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Mixed-integer nonlinear optimization*

  • Pietro Belotti (a1), Christian Kirches (a2) (a3), Sven Leyffer (a3), Jeff Linderoth (a4), James Luedtke (a4) and Ashutosh Mahajan (a5)...
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.

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*

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.

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References
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Abhishek K., Leyffer S. and Linderoth J. T. (2010), ‘FilMINT: An outer-approximation-based solver for nonlinear mixed integer programs’, INFORMS J. Comput. 22, 555567.
Abichandani P., Benson H. Y. and Kam M. (2008), Multi-vehicle path coordination under communication constraints. In American Control Conference, IEEE Conference Publications, pp. 650656.
Abramson M. A. (2004), ‘Mixed variable optimization of a load-bearing thermal insulation system using a filter pattern search algorithm’, Optim. Engng 5, 157177.
Abramson M., Audet C., Chrissis J. and Walston J. (2009), ‘Mesh adaptive direct search algorithms for mixed variable optimization’, Optim. Lett. 3, 3547.
Achterberg T. (2005), SCIP: A framework to integrate constraint and mixed integer programming. ZIB-Report 04-19, Zuse Institut Berlin.
Achterberg T. and Berthold T. (2007), ‘Improving the feasibility pump’, Discrete Optim. 4, 7786.
Achterberg T., Koch T. and Martin A. (2004), ‘Branching rules revisited’, Oper. Res. Lett. 33, 4254.
Adams W. (2011), Use of Lagrange interpolating polynomials in the RLT. In Wiley Encyclopedia of Operations Research and Management Science.
Adams W. and Sherali H. (1986), ‘A tight linearization and an algorithm for zero-one quadratic programming problems’, Management Sci. 32, 12741290.
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, 2147.
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, 11591179.
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, 2147.
Al-Khayyal F. A. and Falk J. E. (1983), ‘Jointly constrained biconvex programming’, Math. Oper. Res. 8, 273286.
Altunay M., Leyffer S., Linderoth J. T. and Xie Z. (2011), ‘Optimal security response to attacks on open science grids’, Computer Networks 55, 6173.
Andersen E. D. and Andersen K. D. (1995), ‘Presolving in linear programming’, Math. Program. 71, 221245.
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, 337363.
Anstreicher K. (2012), ‘On convex relaxations for quadratically constrained quadratic programming’, Math. Program. 136, 233251.
Anstreicher K. M. (2009), ‘Semidefinite programming versus the reformulation–linearization technique for nonconvex quadratically constrained quadratic programming’, J. Global Optim. 43, 471484.
Atamtürk A. and Narayanan V. (2010), ‘Conic mixed-integer rounding cuts’, Math. Program. A 122, 120.
Audet C. and Dennis J. E. Jr (2000), ‘Pattern search algorithms for mixed variable programming’, SIAM J. Optim. 11, 573594.
Bacher R. (1997), The Optimal Power Flow (OPF) and its solution by the interior point approach. EES-UETP Madrid, short course.
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, 305322.
Balakrishnan A. and Graves S. (1989), ‘A composite algorithm for a concave-cost network flow problem’, Networks 19, 175202.
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, 21362145.
Balas E., Ceria S. and Cornuéjols G. (1993), ‘A lift-and-project cutting plane algorithm for mixed 0–1 programs’, Math. Program. 58, 295324.
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, 12291246.
Bao X., Sahinidis N. and Tawarmalani M. (2009), ‘Multiterm polyhedral relaxations for nonconvex quadratically constrained quadratic programs’, Optim. Methods Software 24, 485504.
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, 769776.
Bartholomew E. F., O'Neill R. P. and Ferris M. C. (2008), ‘Optimal transmission switching’, IEEE Trans. Power Systems 23, 13461355.
Bauschke H. H. and Borwein J. M. (1996), ‘On projection algorithms for solving convex feasibility problems’, SIAM Rev. 38, 367426.
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. 447454.
Beale E. M. L. and Forrest J. J. H. (1976), ‘Global optimization using special ordered sets’, Math. Program. 10, 5269.
Bellman R. (1961), ‘On the approximation of curves by line segments using dynamic programming’, Commun. Assoc. Comput. Mach. 4, 284.
Belotti P. (2009), Couenne: A user's manual. Technical report, Lehigh University.
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. 117144.
Belotti P. (2013), ‘Bound reduction using pairs of linear inequalities’, J. Global Optim., to appear.
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. 6576.
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.pdf
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, 597634.
Ben-Tal A. and Nemirovski A. (1995), ‘Optimal design of engineering structures’, Optima 47, 48.
Ben-Tal A. and Nemirovski A. (2001), ‘On polyhedral approximations of the second-order cone’, Math. Oper. Res. 26, 193205.
Benson H. Y. (2011), ‘Mixed integer nonlinear programming using interior point methods’, Optim. Methods Software 26, 911931.
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. 225243.
Berthold T. (2012), RENS: The optimal rounding. ZIB-Report 12-17, Zuse Institut Berlin.
Berthold T. and Gleixner A. M. (2012), Undercover: A primal MINLP heuristic exploring a largest sub-MIP. ZIB-Report 12-07, Zuse Institut Berlin.
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.
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. 181196. http://www.lix.polytechnique.fr/~liberti/ewminlp/ewminlp-proceedings.pdf
Bertsekas D. and Gallager R. (1987), Data Networks, Prentice Hall.
Bhatia R., Segall A. and Zussman G. (2006), ‘Analysis of bandwidth allocation algorithms for wireless personal area networks’, Wireless Networks 12, 589603.
Bienstock D. (1996), ‘Computational study of a family of mixed-integer quadratic programming problems’, Math. Program. 74, 121140.
Bienstock D. and Mattia S. (2007), ‘Using mixed-integer programming to solve power grid blackout problems’, Discrete Optim. 4, 115141.
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. 1728.
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, 315323.
Bier V. M., Oliveros S. and Samuelson L. (2007), ‘Choosing what to protect’, J. Public Economic Theory 9, 563587.
Bisschop J. and Entriken R. (1993), AIMMS: The Modeling System, Paragon Decision Technology.
Bock H. and Longman R. (1985), ‘Computation of optimal controls on disjoint control sets for minimum energy subway operation’, Adv. Astronaut. Sci. 50, 949972.
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. 242247.
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. 5264.
Bonami P. and Gonçalves J. P. M. (2012), ‘Heuristics for convex mixed integer nonlinear programs’, Comput. Optim. Appl. 51, 729747.
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, 186204.
Bonami P., Cornuéjols G., Lodi A. and Margot F. (2009), ‘A feasibility pump for mixed integer nonlinear programs’, Math. Program. 119, 331352.
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. 6192.
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.
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, 123160.
Boorstyn R. and Frank H. (1977), ‘Large-scale network topological optimization’, IEEE Trans. Communications 25, 2947.
Borchers B. and Mitchell J. E. (1994), ‘An improved branch and bound algorithm for mixed integer nonlinear programs’, Comput. Oper. Res. 21, 359368.
Boyd S. and Vandenberghe L. (2004), Convex Optimization, Cambridge University Press.
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. 696707.
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, 219246.
Brooke A., Kendrick D., Meeraus A. and Raman R. (1992), GAMS: A User's Guide, GAMS Development Corporation.
Bryson A. and Ho Y.-C. (1975), Applied Optimal Control, Wiley.
Buchheim C. and Wiegele A. (2013), ‘Semidefinite relaxations for non-convex quadratic mixed-integer programming’, Math. Program., to appear.
Burer S. (2009), ‘On the copositive representation of binary and continuous non-convex quadratic programs’, Math. Program. 120, 479495.
Burer S. and Letchford A. (2009), ‘On nonconvex quadratic programming with box constraints’, SIAM J. Optim. 20, 1073–89.
Burer S. and Letchford A. (2012), ‘Non-convex mixed-integer nonlinear programming: A survey’, Surv. Oper. Res. Management Sci. 17, 97106.
Burer S. and Letchford A. (2013), ‘Unbounded convex sets for non-convex mixed-integer quadratic programming’, Math. Program., to appear.
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, 181195.
Burgschweiger J., Gnädig B. and Steinbach M. (2008), ‘Optimization models for operative planning in drinking water networks’, Optim. Engng 10, 4373.
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.
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. 3559.
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, 61266139.
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, 5469.
Çezik M. and Iyengar G. (2005), ‘Cuts for mixed 0–1 conic programming’, Math. Program. A 104, 179202.
Ceria S. and Soares J. (1999), ‘Convex programming for disjunctive optimization’, Math. Program. 86, 595614.
Chi K., Jiang X., Horiguchi S. and Guo M. (2008), ‘Topology design of network-coding-based multicast networks’, IEEE Trans. Mobile Comput. 7, 114.
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.pdf
Chvátal V. (1973), ‘Edmonds polytopes and a hierarchy of combinatorial problems’, Discrete Math. 4, 305337.
COCONUT (2004), The COCONUT benchmark: A benchmark for global optimization and constraint satisfaction. http://www.mat.univie.ac.at/~neum/glopt/coconut/benchmark.html
Cohen J. S. (2003), Computer Algebra and Symbolic Computation: Elementary Algorithms, Universities Press.
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. 277290.
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. 466473.
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.
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.org
Czyzyk J., Mesnier M. and Moré J. (1998), ‘The NEOS server’, IEEE J. Comput. Sci. Engng 5, 6875.
Dadush D., Dey S. and Vielma J. P. (2011 a), ‘The split closure of a strictly convex body’, Oper. Res. Lett. 39, 121126.
Dadush D., Dey S. S. and Vielma J. P. (2011 b), ‘The Chvátal–Gomory closure of a strictly convex body’, Math. Oper. Res. 36, 227239.
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. 130142.
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. 580589.
Dakin R. J. (1965), ‘A tree search algorithm for mixed programming problems’, Comput. J. 8, 250255.
D'Ambrosio C. and Lodi A. (2011), ‘Mixed integer nonlinear programming tools: A practical overview’, 4OR 9, 329349.
D'Ambrosio C., Frangioni A., Liberti L. and Lodi A. (2012), ‘A storm of feasibility pumps for nonconvex MINLP’, Math. Program. B 136, 375402.
D'Ambrosio C., Lodi A. and Martello S. (2010), ‘Piecewise linear approximation of functions of two variables in MILP models’, Oper. Res. Lett. 38, 3946.
Danna E., Rothberg E. and LePape C. (2005), ‘Exploring relaxation induced neighborhoods to improve MIP solutions’, Math. Program. 102, 7190.
Dantzig G. B. (1960), ‘On the significance of solving linear programming problems with some integer variables’, Econometrica 28, 3044.
Dantzig G. B. (1963), Linear Programming and Extensions, Princeton University Press.
Davis E. (1987), ‘Constraint propagation with interval labels’, Artificial Intelligence 32, 281331.
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, 191205.
De Loera J. A., Hemmecke R., Koppe M. and Weismantel R. (2006), ‘Integer polynomial optimization in fixed dimension’, Math. Oper. Res. 31, 147153.
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.
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. 327340.
Dolan E. and Moré J. (2002), ‘Benchmarking optimization software with performance profiles’, Math. Program. 91, 201213.
Dolan E., Fourer R., Moré J. and Munson T. (2002), ‘Optimization on the NEOS server’, SIAM News 35, 89.
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.
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, 113.
Drewes S. (2009), Mixed integer second order cone programming. PhD thesis, Technische Universität Darmstadt.
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. 4159.
Duran M. A. and Grossmann I. (1986), ‘An outer-approximation algorithm for a class of mixed-integer nonlinear programs’, Math. Program. 36, 307339.
Eckstein J. (1994), ‘Parallel branch-and-bound algorithms for general mixed integer programming on the CM-5’, SIAM J. Optim. 4, 794814.
Eiger G., Shamir U. and Ben-Tal A. (1994), ‘Optimal design of water distribution networks’, Water Resources Research 30, 26372646.
Elhedhli S. (2006), ‘Service system design with immobile servers, stochastic demand, and congestion’, Manufacturing & Service Operations Management 8, 9297.
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, 648656.
Elwalid A., Mitra D. and Wang Q. (2006), ‘Distributed nonlinear integer optimization for data-optical internetworking’, IEEE J. Selected Areas in Communications 24, 15021513.
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.
Exler O. and Schittkowski K. (2007), ‘A trust region SQP algorithm for mixed-integer nonlinear programming’, Optim. Lett. 1, 269280.
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.
FICO Xpress (2009), FICO Xpress optimization suite: Xpress-BCL reference manual, Fair Isaac Corporation.
Fischetti M. and Lodi A. (2003), ‘Local branching’, Math. Program. 98, 2347.
Fischetti M. and Salvagnin D. (2009), ‘Feasibility pump 2.0’, Math. Program. Comput. 1, 201222.
Fischetti M., Glover F. and Lodi A. (2005), ‘The feasibility pump’, Math. Program. 104, 91104.
Fletcher R. (1987), Practical Methods of Optimization, Wiley.
Fletcher R. and Leyffer S. (1994), ‘Solving mixed integer nonlinear programs by outer approximation’, Math. Program. 66, 327349.
Fletcher R. and Leyffer S. (1998), User manual for filter SQP. University of Dundee Numerical Analysis Report NA-181.
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. 259278.
Flores-Tlacuahuac A. and Biegler L. T. (2007), ‘Simultaneous mixed-integer dynamic optimization for integrated design and control’, Comput. Chem. Engng 31, 648656.
Floudas C. (1995), Nonlinear and Mixed-Integer Optimization, Topics in Chemical Engineering, Oxford University Press.
Floudas C. A. (2000), Deterministic Global Optimization: Theory, Algorithms and Applications, Kluwer.
Fourer R., Gay D. M. and Kernighan B. W. (1993), AMPL: A Modeling Language for Mathematical Programming, The Scientific Press.
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, 743757.
Frangioni A. and Gentile C. (2006), ‘Perspective cuts for a class of convex 0–1 mixed integer programs’, Math. Program. 106, 225236.
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, 11551176.
Fuller A. (1963), ‘Study of an optimum nonlinear control system’, J. Electronics Control 15, 6371.
Garver L. L. (1997), ‘Transmission network estimation using linear programming’, IEEE Trans. Power Apparatus Systems 89, 16881697.
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. 6173.
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. 287314.
Gentilini I., Margot F. and Shimada K. (2013), ‘The travelling salesman problem with neighbourhoods: MINLP solution’, Optim. Methods Software 28, 364378,
Geoffrion A. M. (1972), ‘Generalized Benders decomposition’, J. Optim. Theory Appl. 10, 237260.
Geoffrion A. M. (1977), ‘Objective function approximations in mathematical programming’, Math. Program. 13, 2337.
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, 118.
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. 189212.
Glover F. (1989), ‘Tabu search, part I’, ORSA J. Comput. 1, 190206.
Glover F. (1990), ‘Tabu search, part II’, ORSA J. Comput. 2, 432.
Goldberg D. E. (1989), Genetic Algorithms in Search, Optimization, and Machine Learning, Addison-Wesley.
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.
Gomory R. E. (1958), ‘Outline of an algorithm for integer solutions to linear programs’, Bull. Amer. Math. Monthly 64, 275278.
Gomory R. E. (1960), An algorithm for the mixed integer problem. Technical report RM-2597, The RAND Corporation.
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. 109197.
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, 1738.
Goux J.-P. and Leyffer S. (2003), ‘Solving large MINLPs on computational grids’, Optim. Engng 3, 327354.
Griewank A. (2000), Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation, Vol. 19 of Frontiers in Applied Mathematics, SIAM.
Griewank A. and Toint P. L. (1984), ‘On the existence of convex decompositions of partially separable functions’, Math. Program. 28, 2549.
Griva I., Nash S. G. and Sofer A. (2009), Linear and Nonlinear Optimization, second edition, SIAM.
Grossmann I. E. (2002), ‘Review of nonlinear mixed-integer and disjunctive programming techniques’, Optim. Engng 3, 227252.
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.
Grossmann I. E. and Sargent R. W. H. (1979), ‘Optimal design of multipurpose batch plants’, Indust. Engng Chem. Process Design and Development 18, 343348.
Guerra A., Newman A. M. and Leyffer S. (2011), ‘Concrete structure design using mixed-integer nonlinear programming with complementarity constraints’, SIAM J. Optim. 21, 833863.
Günlük O. and Linderoth J. T. (2010), ‘Perspective relaxation of mixed integer nonlinear programs with indicator variables’, Math. Program. B 104, 186203.
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. 6192.
Gupta O. K. and Ravindran A. (1985), ‘Branch and bound experiments in convex nonlinear integer programming’, Management Sci. 31, 15331546.
Gurobi (2012), Gurobi Optimizer Reference Manual, Version 5.0, Gurobi Optimization.
Hansen E. (1992), Global Optimization Using Interval Analysis, Marcel Dekker.
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, 594603.
Hart W. E., Watson J.-P. and Woodruff D. L. (2011), ‘Pyomo: modeling and solving mathematical programs in Python’, Math. Program. Comput. 3, 219260.
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, 14691479.
Heinz S. (2005), ‘Complexity of integer quasiconvex polynomial optimization’, J. Complexity 21, 543556.
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, 245254.
Hemker T. (2008), Derivative free surrogate optimization for mixed-integer nonlinear black box problems in engineering. PhD thesis, Technischen Universität Darmstadt, Darmstadt, Germany.
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, 341360.
Hemmecke R., Onn S. and Weismantel R. (2011), ‘A polynomial oracle-time algorithm for convex integer minimization’, Math. Program. 126, 97117.
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.
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, 6984.
Holmström K. and Edvall M. (2004), The TOMLAB optimization environment. In Modeling Languages in Mathematical Optimization (Kallrath J., ed.), Kluwer Academic, pp. 369378. http://tomopt.com/tomlab/
Holmström K., Göran A. O. and Edvall M. M. (2010), User's Guide for TOMLAB 7, Tomlab Optimization Inc.
Horst H., Pardalos P. M. and Thoai V. (1995), Introduction to Global Optimization, Kluwer.
Horst R. and Tuy H. (1993), Global Optimization, Springer.
IBM Ilog CPLEX (2009), IBM Ilog CPLEX V12.1: User's Manual for CPLEX, IBM.
Jeroslow R. and Lowe J. (1984), ‘Modelling with integer variables’, Math. Program. Studies 22, 167–84.
Jeroslow R. and Lowe J. (1985), ‘Experimental results on the new techniques for integer programming formulations’, J. Oper. Res. Soc. 36, 393403.
Jeroslow R. G. (1973), ‘There cannot be any algorithm for integer programming with quadratic constraints’, Oper. Res. 21, 221224.
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, 489501.
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, 89114.
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. 161172.
Karuppiah R. and Grossmann I. E. (2006), ‘Global optimization for the synthesis of integrated water systems in chemical processes’, Comput. Chem. Engng 30, 650673.
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, 847858.
Kelley J. E. (1960), ‘The cutting plane method for solving convex programs’, J. SIAM 8, 703712.
Kennedy J. and Eberhart R. (1995), Particle swarm optimization. In IEEE International Conference on Neural Networks, Vol. 4, pp. 19421948.
Khachiyan L. and Porkolab L. (2000), ‘Integer optimization on convex semialgebraic sets’, Discrete Comput. Geom. 23, 207224.
Kılınç M. (2011), Disjunctive cutting planes and algorithms for convex mixed integer nonlinear programming. PhD thesis, Department of Industrial and Systems Engineering, University of WisconsinMadison.
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.
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.
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.
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, 137153.
Kirkpatrick S., Gelatt C. D. and Vecchi M. P. (1983), ‘Optimization by simulated annealing’, Science 220, 671680.
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, 21192146.
KNITRO (2012), KNITRO Documentation. Ziena Optimization.
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, 14071421.
Krokhmal P. A. and Soberanis P. (2010), ‘Risk optimization with p-order conic constraints: A linear programming approach’, European J. Oper. Res. 201, 653671.
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, 245257.
Land A. H. and Doig A. G. (1960), ‘An automatic method for solving discrete programming problems’, Econometrica 28, 497520.
Lasserre J. (2000), Convergent LMI relaxations for nonconvex quadratic programs. In Proc. 39th IEEE Conference on Decision and Control, Vol. 5, IEEE, pp. 50415046.
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. 293303.
Lawler E. L. and Woods D. E. (1966), ‘Branch-and-bound methods: A survey’, Oper. Res. 14, 699719.
Lee J. and Leyffer S., eds (2012), Mixed Integer Nonlinear Programming, Vol. 154 of The IMA Volumes in Mathematics and its Applications, Springer.
Lee J. and Wilson D. (2001), ‘Polyhedral methods for piecewise-linear functions I: The lambda method’, Discrete Appl. Math. 108, 269285.
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, 157174.
Lenstra J. H. W. (1983), ‘Integer programming with a fixed number of variables’, Math. Oper. Res. 8, 538548.
Leyffer S. (1998), User manual for MINLP-BB. University of Dundee.
Leyffer S. (2001), ‘Integrating SQP and branch-and-bound for mixed integer nonlinear programming’, Comput. Optim. Appl. 18, 295309.
Leyffer S. (2003), MacMINLP: Test problems for mixed integer nonlinear programming. http://www.mcs.anl.gov/~leyffer/macminlp
Liberti L. and Pantelides C. C. (2003), ‘Convex envelopes of monomials of odd degree’, J. Global Optim. 25, 157168.
Liberti L., Mladenović N. and Nannicini G. (2011), ‘A recipe for finding good solutions to MINLPs’, Math. Program. Comput. 3, 349390.
Lin Y. and Schrage L. (2009), ‘The global solver in the LINDO API’, Optim. Methods Software 24, 657668.
Linderoth J. T. (2005), ‘A simplicial branch-and-bound algorithm for solving quadratically constrained quadratic programs’, Math. Program. B 103, 251282.
Linderoth J. T. and Savelsbergh M. W. P. (1999), ‘A computational study of search strategies in mixed integer programming’, INFORMS J. Comput. 11, 173187.
Liuzzi G., Lucidi S. and Rinaldi F. (2012), ‘Derivative-free methods for bound constrained mixed-integer optimization’, Comput. Optim. Appl. 53, 505526.
Löfberg J. (2004), YALMIP: A toolbox for modeling and optimization in MATLAB. In IEEE International Symposium on Computer Aided Control Systems Design, pp. 284289.
Luedtke J., Namazifar M. and Linderoth J. T. (2012), ‘Some results on the strength of relaxations of multilinear functions’, Math. Program. 136, 325351.
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.
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/minotaur
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.
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, 1049310499.
Markowitz H. M. and Manne A. S. (1957), ‘On the solution of discrete programming problems’, Econometrica 25, 84110.
Martin A., Möller M. and Moritz S. (2006), ‘Mixed integer models for the stationary case of gas network optimization’, Math. Program. 105, 563582.
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.html
McCormick G. P. (1976), ‘Computability of global solutions to factorable nonconvex programs I: Convex underestimating problems’, Math. Program. 10, 147175.
Messine F. (2004), ‘Deterministic global optimization using interval constraint propagation techniques’, RAIRO-RO 38, 277294.
Meyer R. (1976), ‘Mixed integer minimization models for piecewise-linear functions of a single variable’, Discrete Math. 16, 163–71.
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, 2074120748.
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, 155182.
Misener R. and Floudas C. (2013), ‘GloMIQO: Global mixed-integer quadratic optimizer’, J. Global Optim., to appear.
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, 444455.
Müller J. (2012), Surrogate model algorithms for computationally expensive blackbox global optimization problems. PhD thesis, Tampere University of Technology, Finland.
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, 13831400.
Nannicini G. and Belotti P. (2012), ‘Rounding-based heuristics for nonconvex MINLPs’, Math. Program. Comput. 4, 131.
Nannicini G., Belotti P. and Liberti L. (2008), A local branching heuristic for MINLPs. http://arxiv.org/abs/0812.2188
Nemhauser G. and Wolsey L. A. (1988), Integer and Combinatorial Optimization, Wiley.
Nemhauser G. L., Savelsbergh M. W. P. and Sigismondi G. C. (1994), ‘MINTO: A Mixed INTeger Optimizer’, Oper. Res. Lett. 15, 4758.
Nocedal J. and Wright S. (1999), Numerical Optimization, Springer.
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. 3242.
Oldenburg J., Marquardt W., Heinz D. and Leineweber D. (2003), ‘Mixed logic dynamic optimization applied to batch distillation process design’, AIChE J. 49, 29002917.
Padberg M. (1989), ‘The Boolean Quadric Polytope: Some characteristics, facets and relatives’, Math. Program. B 45, 139172.
Padberg M. (2000), ‘Approximating separable nonlinear functions via mixed zero-one programs’, Oper. Res. Lett. 27, 15.
Powell R. (2007), ‘Defending against terrorist attacks with limited resources’, Amer. Political Sci. Review 101, 527541.
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, 463476.
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.
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. 407426.
Quesada I. and Grossmann I. E. (1992), ‘An LP/NLP based branch-and-bound algorithm for convex MINLP optimization problems’, Comput. Chem. Engng 16, 937947.
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, 423448.
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, 185206.
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, 2736.
Rote G. (1992), ‘The convergence rate of the sandwich algorithm for approximating convex functions’, Computing 48, 337–61.
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.
Ryoo H. S. and Sahinidis N. V. (1995), ‘Global optimization of nonconvex NLPs and MINLPs with applications in process design’, Comput. Chem. Engng 19, 552566.
Ryoo H. S. and Sahinidis N. V. (1996), ‘A branch-and-reduce approach to global optimization’, J. Global Optim. 8, 107139.
Sager S. (2005), Numerical Methods for Mixed-Integer Optimal Control Problems, Der Andere Verlag.
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. 631670.
Sager S., Bock H. and Diehl M. (2012), ‘The integer approximation error in mixedinteger optimal control’, Math. Program. A 133, 123.
Sager S., Diehl M., Singh G., A. Küpper and Engell S. (2007), Determining SMB superstructures by mixed-integer control. In Proc. OR2006, Springer, pp. 3744.
Sager S., Jung M. and Kirches C. (2011), ‘Combinatorial integral approximation’, Math. Methods Oper. Res. 73, 363380.
Sager S., Reinelt G. and Bock H. (2009), ‘Direct methods with maximal lower bound for mixed-integer optimal control problems’, Math. Program. 118, 109149.
Sahinidis N. V. (1996), ‘BARON: A general purpose global optimization software package’, J. Global Optim. 8, 201205.
Sandler T. and Arce D. G. M (2003), ‘Terrorism and game theory’, Simulation Gaming 34, 319337.
Sandler T. and Siqueira K. (2006), ‘Global terrorism: Deterrence versus preemption’, Canad. J. Economics 39, 13701387.
Savelsbergh M. W. P. (1994), ‘Preprocessing and probing techniques for mixed integer programming problems’, ORSA J. Comput. 6, 445454.
Saxena A., Bonami P. and Lee J. (2010), ‘Convex relaxations of non-convex mixed integer quadratically constrained programs: Extended formulations’, Math. Program. 124, 383411.
Saxena A., Bonami P. and Lee J. (2011), ‘Convex relaxations of non-convex mixed integer quadratically constrained programs: Projected formulations’, Math. Program. 130, 359413.
Schichl H. (2004), Global optimization in the COCONUT project. In Numerical Software with Result Verification, Springer, pp. 243249.
Schrijver A. (1986), Theory of Linear and Integer Programming, Wiley.
Schweiger C. A. (1999), Process synthesis, design, and control: Optimization with dynamic models and discrete decisions. PhD thesis, Princeton University.
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, 1623.
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, 733747.
Sherali H. and Adams W. (1998), A Reformulation–Linearization Technique for Solving Discrete and Continuous Nonconvex Problems, Kluwer.
Sherali H. and Alameddine A. (1992), ‘A new reformulation–linearization technique for bilinear programming problems’, J. Global Optim. 2, 379410.
Sherali H. and Smith E. (1997), ‘A global optimization approach to a water distribution network design problem’, J. Global Optim. 11, 107132.
Sherali H. D. (2001), ‘On mixed-integer zero-one representations for separable lower-semicontinuous piecewise-linear functions’, Oper. Res. Lett. 28, 155160.
Sherali H. D. and Fraticelli B. M. P. (2002), ‘Enhancing RLT relaxations via a new class of semidefinite cuts’, J. Global Optim. 22, 233261.
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, 126.
Sinha R., Yener A. and Yates R. D. (2002), ‘Noncoherent multiuser communications: Multistage detection and selective filtering’, EURASIP J. Appl. Signal Processing 12, 14151426.
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, 18291835.
Smith E. M. B. and Pantelides C. C. (1997), ‘Global optimization of nonconvex MINLPs’, Comput. Chem. Engng 21, S791S796.
Soleimanipour M., Zhuang W. and Freeman G. H. (2002), ‘Optimal resource management in wireless multimedia wideband CDMA systems’, IEEE Trans. Mobile Computing 1, 143160.
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.
Still C. and Westerlund T. (2006), ‘Solving convex MINLP optimization problems using a sequential cutting plane algorithm’, Comput. Optim. Appl. 34, 6383.
Stubbs R. and Mehrotra S. (1999), ‘A branch-and-cut method for 0–1 mixed convex programming’, Math. Program. 86, 515532.
Stubbs R. and Mehrotra S. (2002), ‘Generating convex polynomial inequalities for mixed 0–1 programs’, J. Global Optim. 24, 311332.
Tawarmalani M. and Sahinidis N. V. (2002), Convexification and Global Optimization in Continuous and Mixed-Integer Nonlinear Programming: Theory, Algorithms, Software, and Applications, Kluwer.
Tawarmalani M. and Sahinidis N. V. (2004), ‘Global optimization of mixed integer nonlinear programs: A theoretical and computational study’, Math. Program. 99, 563591.
Tawarmalani M. and Sahinidis N. V. (2005), ‘A polyhedral branch-and-cut approach to global optimization’, Math. Program. 103, 225249.
Tawarmalani M., Richard J.-P. and Chung K. (2010), ‘Strong valid inequalities for orthogonal disjunctions and bilinear covering sets’, Math. Program. 124, 481512.
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. 451457.
Tomlin J. (1981), ‘A suggested extension of special ordered sets to non-separable non-convex programming problems’, Ann. Discrete Math. 11, 359370.
Toriello A. and Vielma J. P. (2012), ‘Fitting piecewise linear continuous functions’, European J. Oper. Res. 219, 8695.
Türkay M. and Grossmann I. E. (1996), ‘Logic-based MINLP algorithms for the optimal synthesis of process networks’, Comput. Chem. Engng 20, 959978.
Van Roy T. J. (1983), ‘Cross decomposition for mixed integer programming’, Math. Program. 25, 145163.
Vandenbussche D. and Nemhauser G. L. (2005 a), ‘A branch-and-cut algorithm for nonconvex quadratic programs with box constraints’, Math. Program. 102, 559575.
Vandenbussche D. and Nemhauser G. L. (2005 b), ‘A polyhedral study of nonconvex quadratic programs with box constraints’, Math. Program. 102, 531557.
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, 303315.
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, 438450.
Vielma J. P. and Nemhauser G. (2011), ‘Modeling disjunctive constraints with a logarithmic number of binary variables and constraints’, Math. Program. 128, 4972.
Viswanathan J. and Grossmann I. E. (1990), ‘A combined penalty function and outer-approximation method for MINLP optimization’, Comput. Chem. Engng 14, 769782.
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, 2557.
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.
Westerlund T. and Pettersson F. (1995), ‘A cutting plane method for solving convex MINLP problems’, Comput. Chem. Engng 19, s131s136.
Westerlund T. and Pörn R. (2002), ‘Solving pseudo-convex mixed integer optimization problems by cutting plane techniques’, Optim. Engng 3, 253280.
Williams H. P. (1999), Model Building in Mathematical Programming, Wiley.
Wilson D. L. (1998), Polyhedral methods for piecewise-linear functions. PhD thesis, University of Kentucky.
Wolf D. D. and Smeers Y. (2000), ‘The gas transmission problem solved by an extension of the simplex algorithm’, Management Sci. 46, 14541465.
Wolsey L. A. (1998), Integer Programming, Wiley.
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. 391399.
Yajima Y. and Fujie T. (1998), ‘Polyhedral approach for nonconvex quadratic programming problems with box constraints’, J. Global Optim. 13, 151170.
You F. and Leyffer S. (2010), ‘Oil spill response planning with MINLP’, SIAG/OPT Views-and-News 21, 18.
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, 35553564.
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, 187196.
Zhuang J. (2008), Modeling secrecy and deception in homeland security resource allocation. PhD thesis, University of Wisconsin–Madison.
Zhuang J. and Bier V. M. (2007 a), ‘Balancing terrorism and natural disasters: Defensive strategy with endogenous attacker effort’, Oper. Res. 55, 976991.
Zhuang J. and Bier V. M. (2007 b), ‘Investment in security’, Industrial Engineer 39, 5354.
Zhuang J. and Bier V. M. (2010), ‘Reasons for secrecy and deception in homeland-security resource allocation’, Risk Analysis 30, 17371743.
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