## CMA-ES

 swMATH ID: 5063 Software Authors: Hansen Description: CMA-ES stands for Covariance Matrix Adaptation Evolution Strategy. Evolution strategies (ES) are stochastic, derivative-free methods for numerical optimization of non-linear or non-convex continuous optimization problems. They belong to the class of evolutionary algorithms and evolutionary computation. An evolutionary algorithm is broadly based on the principle of biological evolution, namely the repeated interplay of variation (via mutation and recombination) and selection: in each generation (iteration) new individuals (candidate solutions, denoted as x) are generated by variation, usually in a stochastic way, and then some individuals are selected for the next generation based on their fitness or objective function value f(x). Like this, over the generation sequence, individuals with better and better f-values are generated. In an evolution strategy, new candidate solutions are sampled according to a multivariate normal distribution in the mathbb{R}^n. Pairwise dependencies between the variables in this distribution are represented by a covariance matrix. The covariance matrix adaptation (CMA) is a method to update the covariance matrix of this distribution. This is particularly useful, if the function f is ill-conditioned. Adaptation of the covariance matrix amounts to learning a second order model of the underlying objective function similar to the approximation of the inverse Hessian matrix in the Quasi-Newton method in classical optimization. In contrast to most classical methods, fewer assumptions on the nature of the underlying objective function are made. Only the ranking between candidate solutions is exploited for learning the sample distribution and neither derivatives nor even the function values themselves are required by the method. Homepage: https://www.lri.fr/~hansen/cmaes_inmatlab.html Programming Languages: Matlab Dependencies: Matlab Keywords: optimization; The Unconstrained NLO Problem; Global Optimization Related Software: CEC 05; EGO; MultiMin; Genocop; GWO; JADE; GSA; SSA; SPOT; AOA; ABC; WOA; GitHub; SNOBFIT; PSwarm; irace; CEC 13; Aquila optimizer; ALO; AHA Cited in: 122 Documents Further Publications: https://www.lri.fr/~hansen/publications.html#hansenReview2006
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### Cited by 324 Authors

 6 Koumoutsakos, Petros D. 5 Hansen, Nikolaus 5 Hu, Gang 5 Wei, Guo 4 Akimoto, Youhei 4 Angelikopoulos, Panagiotis 4 Auger, Anne 4 Gonzalez, Luis Felipe 4 Périaux, Jacques F. 4 Srinivas, Karkenahalli 4 Zhong, Jingyu 3 Beyer, Hans-Georg 3 Papadimitriou, Costas 3 Sahinidis, Nikolaos V. 2 Ahrari, Ali 2 Amaran, Satyajith 2 Arampatzis, Georgios 2 Balesdent, Mathieu 2 Bartz-Beielstein, Thomas 2 Breunig, U. 2 Bury, Scott J. 2 Chang, Ching-Ter 2 Fernandes, Edite M. G. P. 2 Gandomi, Amir Hossein 2 García-Martínez, Carlos 2 Hadjidoukas, Panagiotis E. 2 Hartl, Richard F. 2 Herrera, Francisco 2 Igel, Christian 2 Lozano, Manuel 2 Marsden, Alison L. 2 Mirams, Gary R. 2 Molina, Daniel 2 Preston, Simon P. 2 Rocha, Ana Maria A. C. 2 Sharda, Bikram 2 Telek, Miklós 2 Vidal, Thibaut 2 Wälchli, Daniel 2 Whittaker, Dominic G. 1 Abd Elaziz, Mohamed 1 Abdulsamad, Hany 1 Abualigah, Laith 1 Agathos, Konstantinos 1 Ahrari, Reza 1 Almaraashi, Majid 1 Amin, Hesham H. 1 Andreasen, Martin Møller 1 Atai, Ali Asghar 1 Audet, Charles 1 Aulig, Nikola 1 Bader, Samuel B. 1 Baldacci, Roberto 1 Baldi, Pierre 1 Band, Leah R. 1 Barbaresco, Frédéric 1 Bartoszuk, Maciej 1 Beliakov, Gleb 1 Belousov, Boris 1 Bigeon, Jean 1 Billy, Frédérique 1 Blanco-Cocom, Luis D. 1 Bolton, Kirsty J. 1 Bordas, Stéphane Pierre Alain 1 Boria, Simonetta 1 Borràs, Agnés 1 Bosman, Peter A. N. 1 Botello Rionda, Salvador 1 Boussaïd, Ilhem 1 Brevault, Loïc 1 Brossier, Romain 1 Büche, Dirk 1 Buet, Blanche 1 Bujny, Mariusz 1 Bureerat, Sujin 1 Byrne, Helen Mary 1 Cai, Jia 1 Cai, Zhihua 1 Caleb-Solly, Praminda 1 Celora, Giulia L. 1 Chatzi, Eleni 1 Chauhan, Dikshit 1 Chen, Huangke 1 Chen, Jing 1 Cheng, Junrui 1 Cheng, Ran 1 Cheng, Xiang 1 Çınar, Didem 1 Clairambault, Jean 1 Congedo, Pietro Marco 1 Costa, M. Fernanda P. 1 Couderc, Romain 1 Cuadrado, Javier 1 Deabes, Wael 1 Dennis, John E. jun. 1 D’eramo, Carlo 1 Dery, Cosmas 1 Desai, Siddharth 1 Desquilbet, François 1 Diabat, Ali H. ...and 224 more Authors
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### Cited in 58 Serials

 14 Computer Methods in Applied Mechanics and Engineering 9 Information Sciences 6 Journal of Global Optimization 5 European Journal of Operational Research 4 Machine Learning 3 Computers and Fluids 3 Journal of Computational and Applied Mathematics 3 Mathematics and Computers in Simulation 3 Theoretical Computer Science 3 Algorithmica 3 Computers & Operations Research 3 Computational Optimization and Applications 3 Optimization and Engineering 2 Computer Physics Communications 2 Applied Mathematics and Computation 2 Computational Mechanics 2 Applied Mathematical Modelling 2 Mathematical Problems in Engineering 2 Natural Computing 2 Mathematical Programming Computation 2 Journal of Computational and Graphical Statistics 2 Journal of Theoretical Biology 1 Artificial Intelligence 1 Inverse Problems 1 Journal of Computational Physics 1 Bulletin of Mathematical Biology 1 Journal of Optimization Theory and Applications 1 Journal of Algorithms 1 Circuits, Systems, and Signal Processing 1 AI Communications 1 Annals of Operations Research 1 Pattern Recognition 1 Stochastic Processes and their Applications 1 Computational Economics 1 SIAM Journal on Scientific Computing 1 Open Economies Review 1 Bernoulli 1 Multibody System Dynamics 1 Optimization Methods & Software 1 CEJOR. Central European Journal of Operations Research 1 Probability in the Engineering and Informational Sciences 1 Journal of High Energy Physics 1 Methodology and Computing in Applied Probability 1 Stochastic Models 1 Journal of Machine Learning Research (JMLR) 1 4OR 1 International Journal of Wavelets, Multiresolution and Information Processing 1 Mathematical Biosciences and Engineering 1 Optimization Letters 1 Mathematical Geosciences 1 Algorithms 1 Journal of Control Science and Engineering 1 Arabian Journal for Science and Engineering 1 EURO Journal on Computational Optimization 1 International Journal of Systems Science. Principles and Applications of Systems and Integration 1 SMAI Journal of Computational Mathematics 1 Electronic Research Archive 1 SN Operations Research Forum
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### Cited in 19 Fields

 68 Operations research, mathematical programming (90-XX) 34 Computer science (68-XX) 20 Numerical analysis (65-XX) 16 Statistics (62-XX) 14 Mechanics of deformable solids (74-XX) 6 Fluid mechanics (76-XX) 6 Biology and other natural sciences (92-XX) 4 Systems theory; control (93-XX) 3 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 2 Calculus of variations and optimal control; optimization (49-XX) 2 Probability theory and stochastic processes (60-XX) 2 Quantum theory (81-XX) 2 Statistical mechanics, structure of matter (82-XX) 2 Geophysics (86-XX) 1 Combinatorics (05-XX) 1 Partial differential equations (35-XX) 1 Dynamical systems and ergodic theory (37-XX) 1 Mechanics of particles and systems (70-XX) 1 Relativity and gravitational theory (83-XX)