# Covariance Matrix Adaptation Evolution Strategy The CMA-ES (Covariance Matrix Adaptation Evolution Strategy) is an evolutionary algorithm for difficult non-linear non-convex optimization problems in continuous domain. The CMA-ES is typically applied to unconstrained or bounded constraint optimization problems, and search space dimensions between three and fifty. ```r pureCMAES(par, fun, lower = NULL, upper = NULL, sigma = 0.5, stopfitness = -Inf, stopeval = 1000*length(par)^2, ...) ``` ## Arguments - `par`: objective variables initial point. - `fun`: objective/target/fitness function. - `lower,upper`: lower and upper bounds for the parameters. - `sigma`: coordinate wise standard deviation (step size). - `stopfitness`: stop if fitness < stopfitness (minimization). - `stopeval`: stop after stopeval number of function evaluations - `...`: additional parameters to be passed to the function. ## Details The CMA-ES implements a stochastic variable-metric method. In the very particular case of a convex-quadratic objective function the covariance matrix adapts to the inverse of the Hessian matrix, up to a scalar factor and small random fluctuations. The update equations for mean and covariance matrix maximize a likelihood while resembling an expectation-maximization algorithm. ## Returns Returns a list with components `xmin` and `fmin`. Be patient; for difficult problems or high dimensions the function may run for several minutes; avoid problem dimensions of 30 and more! ## References Hansen, N. (2011). The CMA Evolution Strategy: A Tutorial. [https://arxiv.org/abs/1604.00772](https://arxiv.org/abs/1604.00772) Hansen, N., D.V. Arnold, and A. Auger (2013). Evolution Strategies. J. Kacprzyk and W. Pedrycz (Eds.). Handbook of Computational Intelligence, Springer-Verlag, 2015. ## Author(s) Copyright (c) 2003-2010 Nikolas Hansen for Matlab code PURECMAES; converted to R by Hans W Borchers. (Hansen's homepage: www.cmap.polytechnique.fr/~nikolaus.hansen/) ## Note There are other implementations of Hansen's CMAES in package `cmaes' (simplified form) and in package `parma' as cmaes() (extended form). ## See Also `cmaes::cmaes`, `parma::cmaes` ## Examples ```r ## Not run: ## Polynomial minimax approximation of data points ## (see the Remez algorithm) n <- 10; m <- 101 # polynomial of degree 10; no. of data points xi <- seq(-1, 1, length = m) yi <- 1 / (1 + (5*xi)^2) # Runge's function pval <- function(p, x) # Horner scheme outer(x, (length(p) - 1):0, "^") %*% p pfit <- function(x, y, n) # polynomial fitting of degree n qr.solve(outer(x, seq(n, 0), "^"), y) fn1 <- function(p) # objective function max(abs(pval(p, xi) - yi)) pf <- pfit(xi, yi, 10) # start with a least-squares fitting sol1 <- pureCMAES(pf, fn1, rep(-200, 11), rep(200, 11)) zapsmall(sol1$xmin) # [1] -50.24826 0.00000 135.85352 0.00000 -134.20107 0.00000 # [7] 59.19315 0.00000 -11.55888 0.00000 0.93453 print(sol1$fmin, digits = 10) # [1] 0.06546780411 ## Polynomial fitting in the L1 norm ## (or use LP or IRLS approaches) fn2 <- function(p) sum(abs(pval(p, xi) - yi)) sol2 <- pureCMAES(pf, fn2, rep(-100, 11), rep(100, 11)) zapsmall(sol2$xmin) # [1] -21.93238 0.00000 62.91083 0.00000 -67.84847 0.00000 # [7] 34.14398 0.00000 -8.11899 0.00000 0.84533 print(sol2$fmin, digits = 10) # [1] 3.061810639 ## End(Not run) ```

cmaes function