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Benchmarking nonlinear optimization software in technical computing environments

Global optimization in Mathematica with MathOptimizer Professional

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Abstract

Our strategic objective is to develop a broadly categorized, expandable collection of test problems, to support the benchmarking of nonlinear optimization software packages in integrated technical computing environments (ITCEs). ITCEs—such as Maple, Mathematica, and MATLAB—support concise, modular and scalable model development: their built-in documentation and visualization features can be put to good use also in test model selection and analysis. ITCEs support the flexible inclusion of both new models and general-purpose solver engines for future studies. Within this broad context, in this article we review a collection of global optimization problems coded in Mathematica, and present illustrative and summarized numerical results obtained using the MathOptimizer Professional software package.

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References

  • Addis B, Locatelli M (2007) A new class of test functions for global optimization. J Glob Optim 38:479–501

    Article  Google Scholar 

  • Addis B, Locatelli M, Schoen F (2008) Efficiently packing unequal disks in a circle. Oper Res Lett 36(1):37–42

    Article  Google Scholar 

  • Ali MM, Khompatraporn Ch, Zabinsky ZB (2005) A numerical evaluation of several global optimization algorithms on selected test problems. J Glob Optim 31:635–672

    Article  Google Scholar 

  • AMPL LLC (2011) AMPL; www.ampl.com

  • Averick BM, Moré JJ (1991) The Minpack-2 test problem collection. Technical report ANL/MCS-TM-157, Argonne National Laboratory, Argonne, IL

  • Barr RS, Golden BL, Kelly JP, Resende MGC, Stewart WR (1995) Designing and reporting on computational experiments with heuristic methods. J Heuristics 1:9–32

    Article  Google Scholar 

  • Berna TJ, Locke MH, Westerberg AW (1980) A new approach to optimization of chemical processes. AIChE J 26(1):37–43

    Article  Google Scholar 

  • Bhatti AM (2000) Practical optimization methods with Mathematica applications. Springer, New York

    Book  Google Scholar 

  • Bongartz I, Conn AR, Gould NIM, Toint PhL (1995) CUTE: constrained and unconstrained testing environment. ACM Trans Math Softw 21(1):123–160

    Article  Google Scholar 

  • Buckley AG (1992) Algorithm 709: testing algorithm implementations. ACM Trans Math Softw 18(4):375–391

    Article  Google Scholar 

  • Casado LC, García I, Martínez JA, Sergeyev YD (2003) New interval analysis support functions using gradient information in a global minimization algorithm. J Glob Optim 25:345–362

    Article  Google Scholar 

  • Castillo I, Kampas FJ, Pintér JD (2008) Solving circle packing problems by global optimization: numerical results and industrial applications. Eur J Oper Res 191:786–802

    Article  Google Scholar 

  • Csendes T, Pál L, Sendin OH, Banga JR (2008) The GLOBAL optimization method revisited. Optim Lett 2:445–454

    Article  Google Scholar 

  • Dimmer PR, Cutteridge OPD (1980) Second derivative Gauss–Newton-based method for solving nonlinear simultaneous equations. IEE Proc G, Electron Circuits Syst 127:278–283

    Article  Google Scholar 

  • Dixon L, Szegö G (eds) (1975) Towards global optimisation, vol 1. North-Holland, Amsterdam

    Google Scholar 

  • Dixon L, Szegö G (eds) (1978) Towards global optimisation, vol 2. North-Holland, Amsterdam

    Google Scholar 

  • Dolan ED, Moré JJ (2002) Benchmarking optimization software with performance profiles. Math Program, Ser A 91:201–213

    Article  Google Scholar 

  • Dolan ED, Moré JJ, Munson TS (2004) Benchmarking optimization software with COPS 3.0. Technical report ANL/MCS-TM-273, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL

  • Doye JPK, Leary RH, Locatelli M, Schoen F (2004) The global optimization of Morse clusters by potential energy transformations. INFORMS J Comput 16:371–379

    Article  Google Scholar 

  • Ebers JJ, Moll JL (1954) Large signal behaviour of junction transistors. Proc IRE 42:1761–1772

    Article  Google Scholar 

  • Fasano G (2008) MIP-based heuristic for non-standard 3D-packing problems. 4OR 6:291–310

    Article  Google Scholar 

  • Fasano G, Saia D, Piras A (2010) Columbus stowage optimization by CAST (Cargo Accommodation Support Tool). Acta Astronaut 67(3):489–496

    Article  Google Scholar 

  • Floudas CA, Pardalos PM, Adjiman CS, Esposito WR, Gümüş ZH, Harding ST, Klepeis JL, Meyer CA, Schweiger CA (1999) Handbook of test problems in local and global optimization. Kluwer Academic, Dordrecht

    Book  Google Scholar 

  • GAMS Development Corporation (2011) GAMS; www.gams.com

  • GAMS Global World (2011) GLOBAL Library: a collection of nonlinear programming models; www.gamsworld.org/global/globallib.htm

  • GAMS Performance World (2011) www.gamsworld.org/performance/index.htm

  • Gaviano M, Kvasov DE, Lera D, Sergeyev YD (2003) Algorithm 829: software for generation of classes of test functions with known local and global minima for global optimization. ACM Trans Math Softw 29(4):469–480

    Article  Google Scholar 

  • Gould NIM, Orban D, Toint PL (2003) CUTEr and SifDec: a constrained and unconstrained testing environment, revisited. ACM Trans Math Softw 29(4):373–394

    Article  Google Scholar 

  • Grossmann IE (ed) (1996) Global optimization in engineering design. Kluwer Academic, Dordrecht

    Google Scholar 

  • Hansen P, Jaumard B, Lu S-H (1992) Global optimization of univariate Lipschitz functions: II. New algorithms and computational comparison. Math Program 55:273–292

    Article  Google Scholar 

  • Hare WL, Wang Y (2010) Fairer benchmarking of optimization algorithms via derivative free optimization. Available from Optimization Online, www.optimization-online.org/DB_HTML/2010/10/2765.html

  • Hock W, Schittkowski K (1981) Test examples for nonlinear programming codes. Lecture notes in economics and mathematical systems, vol 187. Springer, Berlin

    Book  Google Scholar 

  • Hock W, Schittkowski K (1983) A comparative performance evaluation of 27 nonlinear programming codes. Computing 30:335–358

    Article  Google Scholar 

  • Jackson RHF, Boggs PT, Nash SG, Powell S (1991) Guidelines for reporting results of computation experiments. Report of the ad hoc committee. Math Program 49:413–425

    Article  Google Scholar 

  • Kampas FJ, Pintér JD (2006) Configuration analysis and design by using optimization tools in Mathematica. Math J 10:128–154

    Google Scholar 

  • Khompatraporn Ch, Pintér JD, Zabinsky ZB (2005) Comparative assessment of algorithms and software for global optimization. J Glob Optim 31:613–633

    Article  Google Scholar 

  • Lahey Computer Systems (2004) Lahey/Fujitsu FORTRAN 95 (Release 5.70f), www.lahey.com

  • Lavor C, Liberti L, Maculan N (2006) Computational experience with the molecular distance geometry problem. In: Pintér JD (ed) Global optimization: selected case studies. Springer, New York

    Google Scholar 

  • LINDO Systems (2011) LINGO; www.lindo.com

  • Liu D, Zhang X-S (2000) Test problem generator by neural network for algorithms that try solving nonlinear programming problems globally. J Glob Optim 16:229–243

    Article  Google Scholar 

  • Luksan L, Vlçek J (2000) Test problems for non-smooth unconstrained and linearly constrained optimization. Technical report 798, Institute of Computer Science, Academy of Sciences of the Czech Republic, Prague

  • Maplesoft (2011a) Maple. Maplesoft, Inc, Waterloo, ON; www.maplesoft.com

  • Maplesoft (2011b) Maple Global Optimization Toolbox. Maplesoft, Inc, Waterloo, ON; www.maplesoft.com/products/toolboxes/globaloptimization/

  • Maximal Software (2011) MPL. Maximal Software, Inc, Arlington, VA; www.maximal-usa.com

  • Michalewicz Z (1996) Genetic algorithms + data structures = evolution programs. Springer, Heidelberg

    Book  Google Scholar 

  • Mittelmann HD (2011) Decision tree for optimization software; plato.la.asu.edu/guide.html

  • Moré JJ, Garbow BS, Hillström KE (1981) Testing unconstrained optimization software. ACM Trans Math Softw 7:17–41

    Article  Google Scholar 

  • Moré JJ, Wild SM (2009) Benchmarking derivative-free optimization algorithms. SIAM J Optim 20(1):172–191

    Article  Google Scholar 

  • Müller A, Schneider JJ, Schömer E (2009) Packing a multidisperse system of hard disks in a circular environment. Phys Rev E 79:021102 (14 pages)

    Article  Google Scholar 

  • Neumaier A (2011) Global optimization; www.mat.univie.ac.at/~neum/glopt.html

  • Neumaier A, Shcherbina O, Huyer W, Vinkó T (2005) A comparison of complete global optimization solvers. Math Program, Ser B 103:335–356

    Article  Google Scholar 

  • Papalambros PM, Wilde DJ (2000) Principles of optimal design—modeling and computation, 2nd edn. Cambridge University Press, Cambridge

    Book  Google Scholar 

  • Paragon Decision Technology (2011) AIMMS. Paragon Decision Technology BV, Haarlem, The Netherlands; www.aimms.com

  • Pál L, Csendes T (2009) INTLAB implementation of an interval global optimization algorithm. Optim Methods Softw 24(4):749–759

    Article  Google Scholar 

  • Pintér JD (1996a) Global optimization in action. Kluwer Academic, Dordrecht

    Book  Google Scholar 

  • Pintér JD (1996b) Continuous global optimization software: a brief review. Optima 52:1–8. See also at http://plato.la.asu.edu/gom.html

    Google Scholar 

  • Pintér JD (1997) LGO: A program system for continuous and Lipschitz optimization. In: Bomze IM, Csendes T, Horst R, Pardalos PM (eds) Developments in global optimization. Kluwer Academic, Dordrecht, pp 183–197

    Chapter  Google Scholar 

  • Pintér JD (2001) Globally optimized spherical point arrangements: model variants and illustrative results. Ann Oper Res 104:213–230

    Article  Google Scholar 

  • Pintér JD (2002) Global optimization: software, test problems, and applications. In: Pardalos PM, Romeijn HE (eds) Handbook of global optimization, vol 2. Kluwer Academic, Dordrecht, pp 515–569

    Google Scholar 

  • Pintér JD (2005) Nonlinear optimization in modeling environments: software implementations for compilers, spreadsheets, modeling languages, and integrated computing systems. In: Jeyakumar V, Rubinov AM (eds) Continuous optimization: current trends and applications. Springer, New York, pp 147–173

    Chapter  Google Scholar 

  • Pintér JD (2006a) Global optimization with Maple: an introduction with illustrative examples. An electronic book published by Pintér Consulting Services, Canada

  • Pintér JD (ed) (2006b) Global optimization: selected case studies. Springer, New York

    Google Scholar 

  • Pintér JD (2007) Nonlinear optimization with GAMS/LGO. J Glob Optim 38:79–101

    Article  Google Scholar 

  • Pintér JD (2009a) Global optimization in practice: state-of-the-art and perspectives. In: Gao DY, Sherali HD (eds) Advances in applied mathematics and global optimization. Springer, New York, pp 377–404

    Chapter  Google Scholar 

  • Pintér JD (2009b) Software development for global optimization. In: Pardalos PM, Coleman TF (eds) Global optimization: methods and applications. Fields institute communications, vol 55. American Mathematical Society, Providence, pp 183–204

    Google Scholar 

  • Pintér JD, Bagirov A, Zhang J (2003) An illustrated collection of global optimization test problems. Research report, School of Information Technology & Mathematical Sciences, University of Ballarat, Victoria, Australia

  • Pintér JD, Kampas FJ (2005) Nonlinear optimization in Mathematica with MathOptimizer Professional. Math Educ Res 10(2):1–18

    Google Scholar 

  • Pintér JD, Kampas FJ (2006) MathOptimizer Professional: key features and illustrative applications. In: Liberti L, Maculan N (eds) Global optimization: from theory to implementation. Springer, New York, pp 263–279

    Chapter  Google Scholar 

  • Pintér JD, Kampas FJ (2010a) MathOptimizer—an advanced modeling and optimization system for Mathematica users. User’s guide. (MathOptimizer is developed and maintained since 2002 by Pintér Consulting Services, Inc, Canada; www.pinterconsulting.com)

  • Pintér JD, Kampas FJ (2010b) MathOptimizer Professional—an advanced modeling and optimization system for Mathematica, using the LGO Solver Engine. User’s guide. (MathOptimizer Professional is developed and maintained since 2003 by Pintér Consulting Services, Inc, Canada; www.pinterconsulting.com)

  • Pintér JD, Linder D, Chin P (2006) Global optimization toolbox for Maple: an introduction with illustrative applications. Optim Methods Softw 21(4):565–582

    Article  Google Scholar 

  • Rakhmanov EA, Saff EB, Zhou YM (1995) Electrons on the sphere. In: Ali RM, Ruscheweyh S, Saff EB (eds) Computational methods and function theory. World Scientific, Singapore, pp 111–127

    Google Scholar 

  • Ratschek H, Rokne J (1993) Experiments using interval analysis for solving a circuit design problem. J Glob Optim 3:501–518

    Article  Google Scholar 

  • Rios LM, Sahinidis NV (2010) Derivative-free optimization: a review of algorithms and comparison of software implementations. Technical report, Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, PA (to appear)

  • Shcherbina O, Neumaier A, Sam-Haroud D, Vu X-H, Tuan-Viet N (2003) Benchmarking global optimization and constraint satisfaction codes. In: Bliek Ch et al. (eds) Global optimization and constraint satisfaction. Springer, Berlin, pp 211–222

    Chapter  Google Scholar 

  • Schichl H (2003) Global optimization in the COCONUT project. Manuscript, www.mat.univie.ac.at/~herman/papers.html

  • Schittkowski K (1987) More test examples for nonlinear programming. Lecture notes in economics and mathematical systems, vol 182. Springer, Berlin

    Book  Google Scholar 

  • Schittkowski K (2008) An updated set of 306 test problems for nonlinear programming with validated optimal solutions—user’s guide. Research report, Department of Computer Science, University of Bayreuth

  • Schoen F (1993) A wide class of test functions for global optimization. J Glob Optim 3:133–137

    Article  Google Scholar 

  • Schoen F (2008) Personal communication

  • Stortelder WJH, de Swart JJB, Pintér JD (2001) Finding elliptic Fekete points sets: two numerical solution approaches. J Comput Appl Math 130:205–216

    Article  Google Scholar 

  • Tang K, Li X, Suganthan PN, Yang Z, Weise T (2010) Benchmark functions for the CEC’2010 special session and competition on large-scale global optimization. Online research document, January 8, 2010. For possible updates, visit www.it-weise.de/

  • Tawarmalani M, Sahinidis NV (2002) Convexification and global optimization in continuous and mixed-integer nonlinear programming. Kluwer Academic, Dordrecht

    Google Scholar 

  • The MathWorks (2011a) MATLAB. The MathWorks, Inc, Natick, MA; www.mathworks.com

  • The MathWorks (2011b) MATLAB global optimization toolbox 3 user’s guide. The MathWorks, Inc, Natick, MA; www.mathworks.com

  • TOMLAB Optimization (2011) TOMLAB. TOMLAB Optimization AB, Västerås, Sweden; www.tomopt.com

  • Trott M (2004, 2006) The Mathematica GuideBooks. Programming, graphics, numerics, symbolics, vols 1–4. Springer, New York

    Google Scholar 

  • Vanderbei RJ (2011) Nonlinear optimization models; www.princeton.edu/~rvdb/ampl/nlmodels/

  • Weise T (2011) Global optimization algorithms—theory and application; www.it-weise.de/

  • Wolfram Research (2011) Mathematica. Wolfram Research, Inc, Champaign, IL; www.wolfram.com

  • Wolfram S (2003) The Mathematica book, 5th edn. Wolfram Media, Inc, Champaign. Online reference http://reference.wolfram.com/mathematica/guide/Mathematica.html

    Google Scholar 

  • Zabinsky ZB (2003) Stochastic adaptive search for global optimization. Kluwer Academic, Dordrecht

    Google Scholar 

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Authors and Affiliations

  1. PCS Inc., Halifax, NS, Canada

    János D. Pintér

  2. Department of Industrial Engineering, Özyeğin University, Istanbul, Turkey

    János D. Pintér

  3. Physicist at Large Consulting, Ambler, PA, USA

    Frank J. Kampas

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  1. János D. Pintér
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  2. Frank J. Kampas
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Correspondence to János D. Pintér.

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Pintér, J.D., Kampas, F.J. Benchmarking nonlinear optimization software in technical computing environments. TOP 21, 133–162 (2013). https://doi.org/10.1007/s11750-011-0209-5

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