A new benchmark optimization problem of adaptable difficulty: theoretical considerations and practical testing

  • D. K. KarpouzosEmail author
  • K. L. Katsifarakis
Original Paper


In this paper, we present a new benchmark problem for testing both local and global optimization techniques. This problem is based on ideas from groundwater hydraulics and simple Euclidian geometry and has the following attractive features: (a) known values of the infinite global optima, which can be classified in a restricted number of sets, with known location in the search space (b) simple form and (c) quick computation of objective function values. Moreover, the number of local optima sets, their location in the search space and thus the respective values of the objective function can be easily determined by the user, without affecting the global optimum value. In this way, the difficulty of finding the global optimum can be changed from quite small to almost insurmountable, as demonstrated by applying five widely used optimization methods, namely genetic algorithms, sequential quadratic programming, simulated annealing, Knitro and branch and bound. Moreover, some observations on the different behavior of optimization methods are discussed.


Optimization Benchmark problem Adaptable difficulty Genetic algorithms Sequential quadratic programming Simulated annealing Knitro Branch and bound 


Supplementary material

12351_2019_462_MOESM1_ESM.mod (1 kb)
Supplementary material 1 (mod 1 kb)


  1. Addis B, Locatelli M (2003) A new class of test functions for global optimization. J Glob Optim 38(3):479–501Google Scholar
  2. Ali MM, Khompatraporn C, Zabinsky ZB (2005) A numerical evaluation of several stochastic algorithms on selected continuous global optimization test problems. J Glob Optim. 31(4):635–672Google Scholar
  3. Alotto P, Baumgartner U, Freschi F, Jaindl M, Köstinger A, Magele Ch, Renhart W, Repetto M (2008) SMES optimization benchmark extended: introducing Pareto optimal solutions into TEAM22. IEEE Trans Magn 44(6):1066–1069Google Scholar
  4. AMPL modeling system (2018). Accessed on June 2018
  5. Aristotle, Nicomachean Ethics (Book 2) 1106–1109Google Scholar
  6. Bear J (1979) Hydraulics of groundwater. McGraw-Hill, New York CityGoogle Scholar
  7. Deb K, Gupta S, Dutta J, Ranjan B (2012) Solving dual problems using a coevolutionary optimization algorithm. J Glob Optim. 57(3):891–933Google Scholar
  8. Dixon LCW, Szegö GP (1978) The global optimization problem: an introduction. In: Dixon LCW, Szegö GP (eds) Towards global optimization 2. North-Holland, Amsterdam, pp 1–15Google Scholar
  9. Dolan ED, Moré JJ (2002) Benchmarking optimization software with performance profiles. Math Program 91:201–213Google Scholar
  10. Floudas CA, Pardalos PM (eds) (2008) Encyclopedia of optimization, 2nd edn. Kluwer Academic Publishers, DordrechtGoogle Scholar
  11. Floudas CA, Pardalos PM, Adjiman C, Esposito W, Gümüs Z, Harding S, Klepeis J, Meyer C, Schweiger C (1999) Handbook of test problems in local and global optimization. Kluwer Academic Publishers, DordrechtGoogle Scholar
  12. Fourer R, Gay DM, Kernighan BW (1990) A modeling language for mathematical programming. Manag Sci 36:519–554Google Scholar
  13. Gaviano M, Kvasov DE, Lera D, Sergeyev YD (2003) Algorithm 829: software for generation of classes of test functions with known of local and global minima for global optimization. ACM Trans Math Softw 29(4):469–480Google Scholar
  14. Goldberg DE (1989) Genetic algorithms in search, optimization and machine learning. Addison-Wesley Publishing Company, ReadingGoogle Scholar
  15. Jamil M, Yang XS (2013) A literature survey of benchmark functions for global optimisation problems. Int J Math Model Numer Optim 4(2):150–194Google Scholar
  16. Karpouzos DK, Katsifarakis KL (2013) A set of new benchmark optimization problems for water resources management. Water Resour Manag 27(9):3333–3348Google Scholar
  17. Katsifarakis KL (2012) Optimization and water resources management. In: Katsifarakis KL (ed) Hydrology, hydraulics and water resources management: a heuristic optimisation approach. WIT Press, England, pp 1–5Google Scholar
  18. Katsifarakis KL, Karpouzos DK, Theodossiou N (1999) Combined use of BEM and genetic algorithms in groundwater flow and mass transport problems. Engin Anal Bound Elem. 23(7):555–565Google Scholar
  19. Khadem M, Afshar MH (2015) A hybridized GA with LP-LP model for the management of confined groundwater. Groundwater 53(3):485–492Google Scholar
  20. Kontos YN, Katsifarakis KL (2012) Optimization of management of polluted fractured aquifers using genetic algorithms. Eur Water 40:31–42Google Scholar
  21. Koon GH, Sebald AV (1995) Some interesting functions for evaluating evolutionary programming. Strategies. In: Proc evolutionary programming IV, pp 479–499Google Scholar
  22. Kozma A, Conte C, Diehl M (2015) Benchmarking large-scale distributed convex quadratic programming algorithms. Optim Methods Softw 30(1):191–214Google Scholar
  23. Loukeris N, Donelly D, Khuman A, Peng Y (2009) A numerical evaluation of meta-heuristic techniques in portfolio optimization. Oper Res Int J 9(1):81–103Google Scholar
  24. Mahinthakumar G, Sayeed M (2005) Hybrid genetic algorithm-Local search methods for solving groundwater source identification inverse problems. J Water Resour Plan Manag 131(1):45–57Google Scholar
  25. Marti R, Reinelt G, Duarte A (2012) A benchmark library and a comparison of heuristic methods for the linear ordering problem. Comput Optim Appl. 51:1297–1317Google Scholar
  26. Mathworks Inc. (2017) Optimization toolbox: user’s guide. Accessed 11 Aug 2017
  27. Matott LS, Bartelt-Hunt SL, Rabideau AJ, Fowler KR (2006) Application of heuristic optimization techniques and algorithm tuning to multilayered sorptive barrier design. Environ Sci Technol 40:6354–6360Google Scholar
  28. Michalewicz Z (1996) Genetic algorithms + Data structures = Evolution programs, 3rd edn. Springer, BerlinGoogle Scholar
  29. Parsopoulos KE, Vrahatis MN (2004) On the computation of all global minimizers through particle swarm optimization. IEEE Trans Evol Comput 8(3):211–224Google Scholar
  30. Qu BY, Liang JJ, Wang ZY, Chen Q, Suganthan PN (2016) Novel benchmark functions for continuous multimodal optimization with comparative results. Swarm Evolut Comput 26:23–34Google Scholar
  31. Reeves CR, Raw JE (2003) Genetic algorthms-principles and perspectives. Kluwer Academic Publishers, DordrechtGoogle Scholar
  32. Sahinidis NV (1996) BARON: a general purpose global optimization software package. J Global Optim 8(2):201–205Google Scholar
  33. Schoen F (1993) A wide class of test functions for global optimization. J Glob Optim 3(2):133–137Google Scholar
  34. Shcherbina O, Neumaier A, Sam-Haroud D, Vu XH, Nguyen TV (2003) Benchmarking global optimization and constraint satisfaction codes. In: Global optimization and constraint satisfaction. Lecture notes in computer science 2861, Springer, pp 211–222Google Scholar
  35. Singh A (2012) An overview of the optimization modelling applications. J Hydrol 466–467:167–182Google Scholar
  36. Tang Y, Reed PM, Kollat JB (2007) Parallelization strategies for rapid and robust evolutionary multiobjective optimization in water resources applications. Adv Water Resour 30(3):335–353Google Scholar
  37. Wolpert DH, Macready WG (1997) No free lunch theorems for optimization. IEEE Tran Evol Comput 1(1):67–82Google Scholar
  38. Wu ZY, Walski T (2005) Self-adaptive penalty approach compared with other constraint-handling techniques for pipeline optimization. J Water Resour Plan Manag 131(3):181–192Google Scholar
  39. Yang WY, Cao W, Chung TS, Morris J (2005) Applied numerical methods using Matlab. Wiley, HobokenGoogle Scholar
  40. Younis A, Dong Z (2010) Trends, features, and tests of common and recently introduced global optimization methods. Eng Optim 42(8):691–718Google Scholar
  41. Youssef H, Sait SM, Adiche H (2001) Evolutionary algorithms, simulated annealing and tabu search: a comparative study. Eng Appl Artif Intell 14(2):167–181Google Scholar
  42. Zhang LH, Liao LZ (2012) An alternating variable method for the maximal correlation problem. J Glob Optim 54(1):199–218Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Hydraulics, Soil Science and Agricultural Engineering, School of AgricultureAristotle University of ThessalonikiThessalonikiGreece
  2. 2.Division of Hydraulics and Environmental Engineering, Department of Civil EngineeringAristotle University of ThessalonikiThessalonikiGreece

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