRefGoogle Scholar
Bao, X., Sahinidis, N. and Tawarmalani, M. (2009), ‘Multiterm polyhedral relaxations for nonconvex quadratically constrained quadratic programs’, Optim. Methods Software 24, 485–504.CrossRefGoogle Scholar
Bartelt-Hunt, S., Culver, T., Smith, J., Matott, L. S. and Rabideau, A. (2006), ‘Optimal design of a compacted soil liner containing sorptive amendments’, J. Environmental Engng 132, 769–776.CrossRefGoogle Scholar
Bartholomew, E. F., O'Neill, R. P. and Ferris, M. C. (2008), ‘Optimal transmission switching’, IEEE Trans. Power Systems 23, 1346–1355.Google Scholar
Bauschke, H. H. and Borwein, J. M. (1996), ‘On projection algorithms for solving convex feasibility problems’, SIAM Rev. 38, 367–426.CrossRefGoogle Scholar
Beale, E. and Tomlin, J. (1970), Special facilities in a general mathematical programming system for non-convex problems using ordered sets of variables. In Proc. 5th International Conference on Operations Research (Lawrence, J., ed.), Tavistock Publications, pp. 447–454.Google Scholar
Beale, E. M. L. and Forrest, J. J. H. (1976), ‘Global optimization using special ordered sets’, Math. Program. 10, 52–69.CrossRefGoogle Scholar
Bellman, R. (1961), ‘On the approximation of curves by line segments using dynamic programming’, Commun. Assoc. Comput. Mach. 4, 284.Google Scholar
Belotti, P. (2009), Couenne: A user's manual. Technical report, Lehigh University.Google Scholar
Belotti, P. (2012), Disjunctive cuts for non-convex MINLP. In Mixed Integer Non-linear Programming, Vol. 154 of The IMA Volumes in Mathematics and its Applications, Springer, pp. 117–144.Google Scholar
Belotti, P. (2013), ‘Bound reduction using pairs of linear inequalities’, J. Global Optim., to appear.Google Scholar
Belotti, P., Cafieri, S., Lee, J. and Liberti, L. (2010), Feasibility-based bounds tightening via fixed points. In Combinatorial Optimization and Applications (Wu, W. and Daescu, O., eds), Vol. 6508 of Lecture Notes in Computer Science, Springer, pp. 65–76.CrossRefGoogle Scholar
Belotti, P., Góez, J., Pólik, I., Ralphs, T. and Terlaky, T. (2012), A conic representation of the convex hull of disjunctive sets and conic cuts for integer second order cone optimization. Technical report 12T-009, Department of Industrial and Systems Engineering, Lehigh University. http://www.optimization-online.org/DBFILE/2012/06/3494.pdfGoogle Scholar
Belotti, P., Lee, J., Liberti, L., Margot, F. and Wächter, A. (2009), ‘Branching and bounds tightening techniques for non-convex MINLP’, Optim. Methods Software 24, 597–634.CrossRefGoogle Scholar
Ben-Tal, A. and Nemirovski, A. (1995), ‘Optimal design of engineering structures’, Optima 47, 4–8.Google Scholar
Ben-Tal, A. and Nemirovski, A. (2001), ‘On polyhedral approximations of the second-order cone’, Math. Oper. Res. 26, 193–205.CrossRefGoogle Scholar
Benson, H. Y. (2011), ‘Mixed integer nonlinear programming using interior point methods’, Optim. Methods Software 26, 911–931.CrossRefGoogle Scholar
Benson, H. Y. (2012), Using interior-point methods within an outer approximation framework for mixed integer nonlinear programming. In Mixed Integer Non-linear Programming, Vol. 154 of The IMA Volumes in Mathematics and its Applications, Springer, pp. 225–243.Google Scholar
Berthold, T. (2012), RENS: The optimal rounding. ZIB-Report 12-17, Zuse Institut Berlin.Google Scholar
Berthold, T. and Gleixner, A. M. (2012), Undercover: A primal MINLP heuristic exploring a largest sub-MIP. ZIB-Report 12-07, Zuse Institut Berlin.Google Scholar
Berthold, T., Gamrath, G., Gleixner, A., Heinz, S., Koch, T. and Shinano, Y. (2012), Solving mixed integer linear and nonlinear problems using the SCIP optimization suite. ZIB-Report 12-27, Zuse Institut Berlin.Google Scholar
Berthold, T., Gleixner, A., Heinz, S. and Vigerske, S. (2010), Extending SCIP for solving MIQCPs. In Proc. European Workshop on Mixed Integer Nonlinear Programming, pp. 181–196. http://www.lix.polytechnique.fr/~liberti/ewminlp/ewminlp-proceedings.pdfGoogle Scholar
Bertsekas, D. and Gallager, R. (1987), Data Networks, Prentice Hall.Google Scholar
Bhatia, R., Segall, A. and Zussman, G. (2006), ‘Analysis of bandwidth allocation algorithms for wireless personal area networks’, Wireless Networks 12, 589–603.CrossRefGoogle Scholar
Bienstock, D. (1996), ‘Computational study of a family of mixed-integer quadratic programming problems’, Math. Program. 74, 121–140.CrossRefGoogle Scholar
Bienstock, D. and Mattia, S. (2007), ‘Using mixed-integer programming to solve power grid blackout problems’, Discrete Optim. 4, 115–141.CrossRefGoogle Scholar
Bier, V. M. (2005), Game-theoretic and reliability methods in counterterrorism and security. In Mathematical and Statistical Methods in Reliability, Series on Quality, Reliability and Engineering Statistics (Wilson, A., Limnios, N., Keller-McNulty, S. and Armijo, Y., eds), World Scientific, pp. 17–28.Google Scholar
Bier, V. M., Nagaraj, A. and Abhichandani, V. (2005), ‘Protection of simple series and parallel systems with components of different values’, Reliability Engineering System Safety 87, 315–323.CrossRefGoogle Scholar
Bier, V. M., Oliveros, S. and Samuelson, L. (2007), ‘Choosing what to protect’, J. Public Economic Theory 9, 563–587.CrossRefGoogle Scholar
Bisschop, J. and Entriken, R. (1993), AIMMS: The Modeling System, Paragon Decision Technology.Google Scholar
Bock, H. and Longman, R. (1985), ‘Computation of optimal controls on disjoint control sets for minimum energy subway operation’, Adv. Astronaut. Sci. 50, 949–972.Google Scholar
Bock, H. and Plitt, K. (1984), A multiple shooting algorithm for direct solution of optimal control problems. In Proc. 9th IFAC World Congress, Pergamon Press, pp. 242–247.Google Scholar
Bonami, P. (2011), Lift-and-project cuts for mixed integer convex programs. In Integer Programming and Combinatorial Optimization (Günlük, O. and Woeginger, G., eds), Vol. 6655 of Lecture Notes in Computer Science, Springer, pp. 52–64.Google Scholar
Bonami, P. and Gonçalves, J. P. M. (2012), ‘Heuristics for convex mixed integer nonlinear programs’, Comput. Optim. Appl. 51, 729–747.CrossRefGoogle Scholar
Bonami, P., Biegler, L., Conn, A., Cornuéjols, G., Grossmann, I., Laird, C., Lee, J., Lodi, A., Margot, F., Sawaya, N. and Wächter, A. (2008), ‘An algorithmic framework for convex mixed integer nonlinear programs’, Discrete Optim. 5, 186–204.CrossRefGoogle Scholar
Bonami, P., Cornuéjols, G., Lodi, A. and Margot, F. (2009), ‘A feasibility pump for mixed integer nonlinear programs’, Math. Program. 119, 331–352.CrossRefGoogle Scholar
Bonami, P., Kılınç, M. and Linderoth, J. T. (2012), Algorithms and software for convex mixed integer nonlinear programs. In Mixed Integer Nonlinear Programming, Vol. 154 of The IMA Volumes in Mathematics and its Applications, Springer, pp. 61–92.Google Scholar
Bonami, P., Lee, J., Leyffer, S. and Wächter, A. (2011), More branch-and-bound experiments in convex nonlinear integer programming. Preprint ANL/MCS-P1949-0911, Mathematics and Computer Science Division, Argonne National Laboratory.Google Scholar
Bongartz, I., Conn, A. R., Gould, N. I. M. and Toint, P. L. (1995), ‘CUTE: Constrained and unconstrained testing environment’, ACM Trans. Math. Software 21, 123–160.CrossRefGoogle Scholar
Boorstyn, R. and Frank, H. (1977), ‘Large-scale network topological optimization’, IEEE Trans. Communications 25, 29–47.CrossRefGoogle Scholar
Borchers, B. and Mitchell, J. E. (1994), ‘An improved branch and bound algorithm for mixed integer nonlinear programs’, Comput. Oper. Res. 21, 359–368.CrossRefGoogle Scholar
Boyd, S. and Vandenberghe, L. (2004), Convex Optimization, Cambridge University Press.CrossRefGoogle Scholar
Bragalli, C., D'Ambrosio, C., Lee, J., Lodi, A. and Toth, P. (2006), An MINLP solution method for a water network problem. In Algorithms: ESA 2006, 14th Annual European Symposium, Springer, pp. 696–707.CrossRefGoogle Scholar
Bragalli, C., D'Ambrosio, C., Lee, J., Lodi, A. and Toth, P. (2012), ‘On the optimal design of water distribution networks: A practical MINLP approach’, Optim. Engng 13, 219–246.CrossRefGoogle Scholar
Brooke, A., Kendrick, D., Meeraus, A. and Raman, R. (1992), GAMS: A User's Guide, GAMS Development Corporation.Google Scholar
Bryson, A. and Ho, Y.-C. (1975), Applied Optimal Control, Wiley.Google Scholar
Buchheim, C. and Wiegele, A. (2013), ‘Semidefinite relaxations for non-convex quadratic mixed-integer programming’, Math. Program., to appear.Google Scholar
Burer, S. (2009), ‘On the copositive representation of binary and continuous non-convex quadratic programs’, Math. Program. 120, 479–495.CrossRefGoogle Scholar
Burer, S. and Letchford, A. (2009), ‘On nonconvex quadratic programming with box constraints’, SIAM J. Optim. 20, 1073–89.CrossRefGoogle Scholar
Burer, S. and Letchford, A. (2012), ‘Non-convex mixed-integer nonlinear programming: A survey’, Surv. Oper. Res. Management Sci. 17, 97–106.Google Scholar
Burer, S. and Letchford, A. (2013), ‘Unbounded convex sets for non-convex mixed-integer quadratic programming’, Math. Program., to appear.Google Scholar
Burer, S. and Vandenbussche, D. (2009), ‘Globally solving box-constrained non-convex quadratic programs with semidefinite-based finite branch-and-bound’, Comput. Optim. Appl. 43, 181–195.CrossRefGoogle Scholar
Burgschweiger, J., Gnädig, B. and Steinbach, M. (2008), ‘Optimization models for operative planning in drinking water networks’, Optim. Engng 10, 43–73.CrossRefGoogle Scholar
Bussieck, M. R. and Vigerske, S. (2010), MINLP solver software. In Wiley Encyclopedia of Operations Research and Management Science (Cochran, J. J., Cox, L. A., Keskinocak, P., Kharoufeh, J. P., Jeffrey, P. and Smith, J. C., eds), Wiley.Google Scholar
Byrd, R. H., Nocedal, J. and Richard, W. A. (2006), KNITRO: An integrated package for nonlinear optimization. In Large-Scale Nonlinear Optimization (Pillo, G. and Roma, M., eds), Vol. 83 of Nonconvex Optimization and its Applications, Springer, pp. 35–59.CrossRefGoogle Scholar
Callegari, S., Bizzarri, F., Rovatti, R. and Setti, G. (2010), ‘On the approximate solution of a class of large discrete quadratic programming problems by ΔΣ modulation: The case of circulant quadratic forms’, IEEE Trans. Signal Process. 58, 6126–6139.CrossRefGoogle Scholar
Castillo, I., Westerlund, J., Emet, S. and Westerlund, T. (2005), ‘Optimization of block layout design problems with unequal areas: A comparison of MILP and MINLP optimization methods’, Comput. Chem. Engng 30, 54–69.CrossRefGoogle Scholar
Çezik, M. and Iyengar, G. (2005), ‘Cuts for mixed 0–1 conic programming’, Math. Program. A 104, 179–202.CrossRefGoogle Scholar
Ceria, S. and Soares, J. (1999), ‘Convex programming for disjunctive optimization’, Math. Program. 86, 595–614.CrossRefGoogle Scholar
Chi, K., Jiang, X., Horiguchi, S. and Guo, M. (2008), ‘Topology design of network-coding-based multicast networks’, IEEE Trans. Mobile Comput. 7, 1–14.Google Scholar
Chung, K., Richard, J.-P. and Tawarmalani, M. (2011), Lifted inequalities for 0–1 mixed-integer bilinear covering sets. http://www.optimization-online.org/DBFILE/2011/03/2949.pdfGoogle Scholar
Chvátal, V. (1973), ‘Edmonds polytopes and a hierarchy of combinatorial problems’, Discrete Math. 4, 305–337.CrossRefGoogle Scholar
COCONUT (2004), The COCONUT benchmark: A benchmark for global optimization and constraint satisfaction. http://www.mat.univie.ac.at/~neum/glopt/coconut/benchmark.htmlGoogle Scholar
Cohen, J. S. (2003), Computer Algebra and Symbolic Computation: Elementary Algorithms, Universities Press.CrossRefGoogle Scholar
Colombani, Y. and Heipcke, S. (2002), Mosel: An extensible environment for modeling and programming solutions. In Proc. Fourth International Workshop on Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimisation Problems: CP-AI-OR'02 (Jussien, N. and Laburthe, F., eds), Ecole des Mines, Nantes, pp. 277–290.Google Scholar
Costa-Montenegro, E., González-Castaño, F. J., Rodriguez-Hernández, P. S. and Burguillo-Rial, J. C. (2007), Nonlinear optimization of IEEE 802.11 mesh networks. In ICCS 2007, Part IV, Springer, pp. 466–473.Google Scholar
Croxton, K., Gendron, B. and Magnanti, T. (2003), ‘A comparison of mixed-integer programming models for nonconvex piecewise linear cost minimization problems’, Management Sci. 49, 1268–73.CrossRefGoogle Scholar
Currie, J. and Wilson, D. I. (2012), OPTI: Lowering the barrier between open source optimizers and the industrial MATLAB user. In Foundations of Computer-Aided Process Operations (Sahinidis, N. and Pinto, J., eds). http://focapo.orgGoogle Scholar
Czyzyk, J., Mesnier, M. and Moré, J. (1998), ‘The NEOS server’, IEEE J. Comput. Sci. Engng 5, 68–75.CrossRefGoogle Scholar
Dadush, D., Dey, S. and Vielma, J. P. (2011 a), ‘The split closure of a strictly convex body’, Oper. Res. Lett. 39, 121–126.CrossRefGoogle Scholar
Dadush, D., Dey, S. S. and Vielma, J. P. (2011 b), ‘The Chvátal–Gomory closure of a strictly convex body’, Math. Oper. Res. 36, 227–239.CrossRefGoogle Scholar
Dadush, D., Dey, S. S. and Vielma, J. P. (2011 c), On the Chvátal–Gomory closure of a compact convex set. In Integer Programming and Combinatorial Optimization (Günlük, O. and Woeginger, G., eds), Vol. 6655 of Lecture Notes in Computer Science, Springer, pp. 130–142.Google Scholar
Dadush, D., Peikert, C. and Vempala, S. (2011 d), Enumerative lattice algorithms in any norm via M-ellipsoid coverings. In IEEE 52nd Annual Symposium on Foundations of Computer Science: FOCS, pp. 580–589.CrossRefGoogle Scholar
Dakin, R. J. (1965), ‘A tree search algorithm for mixed programming problems’, Comput. J. 8, 250–255.CrossRefGoogle Scholar
D'Ambrosio, C. and Lodi, A. (2011), ‘Mixed integer nonlinear programming tools: A practical overview’, 4OR 9, 329–349.CrossRefGoogle Scholar
D'Ambrosio, C., Frangioni, A., Liberti, L. and Lodi, A. (2012), ‘A storm of feasibility pumps for nonconvex MINLP’, Math. Program. B 136, 375–402.CrossRefGoogle Scholar
D'Ambrosio, C., Lodi, A. and Martello, S. (2010), ‘Piecewise linear approximation of functions of two variables in MILP models’, Oper. Res. Lett. 38, 39–46.CrossRefGoogle Scholar
Danna, E., Rothberg, E. and LePape, C. (2005), ‘Exploring relaxation induced neighborhoods to improve MIP solutions’, Math. Program. 102, 71–90.CrossRefGoogle Scholar
Dantzig, G. B. (1960), ‘On the significance of solving linear programming problems with some integer variables’, Econometrica 28, 30–44.CrossRefGoogle Scholar
Dantzig, G. B. (1963), Linear Programming and Extensions, Princeton University Press.Google Scholar
Davis, E. (1987), ‘Constraint propagation with interval labels’, Artificial Intelligence 32, 281–331.CrossRefGoogle Scholar
Davis, E. and Ierapetritou, M. (2009), ‘A kriging based method for the solution of mixed-integer nonlinear programs containing black-box functions’, J. Global Optim. 43, 191–205.CrossRefGoogle Scholar
De Loera, J. A., Hemmecke, R., Koppe, M. and Weismantel, R. (2006), ‘Integer polynomial optimization in fixed dimension’, Math. Oper. Res. 31, 147–153.CrossRefGoogle Scholar
Dey, S. S. and Morán, D. A. (2013), ‘Some properties of convex hulls of integer points contained in general convex sets’, Math. Program., to appear.Google Scholar
Dey, S. S. and Vielma, J. P. (2010), The Chvátal–Gomory closure of an ellipsoid is a polyhedron. In Integer Programming and Combinatorial Optimization, Vol. 6080 of Lecture Notes in Computer Science, Springer, pp. 327–340.CrossRefGoogle Scholar
Dolan, E. and Moré, J. (2002), ‘Benchmarking optimization software with performance profiles’, Math. Program. 91, 201–213.CrossRefGoogle Scholar
Dolan, E., Fourer, R., Moré, J. and Munson, T. (2002), ‘Optimization on the NEOS server’, SIAM News 35, 8–9.Google Scholar
Donde, V., Lopez, V., Lesieutre, B., Pinar, A., Yang, C. and Meza, J. (2005), Identification of severe multiple contingencies in electric power networks, in Proc. 37th North American Power Symposium, IEEE.Google Scholar
Dorigo, M., Maniezzo, V. and Colorni, A. (1996), ‘The ant system: Optimization by a colony of cooperating agents’, IEEE Trans. Systems, Man and Cybernetics B 26, 1–13.Google ScholarPubMed
Drewes, S. (2009), Mixed integer second order cone programming. PhD thesis, Technische Universität Darmstadt.Google Scholar
Drewes, S. and Ulbrich, S. (2012), Subgradient based outer approximation for mixed integer second order cone programming. In Mixed Integer Nonlinear Programming, Vol. 154 of The IMA Volumes in Mathematics and its Applications, Springer, pp. 41–59.Google Scholar
Duran, M. A. and Grossmann, I. (1986), ‘An outer-approximation algorithm for a class of mixed-integer nonlinear programs’, Math. Program. 36, 307–339.CrossRefGoogle Scholar
Eckstein, J. (1994), ‘Parallel branch-and-bound algorithms for general mixed integer programming on the CM-5’, SIAM J. Optim. 4, 794–814.CrossRefGoogle Scholar
Eiger, G., Shamir, U. and Ben-Tal, A. (1994), ‘Optimal design of water distribution networks’, Water Resources Research 30, 2637–2646.CrossRefGoogle Scholar
Elhedhli, S. (2006), ‘Service system design with immobile servers, stochastic demand, and congestion’, Manufacturing & Service Operations Management 8, 92–97.CrossRefGoogle Scholar
Eliceche, A. M., Corvalán, S. M. and Martínez, P. (2007), ‘Environmental life cycle impact as a tool for process optimisation of a utility plant’, Comput. Chem. Engng 31, 648–656.CrossRefGoogle Scholar
Elwalid, A., Mitra, D. and Wang, Q. (2006), ‘Distributed nonlinear integer optimization for data-optical internetworking’, IEEE J. Selected Areas in Communications 24, 1502–1513.CrossRefGoogle Scholar
Engelhart, M., Funke, J. and Sager, S. (2013), ‘A decomposition approach for a new test-scenario in complex problem solving’, J. Comput. Sci., in press.CrossRefGoogle Scholar
Exler, O. and Schittkowski, K. (2007), ‘A trust region SQP algorithm for mixed-integer nonlinear programming’, Optim. Lett. 1, 269–280.CrossRefGoogle Scholar
Exler, O., Lehmann, T. and Schittkowski, K. (2012), MISQP: A Fortran subroutine of a trust region SQP algorithm for mixed-integer nonlinear programming, user's guide. Technical report, Department of Computer Science, University of Bayreuth.Google Scholar
FICO Xpress (2009), FICO Xpress optimization suite: Xpress-BCL reference manual, Fair Isaac Corporation.Google Scholar
Fischetti, M. and Lodi, A. (2003), ‘Local branching’, Math. Program. 98, 23–47.CrossRefGoogle Scholar
Fischetti, M. and Salvagnin, D. (2009), ‘Feasibility pump 2.0’, Math. Program. Comput. 1, 201–222.CrossRefGoogle Scholar
Fischetti, M., Glover, F. and Lodi, A. (2005), ‘The feasibility pump’, Math. Program. 104, 91–104.CrossRefGoogle Scholar
Fletcher, R. (1987), Practical Methods of Optimization, Wiley.Google Scholar
Fletcher, R. and Leyffer, S. (1994), ‘Solving mixed integer nonlinear programs by outer approximation’, Math. Program. 66, 327–349.CrossRefGoogle Scholar
Fletcher, R. and Leyffer, S. (1998), User manual for filter SQP. University of Dundee Numerical Analysis Report NA-181.Google Scholar
Fletcher, R. and Leyffer, S. (2003), Filter-type algorithms for solving systems of algebraic equations and inequalities. In High Performance Algorithms and Software for Nonlinear Optimization (di Pillo, G. and Murli, A., eds), Kluwer, pp. 259–278.Google Scholar
Flores-Tlacuahuac, A. and Biegler, L. T. (2007), ‘Simultaneous mixed-integer dynamic optimization for integrated design and control’, Comput. Chem. Engng 31, 648–656.CrossRefGoogle Scholar
Floudas, C. (1995), Nonlinear and Mixed-Integer Optimization, Topics in Chemical Engineering, Oxford University Press.CrossRefGoogle Scholar
Floudas, C. A. (2000), Deterministic Global Optimization: Theory, Algorithms and Applications, Kluwer.CrossRefGoogle Scholar
Fourer, R., Gay, D. M. and Kernighan, B. W. (1993), AMPL: A Modeling Language for Mathematical Programming, The Scientific Press.Google Scholar
Fowler, K. R., Reese, J. P., Kees, C. E., Dennis, J. E., Kelley, C. T., Miller, C. T., Audet, C., Booker, A. J., Couture, G., Darwin, R. W., Farthing, M. W., Finkel, D. E., Gablonsky, J. M., Gray, G. A. and Kolda, T. G. (2008), ‘A comparison of derivative-free optimization methods for water supply and hydraulic capture community problems’, Adv. Water Resources 31, 743–757.CrossRefGoogle Scholar
Frangioni, A. and Gentile, C. (2006), ‘Perspective cuts for a class of convex 0–1 mixed integer programs’, Math. Program. 106, 225–236.CrossRefGoogle Scholar
Fügenschuh, A., Herty, M., Klar, A. and Martin, A. (2006), ‘Combinatorial and continuous models for the optimization of traffic flows on networks’, SIAM J. Optim. 16, 1155–1176.CrossRefGoogle Scholar
Fuller, A. (1963), ‘Study of an optimum nonlinear control system’, J. Electronics Control 15, 63–71.CrossRefGoogle Scholar
Garver, L. L. (1997), ‘Transmission network estimation using linear programming’, IEEE Trans. Power Apparatus Systems 89, 1688–1697.Google Scholar
Gay, D. M. (1991), Automatic differentiation of nonlinear AMPL models. In Automatic Differentiation of Algorithms: Theory, Implementation, and Application (Griewank, A. and Corliss, G. F., eds), SIAM, pp. 61–73.Google Scholar
Geissler, B., Martin, A., Morsi, A. and Schewe, L. (2012), Using piecewise linear functions for solving MINLPs. In Mixed Integer Nonlinear Programming (Lee, J. and Leyffer, S., eds), Vol. 154 of The IMA Volumes in Mathematics and its Applications, Springer, pp. 287–314.CrossRefGoogle Scholar
Gentilini, I., Margot, F. and Shimada, K. (2013), ‘The travelling salesman problem with neighbourhoods: MINLP solution’, Optim. Methods Software 28, 364–378,CrossRefGoogle Scholar
Geoffrion, A. M. (1972), ‘Generalized Benders decomposition’, J. Optim. Theory Appl. 10, 237–260.CrossRefGoogle Scholar
Geoffrion, A. M. (1977), ‘Objective function approximations in mathematical programming’, Math. Program. 13, 23–37.CrossRefGoogle Scholar
Gerdts, M. (2005), ‘Solving mixed-integer optimal control problems by branch&bound: A case study from automobile test-driving with gear shift’, Optimal Control Appl. Methods 26, 1–18.CrossRefGoogle Scholar
Gerdts, M. and Sager, S. (2012), Mixed-integer DAE optimal control problems: Necessary conditions and bounds. In Control and Optimization with Differential-Algebraic Constraints (Biegler, L., Campbell, S. and Mehrmann, V., eds), SIAM, pp. 189–212.CrossRefGoogle Scholar
Glover, F. (1989), ‘Tabu search, part I’, ORSA J. Comput. 1, 190–206.CrossRefGoogle Scholar
Glover, F. (1990), ‘Tabu search, part II’, ORSA J. Comput. 2, 4–32.CrossRefGoogle Scholar
Goldberg, D. E. (1989), Genetic Algorithms in Search, Optimization, and Machine Learning, Addison-Wesley.Google Scholar
Goldberg, N., Leyffer, S. and Safro, I. (2012), Optimal response to epidemics and cyber attacks in networks. Preprint ANL/MCS-1992-0112, Mathematics and Computer Science Division, Argonne National Laboratory.Google Scholar
Gomory, R. E. (1958), ‘Outline of an algorithm for integer solutions to linear programs’, Bull. Amer. Math. Monthly 64, 275–278.CrossRefGoogle Scholar
Gomory, R. E. (1960), An algorithm for the mixed integer problem. Technical report RM-2597, The RAND Corporation.Google Scholar
Gould, N. I. M. and Leyffer, S. (2003), An introduction to algorithms for nonlinear optimization. In Frontiers in Numerical Analysis (Blowey, J., Craig, A. and Shardlow, T., eds), Springer, pp. 109–197.CrossRefGoogle Scholar
Gould, N. I. M., Leyffer, S. and Toint, P. L. (2004), ‘A multidimensional filter algorithm for nonlinear equations and nonlinear least squares’, SIAM J. Optim. 15, 17–38.CrossRefGoogle Scholar
Goux, J.-P. and Leyffer, S. (2003), ‘Solving large MINLPs on computational grids’, Optim. Engng 3, 327–354.CrossRefGoogle Scholar
Griewank, A. (2000), Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation, Vol. 19 of Frontiers in Applied Mathematics, SIAM.Google Scholar
Griewank, A. and Toint, P. L. (1984), ‘On the existence of convex decompositions of partially separable functions’, Math. Program. 28, 25–49.CrossRefGoogle Scholar
Griva, I., Nash, S. G. and Sofer, A. (2009), Linear and Nonlinear Optimization, second edition, SIAM.CrossRefGoogle Scholar
Grossmann, I. E. (2002), ‘Review of nonlinear mixed-integer and disjunctive programming techniques’, Optim. Engng 3, 227–252.CrossRefGoogle Scholar
Grossmann, I. E. and Kravanja, Z. (1997), Mixed-integer nonlinear programming: A survey of algorithms and applications. In Large-Scale Optimization with Applications, Part II: Optimal Design and Control (Biegler, L. T., Coleman, T. F., Conn, A. R. and Santosa, F. N., eds), Springer.Google Scholar
Grossmann, I. E. and Sargent, R. W. H. (1979), ‘Optimal design of multipurpose batch plants’, Indust. Engng Chem. Process Design and Development 18, 343–348.CrossRefGoogle Scholar
Guerra, A., Newman, A. M. and Leyffer, S. (2011), ‘Concrete structure design using mixed-integer nonlinear programming with complementarity constraints’, SIAM J. Optim. 21, 833–863.CrossRefGoogle Scholar
Günlük, O. and Linderoth, J. T. (2010), ‘Perspective relaxation of mixed integer nonlinear programs with indicator variables’, Math. Program. B 104, 186–203.Google Scholar
Günlük, O. and Linderoth, J. T. (2012), Perspective reformulation and applications. In Mixed Integer Nonlinear Programming, Vol. 154 of The IMA Volumes in Mathematics and its Applications, Springer, pp. 61–92.Google Scholar
Gupta, O. K. and Ravindran, A. (1985), ‘Branch and bound experiments in convex nonlinear integer programming’, Management Sci. 31, 1533–1546.CrossRefGoogle Scholar
Gurobi, (2012), Gurobi Optimizer Reference Manual, Version 5.0, Gurobi Optimization.Google Scholar
Hansen, E. (1992), Global Optimization Using Interval Analysis, Marcel Dekker.Google Scholar
Harjunkoski, I., Westerlund, T., Pörn, R. and Skrifvars, H. (1998), ‘Different transformations for solving non-convex trim loss problems by MINLP’, European J. Oper. Res. 105, 594–603.CrossRefGoogle Scholar
Hart, W. E., Watson, J.-P. and Woodruff, D. L. (2011), ‘Pyomo: modeling and solving mathematical programs in Python’, Math. Program. Comput. 3, 219–260.CrossRefGoogle Scholar
Hedman, K. W., O'Neill, R. P., Fisher, E. B. and Oren, S. S. (2008), ‘Optimal transmission switching: Sensitivity analysis and extensions’, IEEE Trans. Power Systems 23, 1469–1479.CrossRefGoogle Scholar
Heinz, S. (2005), ‘Complexity of integer quasiconvex polynomial optimization’, J. Complexity 21, 543–556.CrossRefGoogle Scholar
Hellström, E., Ivarsson, M., Aslund, J. and Nielsen, L. (2009), ‘Look-ahead control for heavy trucks to minimize trip time and fuel consumption’, Control Engng Practice 17, 245–254.CrossRefGoogle Scholar
Hemker, T. (2008), Derivative free surrogate optimization for mixed-integer nonlinear black box problems in engineering. PhD thesis, Technischen Universität Darmstadt, Darmstadt, Germany.Google Scholar
Hemker, T., Fowler, K., Farthing, M. and von Stryk, O. (2008), ‘A mixed-integer simulation-based optimization approach with surrogate functions in water resources management’, Optim. Engng 9, 341–360.CrossRefGoogle Scholar
Hemmecke, R., Onn, S. and Weismantel, R. (2011), ‘A polynomial oracle-time algorithm for convex integer minimization’, Math. Program. 126, 97–117.CrossRefGoogle Scholar
Hijazi, H., Bonami, P. and Ouorou, A. (2010), An outer–inner approximation for separable MINLPs. Technical report, LIF, Faculté des Sciences de Luminy, Université de Marseille.Google Scholar
Hildebrand, R. and Köppe, M. (2013), ‘A new Lenstra-type algorithm for quasi-convex polynomial integer minimization with complexity 2O(n log n)’, Discrete Optim. 10, 69–84.CrossRefGoogle Scholar
Holmström, K. and Edvall, M. (2004), The TOMLAB optimization environment. In Modeling Languages in Mathematical Optimization (Kallrath, J., ed.), Kluwer Academic, pp. 369–378. http://tomopt.com/tomlab/CrossRefGoogle Scholar
Holmström, K., Göran, A. O. and Edvall, M. M. (2010), User's Guide for TOMLAB 7, Tomlab Optimization Inc.Google Scholar
Horst, H., Pardalos, P. M. and Thoai, V. (1995), Introduction to Global Optimization, Kluwer.Google Scholar
Horst, R. and Tuy, H. (1993), Global Optimization, Springer.CrossRefGoogle Scholar
IBM Ilog CPLEX (2009), IBM Ilog CPLEX V12.1: User's Manual for CPLEX, IBM.Google Scholar
Jeroslow, R. and Lowe, J. (1984), ‘Modelling with integer variables’, Math. Program. Studies 22, 167–84.CrossRefGoogle Scholar
Jeroslow, R. and Lowe, J. (1985), ‘Experimental results on the new techniques for integer programming formulations’, J. Oper. Res. Soc. 36, 393–403.CrossRefGoogle Scholar
Jeroslow, R. G. (1973), ‘There cannot be any algorithm for integer programming with quadratic constraints’, Oper. Res. 21, 221–224.CrossRefGoogle Scholar
Jobst, N. J., Horniman, M. D., Lucas, C. A. and Mitra, G. (2001), ‘Computational aspects of alternative portfolio selection models in the presence of discrete asset choice constraints’, Quant. Finance 1, 489–501.CrossRefGoogle Scholar
Júdice, J. J., Sherali, H. D., Ribeiro, I. M. and Faustino, A. M. (2006), ‘A complementarity-based partitioning and disjunctive cut algorithm for mathematical programming problems with equilibrium constraints’, J. Global Optim. 36, 89–114.CrossRefGoogle Scholar
Kannan, R. and Monma, C. (1978), On the computational complexity of integer programming problems. In Optimization and Operations Research (Henn, R., Korte, B. and Oettli, W., eds), Vol. 157 of Lecture Notes in Economics and Mathematical Systems, Springer, pp. 161–172.CrossRefGoogle Scholar
Karuppiah, R. and Grossmann, I. E. (2006), ‘Global optimization for the synthesis of integrated water systems in chemical processes’, Comput. Chem. Engng 30, 650–673.CrossRefGoogle Scholar
Keha, A. B., Farias, I. R. De Jr and Nemhauser, G. L. (2006), ‘A branch-and-cut algorithm without binary variables for nonconvex piecewise linear optimization’, Oper. Res. 54, 847–858.CrossRefGoogle Scholar
Kelley, J. E. (1960), ‘The cutting plane method for solving convex programs’, J. SIAM 8, 703712.Google Scholar
Kennedy, J. and Eberhart, R. (1995), Particle swarm optimization. In IEEE International Conference on Neural Networks, Vol. 4, pp. 1942–1948.Google Scholar
Khachiyan, L. and Porkolab, L. (2000), ‘Integer optimization on convex semialgebraic sets’, Discrete Comput. Geom. 23, 207–224.CrossRefGoogle Scholar
Kılınç, M. (2011), Disjunctive cutting planes and algorithms for convex mixed integer nonlinear programming. PhD thesis, Department of Industrial and Systems Engineering, University of Wisconsin–Madison.Google Scholar
Kılınç, M., Linderoth, J. T. and Luedtke, J. (2010), Effective separation of disjunctive cuts for convex mixed integer nonlinear programs. Technical report 1681, Computer Sciences Department, University of Wisconsin–Madison.Google Scholar
Kirches, C. (2011), Fast numerical methods for mixed-integer nonlinear model-predictive control. In Advances in Numerical Mathematics (Bock, H., Hackbusch, W., Luskin, M. and Rannacher, R., eds), Springer Vieweg. PhD thesis, Ruprecht-Karls-Universität Heidelberg.Google Scholar
Kirches, C. and Leyffer, S. (2011), TACO: A toolkit for AMPL control optimization. Preprint ANL/MCS-P1948-0911, Mathematics and Computer Science Division, Argonne National Laboratory.Google Scholar
Kirches, C., Sager, S., Bock, H. and Schlöder, J. (2010), ‘Time-optimal control of automobile test drives with gear shifts’, Optimal Control Appl. Methods 31, 137–153.CrossRefGoogle Scholar
Kirkpatrick, S., Gelatt, C. D. and Vecchi, M. P. (1983), ‘Optimization by simulated annealing’, Science 220, 671–680.CrossRefGoogle ScholarPubMed
Klepeis, J. L. and Floudas, C. A. (2003), ‘ASTRO-FOLD: A combinatorial and global optimization framework for ab initio prediction of three-dimensional structures of proteins from the amino acid sequence’, Biophysical J. 85, 2119–2146.CrossRefGoogle ScholarPubMed
KNITRO (2012), KNITRO Documentation. Ziena Optimization.Google Scholar
Kocis, G. R. and Grossmann, I. E. (1988), ‘Global optimization of nonconvex mixed-integer nonlinear programming (MINLP) problems in process synthesis’, In-dust. Engng Chem. Research 27, 1407–1421.CrossRefGoogle Scholar
Krokhmal, P. A. and Soberanis, P. (2010), ‘Risk optimization with p-order conic constraints: A linear programming approach’, European J. Oper. Res. 201, 653–671.CrossRefGoogle Scholar
Lakhera, S., Shanbhag, U. V. and McInerney, M. (2011), ‘Approximating electrical distribution networks via mixed-integer nonlinear programming’, Internat. J. Electric Power and Energy Systems 33, 245–257.CrossRefGoogle Scholar
Land, A. H. and Doig, A. G. (1960), ‘An automatic method for solving discrete programming problems’, Econometrica 28, 497–520.CrossRefGoogle Scholar
Lasserre, J. (2000), Convergent LMI relaxations for nonconvex quadratic programs. In Proc. 39th IEEE Conference on Decision and Control, Vol. 5, IEEE, pp. 5041–5046.Google Scholar
Lasserre, J. (2001), An explicit exact SDP relaxation for nonlinear 0–1 programs. In Integer Programming and Combinatorial Optimization 2001 (Aardal, K. and Gerards, A., eds), Vol. 2081 of Lecture Notes in Computer Science, Springer, pp. 293–303.CrossRefGoogle Scholar
Lawler, E. L. and Woods, D. E. (1966), ‘Branch-and-bound methods: A survey’, Oper. Res. 14, 699–719.CrossRefGoogle Scholar
Lee, J. and Leyffer, S., eds (2012), Mixed Integer Nonlinear Programming, Vol. 154 of The IMA Volumes in Mathematics and its Applications, Springer.Google Scholar
Lee, J. and Wilson, D. (2001), ‘Polyhedral methods for piecewise-linear functions I: The lambda method’, Discrete Appl. Math. 108, 269–285.CrossRefGoogle Scholar
Leineweber, D., Bauer, I., Schäfer, A., Bock, H. and Schlöder, J. (2003), ‘An efficient multiple shooting based reduced SQP strategy for large-scale dynamic process optimization (Parts I and II)’, Comput. Chem. Engng 27, 157–174.CrossRefGoogle Scholar
Lenstra, J. H. W. (1983), ‘Integer programming with a fixed number of variables’, Math. Oper. Res. 8, 538–548.CrossRefGoogle Scholar
Leyffer, S. (1998), User manual for MINLP-BB. University of Dundee.Google Scholar
Leyffer, S. (2001), ‘Integrating SQP and branch-and-bound for mixed integer nonlinear programming’, Comput. Optim. Appl. 18, 295–309.CrossRefGoogle Scholar
Leyffer, S. (2003), MacMINLP: Test problems for mixed integer nonlinear programming. http://www.mcs.anl.gov/~leyffer/macminlpGoogle Scholar
Liberti, L. and Pantelides, C. C. (2003), ‘Convex envelopes of monomials of odd degree’, J. Global Optim. 25, 157–168.CrossRefGoogle Scholar
Liberti, L., Mladenović, N. and Nannicini, G. (2011), ‘A recipe for finding good solutions to MINLPs’, Math. Program. Comput. 3, 349–390.CrossRefGoogle Scholar
Lin, Y. and Schrage, L. (2009), ‘The global solver in the LINDO API’, Optim. Methods Software 24, 657–668.CrossRefGoogle Scholar
Linderoth, J. T. (2005), ‘A simplicial branch-and-bound algorithm for solving quadratically constrained quadratic programs’, Math. Program. B 103, 251–282.CrossRefGoogle Scholar
Linderoth, J. T. and Savelsbergh, M. W. P. (1999), ‘A computational study of search strategies in mixed integer programming’, INFORMS J. Comput. 11, 173–187.CrossRefGoogle Scholar
Liuzzi, G., Lucidi, S. and Rinaldi, F. (2012), ‘Derivative-free methods for bound constrained mixed-integer optimization’, Comput. Optim. Appl. 53, 505–526.CrossRefGoogle Scholar
Löfberg, J. (2004), YALMIP: A toolbox for modeling and optimization in MATLAB. In IEEE International Symposium on Computer Aided Control Systems Design, pp. 284–289.Google Scholar
Luedtke, J., Namazifar, M. and Linderoth, J. T. (2012), ‘Some results on the strength of relaxations of multilinear functions’, Math. Program. 136, 325–351.CrossRefGoogle Scholar
Mahajan, A., Leyffer, S. and Kirches, C. (2012), Solving mixed-integer nonlinear programs by QP-diving. Preprint ANL/MCS-2071-0312, Mathematics and Computer Science Division, Argonne National Laboratory.Google Scholar
Mahajan, A., Leyffer, S., Linderoth, J. T., Luedtke, J. and Munson, T. (2011), MINO-TAUR: A toolkit for solving mixed-integer nonlinear optimization. Wiki page. http://wiki.mcs.anl.gov/minotaurGoogle Scholar
Maonan, L. and Wenjun, H. (1991), The study of choosing optimal plan of air quantities regulation of mine ventilation network. In Proc. 5th US Mine Ventilation Symposium, pp. 427–421.Google Scholar
Maria, J., Truong, T. T., Yao, J., Lee, T.-W., Nuzzo, R. G., Leyffer, S., Gray, S. K. and Rogers, J. A. (2009), ‘Optimization of 3D plasmonic crystal structures for refractive index sensing’, J. Phys. Chem. C 113, 10493–10499.Google Scholar
Markowitz, H. M. and Manne, A. S. (1957), ‘On the solution of discrete programming problems’, Econometrica 25, 84–110.CrossRefGoogle Scholar
Martin, A., Möller, M. and Moritz, S. (2006), ‘Mixed integer models for the stationary case of gas network optimization’, Math. Program. 105, 563–582.CrossRefGoogle Scholar
Masihabadi, S., Sanjeevi, S. and Kianfar, K. (2011), ‘n-step conic mixed integer rounding inequalities’, Optimization Online. http://www.optimization-online.org/DB_HTML/2011/11/3251.htmlGoogle Scholar
McCormick, G. P. (1976), ‘Computability of global solutions to factorable nonconvex programs I: Convex underestimating problems’, Math. Program. 10, 147–175.CrossRefGoogle Scholar
Messine, F. (2004), ‘Deterministic global optimization using interval constraint propagation techniques’, RAIRO-RO 38, 277–294.CrossRefGoogle Scholar
Meyer, R. (1976), ‘Mixed integer minimization models for piecewise-linear functions of a single variable’, Discrete Math. 16, 163–71.CrossRefGoogle Scholar
Miller, R., Xie, Z., Leyffer, S., Davis, M. and Gray, S. (2010), ‘Surrogate-based modeling of the optical response of metallic nanostructures’, J. Phys. Chem. C 114, 20741–20748.Google Scholar
Misener, R. and Floudas, C. (2012), ‘Global optimization of mixed-integer quadratically-constrained quadratic programs (MIQCQP) through piecewise-linear and edge-concave relaxations’, Math. Program. 136, 155–182.CrossRefGoogle Scholar
Misener, R. and Floudas, C. (2013), ‘GloMIQO: Global mixed-integer quadratic optimizer’, J. Global Optim., to appear.CrossRefGoogle Scholar
Momoh, J., Koessler, R., Bond, M., Stott, B., Sun, D., Papalexopoulos, A. and Ristanovic, P. (1997), ‘Challenges to optimal power flow’, IEEE Trans. Power Systems 12, 444–455.CrossRefGoogle Scholar
Müller, J. (2012), Surrogate model algorithms for computationally expensive blackbox global optimization problems. PhD thesis, Tampere University of Technology, Finland.Google Scholar
Müller, J., Shoemaker, C. A. and Piché, R. (2013), ‘SO-MI: A surrogate model algorithm for computationally expensive nonlinear mixed-integer black-box global optimization problems’, Comput. Oper. Res. 40, 1383–1400.CrossRefGoogle Scholar
Nannicini, G. and Belotti, P. (2012), ‘Rounding-based heuristics for nonconvex MINLPs’, Math. Program. Comput. 4, 1–31.CrossRefGoogle Scholar
Nannicini, G., Belotti, P. and Liberti, L. (2008), A local branching heuristic for MINLPs. http://arxiv.org/abs/0812.2188Google Scholar
Nemhauser, G. and Wolsey, L. A. (1988), Integer and Combinatorial Optimization, Wiley.CrossRefGoogle Scholar
Nemhauser, G. L., Savelsbergh, M. W. P. and Sigismondi, G. C. (1994), ‘MINTO: A Mixed INTeger Optimizer’, Oper. Res. Lett. 15, 47–58.CrossRefGoogle Scholar
Nocedal, J. and Wright, S. (1999), Numerical Optimization, Springer.CrossRefGoogle Scholar
Nowak, I., Alperin, H. and Vigerske, S. (2003), LaGO: An object oriented library for solving MINLPs. In Proc. 1st Global Optimization and Constraint Satisfaction Workshop: COCOS 2002 (Bliek, C., Jermann, C. and Neumaier, A., eds), Vol. 2861 of Lecture Notes in Computer Science, Springer, pp. 32–42.CrossRefGoogle Scholar
Oldenburg, J., Marquardt, W., Heinz, D. and Leineweber, D. (2003), ‘Mixed logic dynamic optimization applied to batch distillation process design’, AIChE J. 49, 2900–2917.CrossRefGoogle Scholar
Padberg, M. (1989), ‘The Boolean Quadric Polytope: Some characteristics, facets and relatives’, Math. Program. B 45, 139–172.CrossRefGoogle Scholar
Padberg, M. (2000), ‘Approximating separable nonlinear functions via mixed zero-one programs’, Oper. Res. Lett. 27, 1–5.CrossRefGoogle Scholar
Powell, R. (2007), ‘Defending against terrorist attacks with limited resources’, Amer. Political Sci. Review 101, 527–541.CrossRefGoogle Scholar
Prata, A., Oldenburg, J., Kroll, A. and Marquardt, W. (2008), ‘Integrated scheduling and dynamic optimization of grade transitions for a continuous polymerization reactor’, Comput. Chem. Engng 32, 463–476.CrossRefGoogle Scholar
Pruitt, K. A., Leyffer, S., Newman, A. M. and Braun, R. (2012), Optimal design and dispatch of distributed generation systems. Preprint ANL/MCS-2004-0112, Mathematics and Computer Science Division, Argonne National Laboratory.Google Scholar
Qualizza, A., Belotti, P. and Margot, F. (2012), Linear programming relaxations of quadratically constrained quadratic programs. In Mixed Integer Nonlinear Programming, Vol. 154 of The IMA Volumes in Mathematics and its Applications, Springer, pp. 407–426.CrossRefGoogle Scholar
Quesada, I. and Grossmann, I. E. (1992), ‘An LP/NLP based branch-and-bound algorithm for convex MINLP optimization problems’, Comput. Chem. Engng 16, 937–947.CrossRefGoogle Scholar
Quist, A. J., van Gemeert, R., Hoogenboom, J. E., Ílles, T., Roos, C. and Terlaky, T. (1998), ‘Application of nonlinear optimization to reactor core fuel reloading’, Ann. Nuclear Energy 26, 423–448.CrossRefGoogle Scholar
Rashid, K., Ambani, S. and Cetinkaya, E. (2013), ‘An adaptive multiquadric radial basis function method for expensive black-box mixed-integer nonlinear constrained optimization’, Engng Optim. 45, 185–206.CrossRefGoogle Scholar
Romero, R., Monticelli, A., Garcia, A. and Haffner, S. (2002), ‘Test systems and mathematical models for transmission network expansion planning’, IEEE Proceedings: Generation, Transmission and Distribution 149, 27–36.Google Scholar
Rote, G. (1992), ‘The convergence rate of the sandwich algorithm for approximating convex functions’, Computing 48, 337–61.CrossRefGoogle Scholar
Rubinstein, R. Y. and Kroese, D. P. (2004), The Cross-Entropy Method: A Unified Approach to Combinatorial Optimization, Monte-Carlo Simulation and Machine Learning, Springer.CrossRefGoogle Scholar
Ryoo, H. S. and Sahinidis, N. V. (1995), ‘Global optimization of nonconvex NLPs and MINLPs with applications in process design’, Comput. Chem. Engng 19, 552–566.CrossRefGoogle Scholar
Ryoo, H. S. and Sahinidis, N. V. (1996), ‘A branch-and-reduce approach to global optimization’, J. Global Optim. 8, 107–139.CrossRefGoogle Scholar
Sager, S. (2005), Numerical Methods for Mixed-Integer Optimal Control Problems, Der Andere Verlag.Google Scholar
Sager, S. (2012), A benchmark library of mixed-integer optimal control problems. In Mixed Integer Nonlinear Programming (Lee, J. and Leyffer, S., eds), Springer, pp. 631–670.CrossRefGoogle Scholar
Sager, S., Bock, H. and Diehl, M. (2012), ‘The integer approximation error in mixedinteger optimal control’, Math. Program. A 133, 1–23.CrossRefGoogle Scholar
Sager, S., Diehl, M., Singh, G., A. Küpper and Engell, S. (2007), Determining SMB superstructures by mixed-integer control. In Proc. OR2006, Springer, pp. 37–44.Google Scholar
Sager, S., Jung, M. and Kirches, C. (2011), ‘Combinatorial integral approximation’, Math. Methods Oper. Res. 73, 363–380.CrossRefGoogle Scholar
Sager, S., Reinelt, G. and Bock, H. (2009), ‘Direct methods with maximal lower bound for mixed-integer optimal control problems’, Math. Program. 118, 109–149.CrossRefGoogle Scholar
Sahinidis, N. V. (1996), ‘BARON: A general purpose global optimization software package’, J. Global Optim. 8, 201–205.CrossRefGoogle Scholar
Sandler, T. and Arce, D. G. M (2003), ‘Terrorism and game theory’, Simulation Gaming 34, 319–337.CrossRefGoogle Scholar
Sandler, T. and Siqueira, K. (2006), ‘Global terrorism: Deterrence versus preemption’, Canad. J. Economics 39, 1370–1387.CrossRefGoogle Scholar
Savelsbergh, M. W. P. (1994), ‘Preprocessing and probing techniques for mixed integer programming problems’, ORSA J. Comput. 6, 445–454.CrossRefGoogle Scholar
Saxena, A., Bonami, P. and Lee, J. (2010), ‘Convex relaxations of non-convex mixed integer quadratically constrained programs: Extended formulations’, Math. Program. 124, 383–411.CrossRefGoogle Scholar
Saxena, A., Bonami, P. and Lee, J. (2011), ‘Convex relaxations of non-convex mixed integer quadratically constrained programs: Projected formulations’, Math. Program. 130, 359–413.CrossRefGoogle Scholar
Schichl, H. (2004), Global optimization in the COCONUT project. In Numerical Software with Result Verification, Springer, pp. 243–249.CrossRefGoogle Scholar
Schrijver, A. (1986), Theory of Linear and Integer Programming, Wiley.Google Scholar
Schweiger, C. A. (1999), Process synthesis, design, and control: Optimization with dynamic models and discrete decisions. PhD thesis, Princeton University.CrossRefGoogle Scholar
Shaik, O., Sager, S., Slaby, O. and Lebiedz, D. (2008), ‘Phase tracking and restoration of circadian rhythms by model-based optimal control’, IET Systems Biology 2, 16–23.CrossRefGoogle ScholarPubMed
Sheikh, W. and Ghafoor, A. (2010), ‘An optimal bandwidth allocation and data droppage scheme for differentiated services in a wireless network’, Wireless Communications and Mobile Comput. 10, 733–747.CrossRefGoogle Scholar
Sherali, H. and Adams, W. (1998), A Reformulation–Linearization Technique for Solving Discrete and Continuous Nonconvex Problems, Kluwer.Google Scholar
Sherali, H. and Alameddine, A. (1992), ‘A new reformulation–linearization technique for bilinear programming problems’, J. Global Optim. 2, 379–410.CrossRefGoogle Scholar
Sherali, H. and Smith, E. (1997), ‘A global optimization approach to a water distribution network design problem’, J. Global Optim. 11, 107–132.CrossRefGoogle Scholar
Sherali, H. D. (2001), ‘On mixed-integer zero-one representations for separable lower-semicontinuous piecewise-linear functions’, Oper. Res. Lett. 28, 155–160.CrossRefGoogle Scholar
Sherali, H. D. and Fraticelli, B. M. P. (2002), ‘Enhancing RLT relaxations via a new class of semidefinite cuts’, J. Global Optim. 22, 233–261.CrossRefGoogle Scholar
Sherali, H. D., Subramanian, S. and Loganathan, G. V. (2001), ‘Effective relaxations and partitioning schemes for solving water distribution network design problems to global optimality’, J. Global Optim. 19, 1–26.CrossRefGoogle Scholar
Sinha, R., Yener, A. and Yates, R. D. (2002), ‘Noncoherent multiuser communications: Multistage detection and selective filtering’, EURASIP J. Appl. Signal Processing 12, 1415–1426.Google Scholar
Skrifvars, H., Leyffer, S. and Westerlund, T. (1998), ‘Comparison of certain MINLP algorithms when applied to a model structure determination and parameter estimation problem’, Comput. Chem. Engng 22, 1829–1835.CrossRefGoogle Scholar
Smith, E. M. B. and Pantelides, C. C. (1997), ‘Global optimization of nonconvex MINLPs’, Comput. Chem. Engng 21, S791–S796.CrossRefGoogle Scholar
Soleimanipour, M., Zhuang, W. and Freeman, G. H. (2002), ‘Optimal resource management in wireless multimedia wideband CDMA systems’, IEEE Trans. Mobile Computing 1, 143–160.CrossRefGoogle Scholar
Soler, M., Olivares, A., Staffetti, E. and Bonami, P. (2011), En-route optimal flight planning constrained to pass through waypoints using MINLP. In Proc. 9th USA/Europe Air Traffic Management Research and Development Seminar, Berlin.Google Scholar
Still, C. and Westerlund, T. (2006), ‘Solving convex MINLP optimization problems using a sequential cutting plane algorithm’, Comput. Optim. Appl. 34, 63–83.CrossRefGoogle Scholar
Stubbs, R. and Mehrotra, S. (1999), ‘A branch-and-cut method for 0–1 mixed convex programming’, Math. Program. 86, 515–532.CrossRefGoogle Scholar
Stubbs, R. and Mehrotra, S. (2002), ‘Generating convex polynomial inequalities for mixed 0–1 programs’, J. Global Optim. 24, 311–332.CrossRefGoogle Scholar
Tawarmalani, M. and Sahinidis, N. V. (2002), Convexification and Global Optimization in Continuous and Mixed-Integer Nonlinear Programming: Theory, Algorithms, Software, and Applications, Kluwer.CrossRefGoogle Scholar
Tawarmalani, M. and Sahinidis, N. V. (2004), ‘Global optimization of mixed integer nonlinear programs: A theoretical and computational study’, Math. Program. 99, 563–591.CrossRefGoogle Scholar
Tawarmalani, M. and Sahinidis, N. V. (2005), ‘A polyhedral branch-and-cut approach to global optimization’, Math. Program. 103, 225–249.CrossRefGoogle Scholar
Tawarmalani, M., Richard, J.-P. and Chung, K. (2010), ‘Strong valid inequalities for orthogonal disjunctions and bilinear covering sets’, Math. Program. 124, 481–512.CrossRefGoogle Scholar
Terwen, S., Back, M. and Krebs, V. (2004), Predictive powertrain control for heavy duty trucks. In Proc. IFAC Symposium in Advances in Automotive Control (Rizzo, G., Glielmo, L., Pianese, C. and Vasca, F., eds), Elsevier, pp. 451–457.Google Scholar
Tomlin, J. (1981), ‘A suggested extension of special ordered sets to non-separable non-convex programming problems’, Ann. Discrete Math. 11, 359–370.Google Scholar
Toriello, A. and Vielma, J. P. (2012), ‘Fitting piecewise linear continuous functions’, European J. Oper. Res. 219, 86–95.CrossRefGoogle Scholar
Türkay, M. and Grossmann, I. E. (1996), ‘Logic-based MINLP algorithms for the optimal synthesis of process networks’, Comput. Chem. Engng 20, 959–978.CrossRefGoogle Scholar
Van Roy, T. J. (1983), ‘Cross decomposition for mixed integer programming’, Math. Program. 25, 145–163.CrossRefGoogle Scholar
Vandenbussche, D. and Nemhauser, G. L. (2005 a), ‘A branch-and-cut algorithm for nonconvex quadratic programs with box constraints’, Math. Program. 102, 559–575.CrossRefGoogle Scholar
Vandenbussche, D. and Nemhauser, G. L. (2005 b), ‘A polyhedral study of nonconvex quadratic programs with box constraints’, Math. Program. 102, 531–557.CrossRefGoogle Scholar
Vielma, J. P., Ahmed, S. and Nemhauser, G. (2010), ‘Mixed-integer models for non-separable piecewise-linear optimization: Unifying framework and extensions’, Oper. Res. 58, 303–315.CrossRefGoogle Scholar
Vielma, J. P., Ahmed, S. and Nemhauser, G. L. (2008), ‘A lifted linear programming branch-and-bound algorithm for mixed integer conic quadratic programs’, INFORMS J. Comput. 20, 438–450.CrossRefGoogle Scholar
Vielma, J. P. and Nemhauser, G. (2011), ‘Modeling disjunctive constraints with a logarithmic number of binary variables and constraints’, Math. Program. 128, 49–72.CrossRefGoogle Scholar
Viswanathan, J. and Grossmann, I. E. (1990), ‘A combined penalty function and outer-approximation method for MINLP optimization’, Comput. Chem. Engng 14, 769–782.CrossRefGoogle Scholar
Wächter, A. and Biegler, L. T. (2006), ‘On the implementation of a primal–dual interior point filter line search algorithm for large-scale nonlinear programming’, Math. Program. 106, 25–57.CrossRefGoogle Scholar
Westerlund, T. and Lundqvist, K. (2005), Alpha-ECP, version 5.101: An interactive MINLP-solver based on the extended cutting plane method. Technical report 01-178-A, Process Design Laboratory at Åbo University.Google Scholar
Westerlund, T. and Pettersson, F. (1995), ‘A cutting plane method for solving convex MINLP problems’, Comput. Chem. Engng 19, s131–s136.CrossRefGoogle Scholar
Westerlund, T. and Pörn, R. (2002), ‘Solving pseudo-convex mixed integer optimization problems by cutting plane techniques’, Optim. Engng 3, 253–280.CrossRefGoogle Scholar
Williams, H. P. (1999), Model Building in Mathematical Programming, Wiley.Google Scholar
Wilson, D. L. (1998), Polyhedral methods for piecewise-linear functions. PhD thesis, University of Kentucky.Google Scholar
Wolf, D. D. and Smeers, Y. (2000), ‘The gas transmission problem solved by an extension of the simplex algorithm’, Management Sci. 46, 1454–1465.CrossRefGoogle Scholar
Wolsey, L. A. (1998), Integer Programming, Wiley.Google Scholar
Wu, X., Topuz, E. and Karfakis, M. (1991), Optimization of ventilation control device locations and sizes in underground mine ventilation systems. In Proc. 5th US Mine Ventilation Symposium, Society for Mining, Metallurgy, and Exploration, pp. 391–399.Google Scholar
Yajima, Y. and Fujie, T. (1998), ‘Polyhedral approach for nonconvex quadratic programming problems with box constraints’, J. Global Optim. 13, 151–170.CrossRefGoogle Scholar
You, F. and Leyffer, S. (2010), ‘Oil spill response planning with MINLP’, SIAG/OPT Views-and-News 21, 1–8.Google Scholar
You, F. and Leyffer, S. (2011), ‘Mixed-integer dynamic optimization for oil-spill response planning with integration of a dynamic oil weathering model’, AIChe J. 57, 3555–3564.CrossRefGoogle Scholar
Zhu, Y. and Kuno, T. (2006), ‘A disjunctive cutting-plane-based branch-and-cut algorithm for 0–1 mixed-integer convex nonlinear programs’, Indust. Engng Chem. Research 45, 187–196.CrossRefGoogle Scholar
Zhuang, J. (2008), Modeling secrecy and deception in homeland security resource allocation. PhD thesis, University of Wisconsin–Madison.Google Scholar
Zhuang, J. and Bier, V. M. (2007 a), ‘Balancing terrorism and natural disasters: Defensive strategy with endogenous attacker effort’, Oper. Res. 55, 976–991.CrossRefGoogle Scholar
Zhuang, J. and Bier, V. M. (2007 b), ‘Investment in security’, Industrial Engineer 39, 53–54.Google Scholar
Zhuang, J. and Bier, V. M. (2010), ‘Reasons for secrecy and deception in homeland-security resource allocation’, Risk Analysis 30, 1737–1743.CrossRefGoogle ScholarPubMed