Optimization of constrained mathematical and engineering design problems using chaos game optimization

Abstract In the past few decades, many different metaheuristic algorithms have been developed for optimization purposes each of which have specific advantages and disadvantages due to multiple applications in different optimization fields. The Chaos Game Optimization (CGO) is proposed in this paper as a new metaheuristic algorithm for optimization of constrained mathematical and engineering design problems. The proposed CGO method is formulated based on some principles of chaos theory in which the fractals configuration by chaos game methodology alongside the fractals self-similarity issues are in perspective. A total number of 34 constrained mathematical problems are collected which have been benchmarked and proposed in the Competitions on Evolutionary Computation (CEC) and 15 constrained engineering design problems are selected in order to evaluate the overall performance of the proposed novel CGO method. In order to validate the results of the novel CGO algorithm, the best results of different standard, improved and hybrid metaheuristic algorithms in dealing with the considered constrained problems are selected from the literature for comparative purposes. In addition, the statistical results of the CGO algorithm including the minimum, mean, maximum and the standard deviation values are all calculated and compared to the results of other metaheuristics. The obtained results proved that the proposed algorithm is capable of providing very competitive results and outperforms the other metaheuristics in most of the cases.

[1]  Seyedali Mirjalili,et al.  SCA: A Sine Cosine Algorithm for solving optimization problems , 2016, Knowl. Based Syst..

[2]  Siamak Talatahari,et al.  Upgraded Whale Optimization Algorithm for fuzzy logic based vibration control of nonlinear steel structure , 2019, Engineering Structures.

[3]  Siamak Talatahari,et al.  Optimal design of real‐size building structures using quantum‐behaved developed swarm optimizer , 2020, The Structural Design of Tall and Special Buildings.

[4]  İsmail Şahin,et al.  Optimization of Hydrostatic Thrust Bearing Using Enhanced Grey Wolf Optimizer , 2019 .

[5]  Hans-Paul Schwefel,et al.  Evolution strategies – A comprehensive introduction , 2002, Natural Computing.

[6]  Siamak Talatahari,et al.  Optimal tuning of fuzzy parameters for structural motion control using multiverse optimizer , 2019, The Structural Design of Tall and Special Buildings.

[7]  Ivan Zelinka,et al.  Mechanical engineering problem optimization by SOMA , 2004 .

[8]  Han Tong Loh,et al.  Computational Implementation and Tests of a Sequential Linearization Algorithm for Mixed-Discrete Nonlinear Design Optimization , 1991 .

[9]  Ibrahim Eksin,et al.  A new optimization method: Big Bang-Big Crunch , 2006, Adv. Eng. Softw..

[10]  M.H. Tayarani-N,et al.  Magnetic Optimization Algorithms a new synthesis , 2008, 2008 IEEE Congress on Evolutionary Computation (IEEE World Congress on Computational Intelligence).

[11]  Hossam Faris,et al.  Salp Swarm Algorithm: A bio-inspired optimizer for engineering design problems , 2017, Adv. Eng. Softw..

[12]  Siamak Talatahari,et al.  Tribe–charged system search for parameter configuration of nonlinear systems with large search domains , 2021 .

[13]  Ali Kaveh,et al.  SHAPE AND SIZE OPTIMIZATION OF TRUSS STRUCTURES WITH FREQUENCY CONSTRAINTS USING ENHANCED CHARGED SYSTEM SEARCH ALGORITHM , 2011 .

[14]  R. Venkata Rao,et al.  Teaching-learning-based optimization: A novel method for constrained mechanical design optimization problems , 2011, Comput. Aided Des..

[15]  Ardeshir Bahreininejad,et al.  Mine blast algorithm: A new population based algorithm for solving constrained engineering optimization problems , 2013, Appl. Soft Comput..

[16]  Yanbin Liu,et al.  A hybrid quantum-based PIO algorithm for global numerical optimization , 2018, Science China Information Sciences.

[17]  Najeh Ben Guedria,et al.  Improved accelerated PSO algorithm for mechanical engineering optimization problems , 2016, Appl. Soft Comput..

[18]  S. N. Kramer,et al.  An Augmented Lagrange Multiplier Based Method for Mixed Integer Discrete Continuous Optimization and Its Applications to Mechanical Design , 1994 .

[19]  Hammoudi Abderazek,et al.  Correction to: A Comparative Study of Recent Non-traditional Methods for Mechanical Design Optimization , 2019 .

[20]  Rainer Storn,et al.  Differential Evolution – A Simple and Efficient Heuristic for global Optimization over Continuous Spaces , 1997, J. Glob. Optim..

[21]  Harish Garg,et al.  A hybrid GSA-GA algorithm for constrained optimization problems , 2019, Inf. Sci..

[22]  Marco Dorigo,et al.  Ant system: optimization by a colony of cooperating agents , 1996, IEEE Trans. Syst. Man Cybern. Part B.

[23]  Amir Hossein Gandomi,et al.  Cuckoo search algorithm: a metaheuristic approach to solve structural optimization problems , 2011, Engineering with Computers.

[24]  John H. Holland,et al.  Adaptation in Natural and Artificial Systems: An Introductory Analysis with Applications to Biology, Control, and Artificial Intelligence , 1992 .

[25]  Seyed Mohammad Mirjalili,et al.  Multi-Verse Optimizer: a nature-inspired algorithm for global optimization , 2015, Neural Computing and Applications.

[26]  Singiresu S. Rao.  Engineering Optimization : Theory and Practice , 2010 .

[27]  Abdolreza Hatamlou,et al.  Black hole: A new heuristic optimization approach for data clustering , 2013, Inf. Sci..

[28]  Haibin Duan,et al.  Mobile Robot ADRC With an Automatic Parameter Tuning Mechanism via Modified Pigeon-Inspired Optimization , 2019, IEEE/ASME Transactions on Mechatronics.

[29]  Jing J. Liang,et al.  Dynamic Multi-Swarm Particle Swarm Optimizer with a Novel Constraint-Handling Mechanism , 2006, 2006 IEEE International Conference on Evolutionary Computation.

[30]  R. Venkata Rao,et al.  A new optimization algorithm for solving complex constrained design optimization problems , 2017 .

[31]  A. Kaveh,et al.  A novel heuristic optimization method: charged system search , 2010 .

[32]  Xiaodong Wu,et al.  Small-World Optimization Algorithm for Function Optimization , 2006, ICNC.

[33]  Andrew Lewis,et al.  Grey Wolf Optimizer , 2014, Adv. Eng. Softw..

[34]  Dervis Karaboga,et al.  A powerful and efficient algorithm for numerical function optimization: artificial bee colony (ABC) algorithm , 2007, J. Glob. Optim..

[35]  Xiaojun Zhou,et al.  A Two-stage State Transition Algorithm for Constrained Engineering Optimization Problems , 2018 .

[36]  Tapabrata Ray,et al.  Society and civilization: An optimization algorithm based on the simulation of social behavior , 2003, IEEE Trans. Evol. Comput..

[37]  Liang Gao,et al.  Queuing search algorithm: A novel metaheuristic algorithm for solving engineering optimization problems , 2018, Applied Mathematical Modelling.

[38]  Erwie Zahara,et al.  Hybrid Nelder-Mead simplex search and particle swarm optimization for constrained engineering design problems , 2009, Expert Syst. Appl..

[39]  Xin-She Yang,et al.  Cuckoo Search via Lévy flights , 2009, 2009 World Congress on Nature & Biologically Inspired Computing (NaBIC).

[40]  Zong Woo Geem,et al.  A New Heuristic Optimization Algorithm: Harmony Search , 2001, Simul..

[41]  Hammoudi Abderazek,et al.  A Comparative Study of Recent Non-traditional Methods for Mechanical Design Optimization , 2019, Archives of Computational Methods in Engineering.

[42]  Jing J. Liang,et al.  Problem Deflnitions and Evaluation Criteria for the CEC 2006 Special Session on Constrained Real-Parameter Optimization , 2006 .

[43]  R. Venkata Rao,et al.  Mechanical Design Optimization Using Advanced Optimization Techniques , 2012 .

[44]  Ales Zamuda,et al.  Adaptive constraint handling and Success History Differential Evolution for CEC 2017 Constrained Real-Parameter Optimization , 2017, 2017 IEEE Congress on Evolutionary Computation (CEC).

[45]  Andrew Lewis,et al.  The Whale Optimization Algorithm , 2016, Adv. Eng. Softw..

[46]  E. Sandgren,et al.  Nonlinear Integer and Discrete Programming in Mechanical Design Optimization , 1990 .

[47]  Bilal Alatas,et al.  ACROA: Artificial Chemical Reaction Optimization Algorithm for global optimization , 2011, Expert Syst. Appl..

[48]  Janez Brest,et al.  Self-Adaptive Differential Evolution Algorithm in Constrained Real-Parameter Optimization , 2006, 2006 IEEE International Conference on Evolutionary Computation.

[49]  Haibin Duan,et al.  Pigeon-inspired optimization: a new swarm intelligence optimizer for air robot path planning , 2014, Int. J. Intell. Comput. Cybern..

[50]  N. Siddique,et al.  Central Force Optimization , 2017 .

[51]  Tapabrata Ray,et al.  A socio-behavioural simulation model for engineering design optimization , 2002 .

[52]  Xin-She Yang,et al.  Nature-Inspired Metaheuristic Algorithms , 2008 .

[53]  Fred W. Glover,et al.  Future paths for integer programming and links to artificial intelligence , 1986, Comput. Oper. Res..

[54]  Ling Wang,et al.  An effective co-evolutionary particle swarm optimization for constrained engineering design problems , 2007, Eng. Appl. Artif. Intell..

[55]  C. D. Gelatt,et al.  Optimization by Simulated Annealing , 1983, Science.

[56]  James N. Siddall,et al.  Optimal Engineering Design: Principles and Applications , 1982 .

[57]  Luciano Lamberti,et al.  Move limits definition in structural optimization with sequential linear programming. Part I: Optimization algorithm , 2003 .

[58]  Kalyanmoy Deb,et al.  Optimizing Engineering Designs Using a Combined Genetic Search , 1997, ICGA.

[59]  Amir Hossein Alavi,et al.  Krill herd: A new bio-inspired optimization algorithm , 2012 .

[60]  Vimal Savsani,et al.  Passing vehicle search (PVS): A novel metaheuristic algorithm , 2016 .

[61]  Amir Hossein Gandomi,et al.  Bat algorithm for constrained optimization tasks , 2012, Neural Computing and Applications.

[62]  Hamed Shah-Hosseini,et al.  Principal components analysis by the galaxy-based search algorithm: a novel metaheuristic for continuous optimisation , 2011, Int. J. Comput. Sci. Eng..

[63]  Dan Simon,et al.  Biogeography-Based Optimization , 2022 .

[64]  Min-Yuan Cheng,et al.  Symbiotic Organisms Search: A new metaheuristic optimization algorithm , 2014 .

[65]  Seyed Mohammad Mirjalili,et al.  Moth-flame optimization algorithm: A novel nature-inspired heuristic paradigm , 2015, Knowl. Based Syst..

[66]  G. G. Wang,et al.  Adaptive Response Surface Method Using Inherited Latin Hypercube Design Points , 2003 .

[67]  Carlos A. Coello Coello,et al.  Engineering optimization using simple evolutionary algorithm , 2003, Proceedings. 15th IEEE International Conference on Tools with Artificial Intelligence.

[68]  Leandro dos Santos Coelho,et al.  Gaussian quantum-behaved particle swarm optimization approaches for constrained engineering design problems , 2010, Expert Syst. Appl..

[69]  Ling Wang,et al.  An effective co-evolutionary differential evolution for constrained optimization , 2007, Appl. Math. Comput..

[70]  A Baghlani,et al.  TEACHING-LEARNING-BASED OPTIMIZATION ALGORITHM FOR SHAPE AND SIZE OPTIMIZATION OF TRUSS STRUCTURES WITH DYNAMIC FREQUENCY CONSTRAINTS , 2013 .

[71]  Antônio José da Silva Neto,et al.  A constrained ITGO heuristic applied to engineering optimization , 2018, Expert Syst. Appl..

[72]  Carlos A. Coello Coello,et al.  Use of a self-adaptive penalty approach for engineering optimization problems , 2000 .

[73]  Carlos A. Coello Coello,et al.  Modified Differential Evolution for Constrained Optimization , 2006, 2006 IEEE International Conference on Evolutionary Computation.

[74]  Fred W. Glover,et al.  A History of Metaheuristics , 2015, Handbook of Heuristics.

[75]  Russell C. Eberhart,et al.  A new optimizer using particle swarm theory , 1995, MHS'95. Proceedings of the Sixth International Symposium on Micro Machine and Human Science.

[76]  A. Kaveh,et al.  A new meta-heuristic method: Ray Optimization , 2012 .

[77]  Wenjian Luo,et al.  Differential evolution with dynamic stochastic selection for constrained optimization , 2008, Inf. Sci..

[78]  Ali Kaveh,et al.  Colliding bodies optimization: A novel meta-heuristic method , 2014 .

[79]  Amir Hossein Gandomi,et al.  A chaotic particle-swarm krill herd algorithm for global numerical optimization , 2013, Kybernetes.

[80]  Pei-wei Tsai,et al.  Cat Swarm Optimization , 2006, PRICAI.

[81]  Josef Tvrdík,et al.  A simple framework for constrained problems with application of L-SHADE44 and IDE , 2017, 2017 IEEE Congress on Evolutionary Computation (CEC).

[82]  Aravind Srinivasan,et al.  Innovization: innovating design principles through optimization , 2006, GECCO.

[83]  Caro Lucas,et al.  Imperialist competitive algorithm: An algorithm for optimization inspired by imperialistic competition , 2007, 2007 IEEE Congress on Evolutionary Computation.

[84]  Rajiv Tiwari,et al.  Multi-objective design optimisation of rolling bearings using genetic algorithms , 2007 .

[85]  Radka Polakova,et al.  L-SHADE with competing strategies applied to constrained optimization , 2017, 2017 IEEE Congress on Evolutionary Computation (CEC).

[86]  Ardeshir Bahreininejad,et al.  Water cycle algorithm - A novel metaheuristic optimization method for solving constrained engineering optimization problems , 2012 .

[87]  Siamak Talatahari,et al.  Optimum design of fuzzy controller using hybrid ant lion optimizer and Jaya algorithm , 2019, Artificial Intelligence Review.

[88]  Hossein Nezamabadi-pour,et al.  GSA: A Gravitational Search Algorithm , 2009, Inf. Sci..