Skip to main content
SpringerLink
Log in

A chaos-based constrained optimization algorithm

  • Original Research
  • Published:
Journal of Ambient Intelligence and Humanized Computing Aims and scope Submit manuscript

Abstract

This paper presents a novel chaotic augmented Lagrange method for solving constrained optimization problems. The algorithm employs chaotic maps to reduce the search space and to get the best parameters for handling the problem constraints. Then, the first carrier wave method can be applied to obtain a solution as an initial point of simplex method to find optimal solution. To verify the efficiency of the proposed algorithm, an empirical study is conducted in three groups: mathematical, challenging, and structural optimization problems. The experimental results show that the proposed method can solve different kinds of constrained optimization problems with great precision.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

  • Abdechiri M, Meybodi MR, Bahrami H (2013) Gases brownian motion optimization: an algorithm for optimization (GBMO). Appl Soft Comput 13(5):2932–2946

    Article  Google Scholar 

  • Abdechiri M, Faez K, Amindavar H, Bilotta E (2017) The chaotic dynamics of high-dimensional systems. Nonlinear Dyn 87(4):2597–2610

    Article  MathSciNet  Google Scholar 

  • Aihara K, Takabe T, Toyoda M (1990) Chaotic neural networks. Phys Lett A 144(6):333–340

    Article  MathSciNet  Google Scholar 

  • Alatas B (2010) Chaotic harmony search algorithms. Appl Math Comput 216(9):2687–2699

    MATH  Google Scholar 

  • Alatas B (2011) Uniform Big Bang–chaotic Big Crunch optimization. Commun Nonlinear Sci Numer Simul 16(9):3696–3703

    Article  MATH  Google Scholar 

  • Alatas B, Akin E (2009) Chaotically encoded particle swarm optimization algorithm and its applications. Chaos Soliton & Fractals 41(2):939–950

    Article  Google Scholar 

  • Alikhani JK, Hosseini SMM (2015) A new hybrid algorithm based on chaotic maps for solving systems of nonlinear equations. Chaos Solitons Fractals 81:233–245

    Article  MathSciNet  MATH  Google Scholar 

  • Alikhani JK, Hosseini SMM, Ghaini FM (2016) A new optimization algorithm based on chaotic maps and golden section search method. Eng Appl Artif Intell 50:201–214

    Article  Google Scholar 

  • Amirjanov A (2006) The development of a changing range genetic algorithm. Comput Methods Appl Mech Eng 195:2495–2508

    Article  MathSciNet  MATH  Google Scholar 

  • Arora JS (1989) Introduction to optimum design. McGraw-Hill, New York

    Google Scholar 

  • Arora JS, Elwakeil OA, Chahande AI, Hsieh CC (1995) Global optimization methods for engineering applications: a review. Struct Optim 9(3–4):137–159

    Article  Google Scholar 

  • Ashrafi SM, Dariane AB (2013) Performance evaluation of an improved harmony search algorithm for numerical optimization: Melody Search (MS). Eng Appl Artif Intell 26(4):1301–1321

    Article  Google Scholar 

  • Aslimani N, Talbi E-G, Ellaia R (2020) Tornado: an autonomous chaotic algorithm for large scale global optimization, HAL Id: hal-02499326, https://hal.inria.fr/hal-02499326v2

  • Belegundu AD (1982) A study of mathematical programming methods for structural optimization. PhD thesis, Department of civil and Environmental Engineering, University of Iowa.

  • Bernardino HS, Barbosa HJC, Lemonge ACC (2007) A hybrid genetic algorithm for constrained optimization problems in mechanical engineering. In: 2007 IEEE Congress on evolutionary computation, pp 646–53

  • Bing L, Weisun J (1997) Chaos optimization method and its application. Control Theory Appl 14(4):613–615

    Google Scholar 

  • Bucolo M, Caponetto R, Fortuna L, Frasca M, Rizzo A (2002) Does chaos work better than noise? Circuits Syst Mag 2(3):4–19

    Article  Google Scholar 

  • Chen TY, Chen HC (2009) Mixed–discrete structural optimization using a rank-niche evolution strategy. Eng Optim 41(1):39–58

    Article  Google Scholar 

  • Chen H, Xu Y, Wang M, Zhao X (2019) A balanced whale optimization algorithm for constrained engineering design problems. Appl Math Model 71:45–59

    Article  MathSciNet  MATH  Google Scholar 

  • Coello CAC (2000) Use of a self-adaptive penalty approach for engineering optimization problems. Comput Ind 41(2):113–127

    Article  Google Scholar 

  • Coello CAC, Montes EM (2002) Constraint-handling in genetic algorithms through the use of dominance-based tournament selection. Adv Eng Inform 16(3):193–203

    Article  Google Scholar 

  • Cuevas E, Cienfuegos M (2014) A new algorithm inspired in the behavior of the social-spider for constrained optimization. Expert Syst Appl 41(2):412–425

    Article  Google Scholar 

  • Deb K (1995) Optimization for engineering design: algorithms and examples. Prentice-Hall, New Delhi

    Google Scholar 

  • Deb K (2000) Anefficient constraint handling method for genetic algorithms. Comput Methods Appl Mech Engrg 186:311–338

    Article  MATH  Google Scholar 

  • Demir FB, Tuncer T, Kocamaz AF (2020) A chaotic optimization method based on logistic-sine map for numerical function optimization. Neural Comput Appl. https://doi.org/10.1007/s00521-020-04815-9

    Article  Google Scholar 

  • Devaney RL (1987) An introduction to chaotic dynamical systems. Addison-Wesley, Boston

    Google Scholar 

  • Erbatur F, Hasançebi O, Tütüncü I, Kılıç H (2000) Optimal design of planar and space structures with genetic algorithms. Comput Struct 75(2):209–224

    Article  Google Scholar 

  • Fesanghary M, Mahdavi M, Minary-Jolandan M, Alizadeh Y (2008) Hybridizing harmony search algorithm with sequential quadratic programming for engineering optimization problems. Comput Methods Appl Mech Eng 197(33):3080–3091

    Article  MATH  Google Scholar 

  • Firouznia M, Faez K, Amindavar H, Alikhani Koupaei J (2018) Chaotic particle filter for visual object tracking. J Vis Commun Image Rep 53:1–12

    Article  Google Scholar 

  • Floundas C, Pardalos P (1987) A Collection of test problems for constrained Global optimization. Lecture notes in computer Science. Springer-Verlag, Berlin

    Google Scholar 

  • Gandomi AH, Yang XS, Alavi AH (2011) Mixed variable structural optimization using firefly algorithm. Comput Struct 89(23):2325–2336

    Article  Google Scholar 

  • Gandomi AH, Yang XS, Talatahari S, Alavi AH (2013) Firefly algorithm with chaos. Commun Nonlinear Sci Numer Simul 18(1):89–98

    Article  MathSciNet  MATH  Google Scholar 

  • Garg H (2016) A hybrid PSO-GA algorithm for constrained optimization problems. Appl Math Comput 274:292–305

    MathSciNet  MATH  Google Scholar 

  • Garg H (2019) A hybrid GSA-GA algorithm for constrained optimization problems. Inf Sci 478:499–523

    Article  Google Scholar 

  • Gong W, Cai Z, Liang D (2014) Engineering optimization by means of an improved constrained differential evolution. Comput Methods Appl Mech Eng 268:884–904

    Article  MathSciNet  MATH  Google Scholar 

  • Gu L, Yang RJ, Tho CH, Makowskit M, Faruquet OY, Li YL (2001) Optimisation and robustness for crashworthiness of side impact. Int J Veh Des 26(4):348–360

    Article  Google Scholar 

  • Hamaizia T, Lozi R, Hamri NE (2012) Fast chaotic optimization algorithm based on locally averaged strategy and multifold chaotic attractor. Appl Math Comput 219(1):188–196

    MathSciNet  MATH  Google Scholar 

  • Hasancebi O, Azad SK (2012) An efficient metaheuristic algorithm for engineering optimization: SOPT. Int J Optim Civil Eng 2(4):479–487

    Google Scholar 

  • He D, He C, Jiang LG, Zhu HW, Hu GR (2001) Chaotic characteristics of a one-dimensional iterative map with infinite collapses. Circuits Syst 48(7):900–906

    MathSciNet  MATH  Google Scholar 

  • He YY, Zhou JZ, Xiang XQ, Chen H, Qin H (2009) Comparison of different chaotic maps in particle swarm optimization algorithm for long-term cascaded hydroelectric system scheduling. Chaos Solitons Fractals 42(5):3169–3176

    Article  MATH  Google Scholar 

  • Hilborn RC (2004) Chaos and nonlinear dynamics: an introduction for scientists and engineers. Oxford University Press, Oxford

    MATH  Google Scholar 

  • Himmelblau DM (1972) Applied nonlinear programming. McGraw-Hill, New York

    MATH  Google Scholar 

  • Huang FZ, Wang L, He Q (2007) An effective co-evolutionary differential evolution for constrained optimization. Appl Math Comput 186(1):340–356

    MathSciNet  MATH  Google Scholar 

  • Jia G, Wang Y, Cai Z, Jin Y (2013) An improved (μ+ λ)-constrained differential evolution for constrained optimization. Inf Sci 222:302–322

    Article  MathSciNet  MATH  Google Scholar 

  • Jiang BLW (1998) Optimizing complex functions by chaos search. Cybern Syst 29(4):409–419

    Article  MATH  Google Scholar 

  • Kapitaniak T (1995) Continuous control and synchronization in chaotic systems. Chaos Solitons Fractals 6:237–244

    Article  MATH  Google Scholar 

  • Karaboga D, (2005) An idea based on honey bee swarm for numerical optimization, Technical report-tr06, Erciyes University, Engineering Faculty, Computer Engineering Department.

  • Kashan AH (2015) An effective algorithm for constrained optimization based on optics inspired optimization (OIO). Comput Aided Des 63:52–71

    Article  Google Scholar 

  • Kaveh A, Sheikholeslami R, Talatahari S, Keshvari-Ilkhichi M (2014) Chaotic swarming particles: a new method for size optimization of truss structures. Adv Eng Softw 67:136–147

    Article  Google Scholar 

  • Kennedy J, Eberhart R (1995) Particle swarm optimization. IEEE Int Conf Neural Netw 4:1942–1948

    Google Scholar 

  • Kiefer J (1953) Sequential minimax search for a maximum. Proc Am Math Soc 4(3):502–506

    Article  MathSciNet  MATH  Google Scholar 

  • Kirkpatrick S, Vecchi MP (1983) Optimization by simmulated annealing. Science 220(4598):671–680

    Article  MathSciNet  MATH  Google Scholar 

  • Kohda T, Tsuneda A, Sakae T, (1992) Chaotic binary sequences by Chebyshev maps and their correlation properties. In: Spread Spectrum Techniques and Applications, ISSTA 92, IEEE Second International Symposium, pp 63–66

  • Koziel S, Michalewicz Z (1999) Evolutionary algorithms, homomorphous mappings, and constrained parameter optimization. Evol Comput 7(1):19–44

    Article  Google Scholar 

  • Lagarias JC, Reeds JA, Wright MH, Wright PE (1998) Convergence properties of the nelder-mead simplex method in low dimensions. SIAM J Optim 9(1):112–147

    Article  MathSciNet  MATH  Google Scholar 

  • Lee KS, Geem ZW (2005) A new meta-heuristic algorithm for continuous engineering optimization: harmony search theory and practice. Comput Methods Appl Mech Eng 194(36):3902–3933

    Article  MATH  Google Scholar 

  • Lemonge AC, Barbosa HJ (2004) An adaptive penalty scheme for genetic algorithms in structural optimization. Int J Numer Meth Eng 59(5):703–736

    Article  MATH  Google Scholar 

  • Liang JJ, Runarsson TP, Mezura-Montes E, Clerc M, Suganthan PN, Coello CAC, Deb K (2006) Problem definitions and evaluation criteria for the CEC 2006 special session on constrained real parameter optimization. Nanyang Technological University, Singapore

    Google Scholar 

  • Liberti L, Maculan N (2006) Global optimization: from theory to implementation. Springer Science and Business Media, Berlin

    Book  MATH  Google Scholar 

  • Lorenz EN (1963) Deterministic non-periodic flow. J Atmos Sci 20(2):130–141

    Article  MATH  Google Scholar 

  • Lu Z, Shieh LS, Chen G (2003) On robust control of uncertain chaotic systems: a sliding-mode synthesis via chaotic optimization. Chaos Solitons Fractals 18(4):819–827

    Article  MathSciNet  MATH  Google Scholar 

  • Lu HJ, Zhang HM, Ma LH (2006) A new optimization algorithm based on chaos. J Zhejiang Univ Sci A 7(4):539–542

    Article  MATH  Google Scholar 

  • Mahdavi M, Fesanghary M, Damangir E (2007) An improved harmony search algorithm for solving optimization problems. Appl Math Comput 188(2):1567–1579

    MathSciNet  MATH  Google Scholar 

  • Mallipeddi R, Suganthan P, Suganthan N (2010) Ensemble of constraint handling techniques. IEEE Trans Evol Comput 14(4):561–579

    Article  MATH  Google Scholar 

  • May RM (1976) Simple mathematical models with very complicated dynamics. Nature 261(5560):459–467

    Article  MATH  Google Scholar 

  • Mezura-Montes E, Coello CA, Velázquez-Reyes J, Muñoz-Dávila L (2007) Multiple trial vectors in differential evolution for engineering design. Eng Optim 39(5):567–589

    Article  MathSciNet  Google Scholar 

  • Misaghi M, Yaghoobi M (2019) Improved invasive weed optimization algorithm (IWO) based on chaos theory for optimal design of PID controller. J Comput Des Eng 6(3):284–295

    Google Scholar 

  • Niu B, Wang J, Wang H (2015) Bacterial-inspired algorithms for solving constrained optimization problems. Neurocomputing 148:54–62

    Article  Google Scholar 

  • Nocedal J, Wright SJ (2006) Numerical optimization. Springer Science and Business Media, New York

    MATH  Google Scholar 

  • O’Connor KM (2005) Calculus: labs for matlab. Jones and Bartlett publishers, Burlington, pp 63–64

    Google Scholar 

  • Ott E, Grebogi C, Yorke JA (1990) Controlling chaos. Phys Rev Lett 64(11):1196–1199

    Article  MathSciNet  MATH  Google Scholar 

  • Pecora LM, Carroll TL (1990) Synchronization in chaotic systems. Phys Rev Lett 64(8):821–824

    Article  MathSciNet  MATH  Google Scholar 

  • Qiao W, Yang Z (2019) Modified dolphin swarm algorithm based on chaotic maps for solving high-dimensional function optimization problems. IEEE Access 7:110472–110486

    Article  Google Scholar 

  • Raouf AO, Baset AM, El-henawy I (2014) A new hybrid flower pollination algorithm for solving constrained global optimization problems. Int J Appl 4(2):1–13

    Google Scholar 

  • Reklaitis GV, Ravindran A, Ragsdell KM (1983) Engineering optimization: methods and applications. Wiley, New York, pp 43–47

  • Sandgren E (1990) Nonlinear integer and discrete programming in mechanical design optimization. J Mech Des 112(2):223–229

    Article  Google Scholar 

  • Sayed GI, Darwish A, Hassanien AE (2018) A new chaotic multi-verse optimization algorithm for solving engineering optimization problems. J Exp Theor Artif Intell 30(2):293–317

    Article  Google Scholar 

  • Sedlaczek K, Eberhard P (2006) Using augmented Lagrangian particle swarm optimization for constrained problems in engineering. Struct Multidiscip Optim 32(4):277–286

    Article  Google Scholar 

  • Seif Z, Ahmadi MB (2015) An opposition-based algorithm for function optimization. Eng Appl Artif Intell 37:293–306

    Article  Google Scholar 

  • Sun X, Wang F, Wen S (2014) An improved evolutionary strategy of genetic algorithm and a new method on generation of initial population when using genetic algorithms for solving constrained optimization problems. Int J Hybrid Inform Technol 7(4):331–344

    Article  Google Scholar 

  • Tavazoei MS, Haeri M (2007) Comparison of different one-dimensional maps as chaotic search pattern in chaos optimization algorithms. Appl Math Comput 187(2):1076–1085

    MathSciNet  MATH  Google Scholar 

  • Tessema B, Yen G, (2006) A selfadaptive penalty function based algorithm for constrained optimization. In: Proceedings of the 2006 IEEE Congress on Evolutionary Computation, IEEE Press, Vancouver, BC, Canada, pp 246–253

  • Thanedar PB, Vanderplaats GN (1995) Survey of discrete variable optimization for structural design. J Struct Eng 121(2):301–306

    Article  Google Scholar 

  • Wang GG, Deb S, Gandomi AH, Zhang Z, Alavi AH (2016) Chaotic cuckoo search. Soft Comput 20(9):3349–3362

    Article  Google Scholar 

  • Wang L, Liu X, Sun M, Qu J, Wei Y (2018) A new chaotic starling particle swarm optimization algorithm for clustering problems. Math Probl Eng. https://doi.org/10.1155/2018/8250480

    Article  Google Scholar 

  • Wei W, Wang J, Tao M (2015) Constrained differential evolution with multi-objective sorting mutation operators for constrained optimization. Appl Soft Comput 33:207–222

    Article  Google Scholar 

  • Wu SJ, Chow PT (1995) Genetic algorithms for nonlinear mixed discrete-integer optimization problems via meta-genetic parameter optimization. Eng Optim 24(2):137–159

    Article  Google Scholar 

  • Yang XS (2010) Engineering optimization: an introduction with metaheuristic applications. John Wiley and Sons, New Jersey

    Book  Google Scholar 

  • Yang D, Li G, Cheng G (2007) On the efficiency of chaos optimization algorithms for global optimization. Chaos Solitons Fractals 34(4):1366–1375

    Article  MathSciNet  Google Scholar 

  • Yang D, Liu Z, Zhou J (2014) Chaos optimization algorithms based on chaotic maps with different probability distribution and search speed for global optimization. Commun Nonlinear Sci Numer Simul 19(4):1229–1246

    Article  MathSciNet  MATH  Google Scholar 

  • Yi W, Gao L, Pei Z, Lu J, Chen Y (2020) ε Constrained differential evolution using halfspace partition for optimization problems. J Intell Manuf. https://doi.org/10.1007/s10845-020-01565-2

    Article  Google Scholar 

  • Zahara E, Kao YT (2010) Hybrid Nelder-Mead simplex search and particle swarm optimization for constrained engineering design problems. Expert Syst Appl 36(2):3880–3886

    Article  Google Scholar 

  • Zhang M, Luo W, Wang X (2008) Differential evolution with dynamic stochastic selection for constrained optimization. Inf Sci 178(15):3043–3074

    Article  Google Scholar 

  • Zhang C, Lin Q, Gao L, Li X (2015) Backtracking Search Algorithm with three constraint handling methods for constrained optimization problems. Expert Syst Appl 42(21):7831–7845

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Payame Noor University (PNU), P.O. Box 19395-4697, Tehran, Iran

    Javad Alikhani Koupaei

  2. Department of Computer Engineering, University of Khansar, Khansar, Iran

    Marjan Firouznia

Authors
  1. Javad Alikhani Koupaei
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Marjan Firouznia
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Javad Alikhani Koupaei.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Alikhani Koupaei, J., Firouznia, M. A chaos-based constrained optimization algorithm. J Ambient Intell Human Comput 12, 9953–9976 (2021). https://doi.org/10.1007/s12652-020-02746-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12652-020-02746-w

Keywords

Search

Navigation