Abstract
The holy grail of constrained optimization is the development of an efficient, scale invariant and generic constraint handling procedure. To address these, the present paper proposes a unified approach of constraint handling, which is capable of handling all inequality, equality and hybrid constraints in a coherent manner. The proposed method also automatically resolves the issue of constraint scaling which is critical in real world and engineering optimization problems. The proposed unified approach converts the single-objective constrained optimization problem into a multi-objective problem. Evolutionary multi-objective optimization is used to solve the modified bi-objective problem and to estimate the penalty parameter automatically. The constrained optimum is further improved using classical optimization. The efficiency of the proposed method is validated on a set of well-studied constrained test problems and compared against without using normalization technique to show the necessity of normalization. The results establish the importance of scaling , especially in constrained optimization and call for further investigation into its use in constrained optimization research.
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For example, if we have a constraint of the form \(g_i(\mathbf{x}) \ge 0\), then for a candidate \(\mathbf{x}_c\), if \(g_i(\mathbf{x}_c)\) is indeed \(\ge 0\) we take \(v_i = 0\); otherwise, we take \(v_i = - g_i(\mathbf{x}_c)\). Finally, we take \(v = \sum _i v_i\), which is nonnegative because each \(v_i\) is nonnegative. We could also take \(v = \sum _i c_i v_i\), where the \(c_i\) are positive weighting factors, as discussed later in the main text.
It need not have an unconstrained minimum in general. For example, consider \(f(x) = x\) (scalar) with the constraint \(x \ge 0\); if we remove the constraint then the minimum is \(- \infty \) at \(x = -\infty \). However, our design variables are range bound by assumption.
Consider a problem where (say) the 7th largest eigenvalue of a 20 \(\times \) 20 symmetric matrix is constrained to be equal to \(14\). The constraint cannot be expressed as a simple explicit function of the design variables (the matrix elements).
References
Al Jadaan O, Rajamani L, Rao CR (2009) Parameterless penalty function for solving constrained evolutionary optimization. In: Hybrid intelligent models and applications, 2009. HIMA’09. IEEE workshop, pp 56–63. IEEE
Al-Fawzan MA, Haouari M (2005) A bi-objective model for robust resource-constrained project scheduling. Int J Prod Econ 96(2):175–187
Ali MM, Zhu WX (2013) A penalty function-based differential evolution algorithm for constrained global optimization. Comput Optim Appl 54(3):707–739
Barbosa HJC, Lemonge ACC (2003) A new adaptive penalty scheme for genetic algorithms. Inf Sci 156(3):215–251
Bernardino HS, Barbosa HJC, Lemonge ACC (2007) A hybrid genetic algorithm for constrained optimization problems in mechanical engineering. In: IEEE Congress on Evolutionary Computation, 2007. CEC 2007, pp 646–653. IEEE
Bowen J, Dozier G (1996) Constraint satisfaction using a hybrid evolutionary hill-climbing algorithm that performs opportunistic arc and path revision. In: Proceedings of the thirteenth national conference on artificial intelligence, vol 1, pp 326–331. AAAI Press
Brajevic I, Tuba M (2013) An upgraded artificial bee colony (abc) algorithm for constrained optimization problems. J Intell Manuf 24(4):729–740
Cai Z, Wang Y (2006) A multiobjective optimization-based evolutionary algorithm for constrained optimization. IEEE Trans Evol Comput 10(6):658–675
Cai X, Zhenzhou H, Fan Z (2013) A novel memetic algorithm based on invasive weed optimization and differential evolution for constrained optimization. Soft Comput 17(10):1893–1910
Chinneck JW (1995) Analyzing infeasible nonlinear programs. Comput Optim Appl 4(2):167–179
Coello CAC (2000) Use of a self-adaptive penalty approach for engineering optimization problems. Comput Ind 41(2):113–127
Coello CAC (2000) Treating constraints cobjectives for single-objective evolutionary optimization. Eng Optim+ A35 32(3):275–308
Coello CAC (2002) Theoretical and numerical constraint-handling techniques used with evolutionary algorithms: a survey of the state of the art. Comput Methods Appl Mech Eng 191(11–12):1245– linebreak1287
Conn AR, Pietrzykowski T (1977) A penalty function method converging directly to a constrained optimum. SIAM J Numer Anal 14(2):348–375
Costa L, Santo IE, Oliveira P (2013) An adaptive constraint handling technique for evolutionary algorithms. Optimization 62(2):241–253
Courant R (1943) Variational methods for the solution of problems of equilibrium and vibrations. Bull Am Math Soc 49(1):23
Deb K (1995) Optimization for engineering design: algorithms and examples. Prentice-Hall, New Delhi
Deb K (2000) An efficient constraint handling method for genetic algorithms. Comput Methods Appl Mech Eng 186(2–4):311–338
Deb K, Datta R (2013) A bi-objective constrained optimization algorithm using a hybrid evolutionary and penalty function approach. Eng Optim 45(5):503–527
Deb K, Datta R (2010) A fast and accurate solution of constrained optimization problems using a hybrid bi-objective and penalty function approach. In: IEEE Congress on Evolutionary Computation (CEC-2010), pp 1–8. IEEE
Deb K, Lele S, Datta R (2007) A hybrid evolutionary multi-objective and SQP based procedure for constrained optimization. In: Proceedings of the 2nd international conference on advances in computation and intelligence, pp 36–45. Springer, Berlin
Dozier G, Bowen J, Homaifar AA (1998) Solving constraint satisfaction problems using hybrid evolutionary search. IEEE Trans Evol Comput 2(1):23–33
Echeverri MG, Lezama JML, Romero R (2009) An efficient constraint handling methodology for multi-objective evolutionary algorithms. Revista Facultad de Ingenieria-Universidad de Antioquia 49:141–150
Elsayed SM, Sarker RA, Essam DL (2011) Multi-operator based evolutionary algorithms for solving constrained optimization problems. Comput Oper Res 38(12):1877–1896
Elsayed SM, Sarker RA, Essam DL (2013) Self-adaptive differential evolution incorporating a heuristic mixing of operators. Comput Optim Appl 54(3):771–790
Fletcher R (1975) An ideal penalty function for constrained optimization. IMA J Appl Math 15(3):319–342
Gandomi AH, Yang XS, Alavi AH, Talatahari S (2013) Bat algorithm for constrained optimization tasks. Neural Comput Appl 22(6):1239–1255
Gao Z, Xiao T, Fan W (2011) Hybrid differential evolution and Nelder-Mead algorithm with re-optimization. Soft Comput 15(3):581– 594
Haddad OB, Mirmomeni M, Mehrizi MZ, Mariño MA (2010) Finding the shortest path with honey-bee mating optimization algorithm in project management problems with constrained/unconstrained resources. Comput Optim Appl 47(1):97–128
He Q, Wang L (2007) A hybrid particle swarm optimization with a feasibility-based rule for constrained optimization. Appl Math Comput 186(2):1407–1422
Homaifar A, Lai SH-V, Qi X (1994) Constrained optimization via genetic algorithms. Simulation 62(4):242–254
Jan MA, Khanum RA (2013) A study of two penalty-parameterless constraint handling techniques in the framework of moea/d. Appl Soft Comput 13(1):128–148
Joines JA, Houck CR (1994) On the use of non-stationary penalty functions to solve nonlinear constrained optimization problems with ga’s. In: Evolutionary computation, 1994. Proceedings of the First IEEE conference on IEEE world congress on computational intelligence, pp 579–584. IEEE
Keefer DL, Gottfried BS (1970) Differential constraint scaling in penalty function optimization. AIIE Trans 2(4):281–289
Kort BW, Bertsekas DP (1972) A new penalty function method for constrained minimization. In Decision and control, 1972 and 11th symposium on adaptive processes. In: Proceedings of the 1972 IEEE conference, vol 11, pp 162–166. IEEE
Liang JJ, Runarsson TP, Mezura-Montes E, Clerc M, Suganthan PN, Coello CAC, Deb K (2006) Problem definitions and evaluation criteria for the CEC. In: Special session on constrained real-parameter optimization. Nanyang Technological University, Singapore, Technical report
Liao TW (2010) Two hybrid differential evolution algorithms for engineering design optimization. Appl Soft Comput 10(4):1188–1199
Lin YC, Hwang KS, Wang F (2002) Hybrid differential evolution with multiplier updating method for nonlinear constrained optimization problems. In: Evolutionary Computation, 2002. CEC’02. Proceedings of the 2002 Congress on, vol 1, pp 872–877. IEEE
Liu H, Cai Z, Wang Y (2010) Hybridizing particle swarm optimization with differential evolution for constrained numerical and engineering optimization. Appl Soft Comput 10(2):629–640
Long W (2014) Knowledge-base constrained optimization evolutionary algorithm and its applications. Appl Mech Mater 536:476–480
Manoharan PS, Kannan PS, Baskar S, Iruthayarajan MW (2008) Penalty parameter-less constraint handling scheme based evolutionary algorithm solutions to economic dispatch. IET Gener Transm Distrib 2(4):478–490
Masuda K, Kurihara K (2011) A constrained global optimization method based on multi-objective particle swarm optimization. IEEJ Trans Electr Inf Syst 131(5):990–999
Mazhoud I, Hadj-Hamou K, Bigeon J, Joyeux P (2013) Particle swarm optimization for solving engineering problems: a new constraint-handling mechanism. Eng Appl Artif Intell 26(4):1263–1273
Mezura-Montes E (2009) Constraint-handling in evolutionary optimization. Springer, Berlin
Mezura-Montes E, Coello CAC (2011) Constraint-handling in nature-inspired numerical optimization: past, present and future. Swarm Evol Comput 1(4):173–194
Mezura-Montes E, Coello CAC (2008) Constrained optimization via multiobjective evolutionary algorithms. In: Multiobjective problem solving from nature, pp 53–75
Michalewicz Z, Schoenauer M (1996) Evolutionary algorithms for constrained parameter optimization problems. Evol Comput 4(1):1–32
Michalewicz Z, Janikow CZ (1991) Handling constraints in genetic algorithms. In: Proceedings of the 4th international conference on genetic algorithms, pp 151–157
Montemurro M, Vincenti A, Vannucci P (2013) The automatic dynamic penalisation method (adp) for handling constraints with genetic algorithms. Comput Methods Appl Mech Eng 256:70–87
Myung H, Kim JH (1997) Evolian: evolutionary optimization based on lagrangian with constraint scaling. In: Evolutionary Programming VI, pp 177–187. Springer, Berlin
Ong YS, Lum KY, Nair PB (2008) Hybrid evolutionary algorithm with hermite radial basis function interpolants for computationally expensive adjoint solvers. Comput Optim Appl 39(1):97–119
Papa G (2013) Parameter-less algorithm for evolutionary-based optimization. Comput Optim Appl 56(1):209–229
Paredis J (1994) Co-evolutionary constraint satisfaction. In: Parallel problem solving from nature PPSN III, pp 46–55. Springer, Berlin
Powell MJD (1978) A fast algorithm for nonlinearly constrained optimization calculations. In: Numerical analysis, pp 144–157. Springer, Berlin
Ramezani P, Ahangaran M, Yang XS (2013) Constrained optimisation and robust function optimisation with eiwo. Int J Bio-Inspired Comput 5(2):84–98
Ray T, Singh H, Isaacs A, Smith W (2009) Infeasibility driven evolutionary algorithm for constrained optimization. In: Mezura-Montes E (ed) Constraint-handling in evolutionary computation. pp 145–165. Springer, Berlin
Runarsson TP, Yao X (2000) Stochastic ranking for constrained evolutionary optimization. Evol Comput IEEE Trans 4(3):284–294
Segura C, Coello CAC, Miranda G, León C (2013) Using multi-objective evolutionary algorithms for single-objective optimization. 4OR 11(3):201–228
Sha J, Xu M (2011) Applying hybrid genetic algorithm to constrained trajectory optimization. In Electronic and Mechanical Engineering and Information Technology (EMEIT), 2011 International Conference, vol 7, pp 3792–3795. IEEE
Surry PD, Radcliffe NJ, Boyd ID (1995) A multi-objective approach to constrained optimisation of gas supply networks: the COMOGA method. In: Evolutionary computing. AISB Workshop, pp 166–180. Springer, Berlin
Takahama T, Sakai S, Iwane N (2005) Constrained optimization by the \(\varepsilon \) constrained hybrid algorithm of particle swarm optimization and genetic algorithm. AI 2005: Advances in Artificial Intelligence, pp 389–400
Tessema B, Yen GG (2006) A self adaptive penalty function based algorithm for constrained optimization. In: IEEE congress on evolutionary computation (CEC-2006), pp 246–253. IEEE
Venkatraman S, Yen GG (2005) A generic framework for constrained optimization using genetic algorithms. IEEE Trans Evol Comput 9(4):424–435
Venter G, Haftka RT (2010) Constrained particle swarm optimization using a bi-objective formulation. Struct Multidiscip Optim 40(1–6):65–76
Wang L (2001) Intelligent optimization algorithms with applications. Tsinghua University & Springer Press, Beijing
Wang Y, Cai Z (2012) Combining multiobjective optimization with differential evolution to solve constrained optimization problems. Evol Comput IEEE Trans 16(1):117–134
Xiao J, Huang Y, Cheng Z, He J, Niu Y (2014) A hybrid membrane evolutionary algorithm for solving constrained optimization problems. Optik Int J Light Electron Optics 125(2):897–902
Xu YC, Lei B, Hendriks EA (2013) Constrained particle swarm algorithms for optimizing coverage of large-scale camera networks with mobile nodes. Soft Comput 17(6):1047–1057
Zahara E, Kao YT (2009) Hybrid Nelder-Mead simplex search and particle swarm optimization for constrained engineering design problems. Expert Syst Appl 36(2):3880–3886
Zhang Q, Li H (2007) Moea/d: a multiobjective evolutionary algorithm based on decomposition. Evol Comput IEEE Trans 11(6):712–731
Zhao J, Wang L, Zeng P, Fan W (2011) An effective hybrid genetic algorithm with flexible allowance technique for constrained engineering design optimization. Expert Syst Appl 39(5):6041–6051
Zhou Y, Li Y, He J, Kang L (2003) Multi-objective and MGG evolutionary algorithm for constrained optimization. In: IEEE congress on evolutionary computation (CEC-2003), vol 1, pp 1–5. IEEE
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We thank Anindya Chatterjee, Mechanical Engineering, IIT Kanpur for proposing development of this unified approach, and some related suggestions.
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Communicated by V. Loia.
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Datta, R., Deb, K. Uniform adaptive scaling of equality and inequality constraints within hybrid evolutionary-cum-classical optimization. Soft Comput 20, 2367–2382 (2016). https://doi.org/10.1007/s00500-015-1646-0
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DOI: https://doi.org/10.1007/s00500-015-1646-0