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. (Source:

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  1. Abualigah, Laith; Diabat, Ali; Mirjalili, Seyedali; Abd Elaziz, Mohamed; Gandomi, Amir H.: The arithmetic optimization algorithm (2021)
  2. Audet, Charles; Bigeon, Jean; Couderc, Romain: Combining cross-entropy and MADS methods for inequality constrained global optimization (2021)
  3. Blanco-Cocom, Luis; Botello-Rionda, Salvador; Ordoñez, L. C.; Valdez, S. Ivvan: Robust parameter estimation of a PEMFC via optimization based on probabilistic model building (2021)
  4. Dery, Cosmas; Serletis, Apostolos: The relative importance of monetary policy, uncertainty, and financial shocks (2021)
  5. Klink, Pascal; Abdulsamad, Hany; Belousov, Boris; D’eramo, Carlo; Peters, Jan; Pajarinen, Joni: A probabilistic interpretation of self-paced learning with applications to reinforcement learning (2021)
  6. Salgotra, Rohit; Singh, Urvinder; Singh, Gurdeep; Mittal, Nitin; Gandomi, Amir H.: A self-adaptive hybridized differential evolution naked mole-rat algorithm for engineering optimization problems (2021)
  7. Shi, Feng; Neumann, Frank; Wang, Jianxin: Runtime performances of randomized search heuristics for the dynamic weighted vertex cover problem (2021)
  8. Teichmann, Jakob; Menzel, Peter; Heinig, Thomas; van den Boogaart, Karl Gerald: Modeling and fitting of three-dimensional mineral microstructures by multinary random fields (2021)
  9. Zhang, Jiaxin; Tran, Hoang; Zhang, Guannan: Accelerating reinforcement learning with a directional-Gaussian-smoothing evolution strategy (2021)
  10. Chen, Huangke; Cheng, Ran; Wen, Jinming; Li, Haifeng; Weng, Jian: Solving large-scale many-objective optimization problems by covariance matrix adaptation evolution strategy with scalable small subpopulations (2020)
  11. Hellwig, Michael; Beyer, Hans-Georg: On the steady state analysis of covariance matrix self-adaptation evolution strategies on the noisy ellipsoid model (2020)
  12. Horváth, Gábor; Horváth, Illés; Telek, Miklós: High order concentrated matrix-exponential distributions (2020)
  13. Liang, Liang: A fusion multiobjective empire split algorithm (2020)
  14. Li, Zhenhua; Zhang, Qingfu: Variable metric evolution strategies by mutation matrix adaptation (2020)
  15. Razaaly, Nassim; Persico, Giacomo; Gori, Giulio; Congedo, Pietro Marco: Quantile-based robust optimization of a supersonic nozzle for organic rankine cycle turbines (2020)
  16. Verma, Aekaansh; Wong, Kwai; Marsden, Alison L.: A concurrent implementation of the surrogate management framework with application to cardiovascular shape optimization (2020)
  17. Zhu, H.; Hu, Y. M.; Zhu, W. D.; Fan, W.; Zhou, B. W.: Multi-objective design optimization of an engine accessory drive system with a robustness analysis (2020)
  18. Breunig, U.; Baldacci, R.; Hartl, R. F.; Vidal, T.: The electric two-echelon vehicle routing problem (2019)
  19. Buet, Blanche; Mirebeau, Jean-Marie; van Gennip, Yves; Desquilbet, François; Dreo, Johann; Barbaresco, Frédéric; Leonardi, Gian Paolo; Masnou, Simon; Schönlieb, Carola-Bibiane: Partial differential equations and variational methods for geometric processing of images (2019)
  20. Gebhardt, Gregor H. W.; Kupcsik, Andras; Neumann, Gerhard: The kernel Kalman rule. Efficient nonparametric inference by recursive least-squares and subspace projections (2019)

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