Abstract
To solve global optimization problems, nature-inspired metaheuristics have emerged as the best option in many cases. The researchers have mimicked or mapped the optimal behaviors of living organisms, natural phenomena and human behaviors to solve optimization problems. Whenever a new algorithm is proposed, its exploration, exploitation and convergence capabilities are tested; however, in this study, it is highlighted that a metaheuristic should also exhibit invariance to function translation/shifting. Whale optimization algorithm (WOA) is a well-reputed and highly referred algorithm in the literature; however, we find that its performance is strongly affected under certain circumstances involving function shifting. Our analysis reveals that WOA performs really well when the global optimum is situated at the origin (\(0, 0, \ldots , 0\)); on the contrary, the performance amazingly degrades whenever the global optimum is situated away from the origin. Moreover, in most cases, our findings reveal that the performance degradation is directly proportional to the distance of the global optimum from the origin. Furthermore, the root cause of the problem is discovered and discussed. Finally, it is suggested how WOA can be made shift-invariant.
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References
Abd El Aziz M, Ewees AA, Hassanien AE (2018) Multi-objective whale optimization algorithm for content-based image retrieval. Multimed Tools Appl 77:26135–26172
Abd Elaziz M, Oliva D (2018) Parameter estimation of solar cells diode models by an improved opposition-based whale optimization algorithm. Energy Convers Manag 171:1843–1859
Alameer Z, Elaziz MA, Ewees AA, Ye H, Jianhua Z (2019) Forecasting gold price fluctuations using improved multilayer perceptron neural network and whale optimization algorithm. Resour Policy 61:250–260
Alamri HS, Alsariera YA, Zamli KZ (2018) Opposition-based whale optimization algorithm. Adv Sci Lett 24:7461–7464
Askari Q, Saeed M, Younas I (2020) Heap-based optimizer inspired by corporate rank hierarchy for global optimization. Expert Syst Appl 161:113702
Askari Q, Younas I, Saeed M (2020) Critical evaluation of sine cosine algorithm and a few recommendations. In: Proceedings of the 2020 genetic and evolutionary computation conference companion, pp 319–320
Askari Q, Younas I, Saeed M (2020) Political optimizer: a novel socio-inspired meta-heuristic for global optimization. Knowl-Based Syst 195:105709
Askari Q, Younas I (2021) Political optimizer based feedforward neural network for classification and function approximation. Neural Process Lett 53:429–458. https://doi.org/10.1007/s11063-020-10406-5
Awad NH, Suganthan P, Liang J, Qu B, Ali M (2017) Problem definitions and evaluation criteria for the CEC 2017 special session and competition on single objective real-parameter numerical optimization. In: 2017 IEEE congress on evolutionary computation (CEC)
Bianchi L, Dorigo M, Gambardella LM, Gutjahr WJ (2008) A survey on metaheuristics for stochastic combinatorial optimization. Nat Comput 8:239–287
Chen Z, Liu Y, Yang Z, Fu X, Tan J, Yang X (2020) An enhanced teaching-learning-based optimization algorithm with self-adaptive and learning operators and its search bias towards origin. Swarm Evol Comput 60:100766
Dasu B, Sivakumar M, Srinivasarao R (2019) Interconnected multi-machine power system stabilizer design using whale optimization algorithm. Prot Control Mod Power Syst 4:4–111
Dorigo M, Caro GD (1999) Ant colony optimization: a new meta-heuristic. In: Proceedings of the 1999 congress on evolutionary computation-CEC99 (Cat. No. 99TH8406). IEEE
Ghasemi M, Davoudkhani IF, Akbari E, Rahimnejad A, Ghavidel S, Li L (2020) A novel and effective optimization algorithm for global optimization and its engineering applications: turbulent flow of water-based optimization (tfwo). Eng Appl Artif Intell 92:103666
Heidari AA, Mirjalili S, Faris H, Aljarah I, Mafarja M, Chen H (2019) Harris Hawks optimization: algorithm and applications. Futur Gener Comput Syst 97:849–872
Hussien AG, Hassanien AE, Houssein EH, Bhattacharyya S, Amin M (2019) S-shaped binary whale optimization algorithm for feature selection. In: Recent trends in signal and image processing. Springer, pp 79–87
Hussien AG, Hassanien AE, Houssein EH, Amin M, Azar AT (2020) New binary whale optimization algorithm for discrete optimization problems. Eng Optim 52:945–959
Kaur G, Arora S (2018) Chaotic whale optimization algorithm. J Comput Des Eng 5:275–284
Kennedy J, Eberhart R (1995) Particle swarm optimization. In: Proceedings of ICNN95—international conference on neural networks. IEEE
Lampinen J, Storn R (2004) Differential evolution. In: Onwubolu GC, Babu BV (eds) New optimization techniques in engineering. Springer, Berlin, pp 123–166
Li S, Chen H, Wang M, Heidari AA, Mirjalili S (2020) Slime Mould algorithm: a new method for stochastic optimization. Fut Gener Comput Syst 111:300–323
Liang J, Suganthan P, Deb K (2005) Novel composition test functions for numerical global optimization. In: Proceedings 2005 IEEE swarm intelligence symposium, 2005. SIS 2005. IEEE
Liang JJ, Baskar S, Suganthan PN, Qin AK (2006) Performance evaluation of multiagent genetic algorithm. Nat Comput 5:83–96
Liang JJ, Qu BY, Suganthan PN (2013) Problem definitions and evaluation criteria for the CEC 2014 special session and competition on single objective real-parameter numerical optimization. Zhengzhou University, Zhengzhou China and Technical Report, Nanyang Technological University, Singapore, Computational Intelligence Laboratory, p 635
Ling Y, Zhou Y, Luo Q (2017) Lévy flight trajectory-based whale optimization algorithm for global optimization. IEEE Access 5:6168–6186
Mahmoodabadi M, Rasekh M, Zohari T (2018) TGA: team game algorithm. Fut Comput Inf J 3:191–199
Mirjalili S (2016) Sca: a sine cosine algorithm for solving optimization problems. Knowl-Based Syst 96:120–133
Mirjalili S, Lewis A (2016) The whale optimization algorithm. Adv Eng Softw 95:51–67
Mirjalili S, Mirjalili SM, Lewis A (2014) Grey wolf optimizer. Adv Eng Softw 69:46–61
Niu P, Niu S, Chang L et al (2019) The defect of the Grey Wolf optimization algorithm and its verification method. Knowl-Based Syst 171:37–43
Oliva D, Abd El Aziz M, Hassanien AE (2017) Parameter estimation of photovoltaic cells using an improved chaotic whale optimization algorithm. Appl Energy 200:141–154
Pickard JK, Carretero JA, Bhavsar VC (2016) On the convergence and origin bias of the teaching-learning-based-optimization algorithm. Appl Soft Comput 46:115–127
Prasad D, Mukherjee A, Mukherjee V (2017) Transient stability constrained optimal power flow using chaotic whale optimization algorithm. In: Handbook of neural computation. Elsevier, pp 311–332
Qais MH, Hasanien HM, Alghuwainem S (2020) Enhanced whale optimization algorithm for maximum power point tracking of variable-speed wind generators. Appl Soft Comput 86:105937
Rao R, Savsani V, Vakharia D (2011) Teaching-learning-based optimization: a novel method for constrained mechanical design optimization problems. Comput Aid Des 43:303–315
Rashedi E, Nezamabadi-pour H, Saryazdi S (2009) GSA: a gravitational search algorithm. Inf Sci 179:2232–2248
Sayed GI, Darwish A, Hassanien AE (2018) A new chaotic whale optimization algorithm for features selection. J Classif 35:300–344
Simon D (2008) Biogeography-based optimization. IEEE Trans Evol Comput 12:702–713
Sun W, Zhang C (2018) Analysis and forecasting of the carbon price using multi-resolution singular value decomposition and extreme learning machine optimized by adaptive whale optimization algorithm. Appl Energy 231:1354–1371
Sun Y, Wang X, Chen Y, Liu Z (2018) A modified whale optimization algorithm for large-scale global optimization problems. Expert Syst Appl 114:563–577
Tu J, Chen H, Liu J, Heidari AA, Zhang X, Wang M, Ruby R, Pham Q-V (2021) Evolutionary biogeography-based whale optimization methods with communication structure: towards measuring the balance. Knowl-Based Syst 212:106642
Tubishat M, Abushariah MAM, Idris N, Aljarah I (2018) Improved Whale optimization algorithm for feature selection in Arabic sentiment analysis. Appl Intell 49:1688–1707
Wang G-G (2018) Moth search algorithm: a bio-inspired metaheuristic algorithm for global optimization problems. Memet Comput 10:151–164
Wang M, Chen H (2020) Chaotic multi-swarm whale optimizer boosted support vector machine for medical diagnosis. Appl Soft Comput 88:105946
Wang J, Du P, Niu T, Yang W (2017) A novel hybrid system based on a new proposed algorithm-multi-objective whale optimization algorithm for wind speed forecasting. Appl Energy 208:344–360
Wang G-G, Deb S, Cui Z (2019) Monarch butterfly optimization. Neural Comput Appl 31:1995–2014
Yang X-S, Deb S (2009) Cuckoo search via lévy flights. In: 2009 World congress on nature and biologically inspired computing (NaBIC). IEEE, pp 210–214
Yao X, Liu Y, Lin G (1999) Evolutionary programming made faster. IEEE Trans Evol Comput 3:82–102
Yuan P, Guo C, Zheng Q, Ding J (2018) Sidelobe suppression with constraint for MIMO radar via chaotic whale optimisation. Electron Lett 54:311–313
Zervoudakis K, Tsafarakis S (2020) A mayfly optimization algorithm. Comput Ind Eng 145:106559
Zhang X, Hu W, Qu W, Maybank S (2010) Multiple object tracking via species-based particle swarm optimization. IEEE Trans Circuits Syst Video Technol 20:1590–1602
Zhang X, Hu W, Xie N, Bao H, Maybank S (2015) A robust tracking system for low frame rate video. Int J Comput Vis 115:279–304
Zhong W, Liu J, Xue M, Jiao L (2004) A multiagent genetic algorithm for global numerical optimization. IEEE Trans Syst Man Cybern B (Cybern) 34:1128–1141
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Askari, Q., Younas, I. & Saeed, M. Emphasizing the importance of shift invariance in metaheuristics by using whale optimization algorithm as a test bed. Soft Comput 25, 14209–14225 (2021). https://doi.org/10.1007/s00500-021-06101-9
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DOI: https://doi.org/10.1007/s00500-021-06101-9