adagio (version 0.9.2)

CMAES: Covariance Matrix Adaptation Evolution Strategy

Description

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.

Usage

pureCMAES(par, fun, lower = NULL, upper = NULL, sigma = 0.5,
                    stopfitness = -Inf, stopeval = 1000*length(par)^2, ...)

Value

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!

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.

Author

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/)

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.

References

Hansen, N. (2011). The CMA Evolution Strategy: A Tutorial.
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.

See Also

cmaes::cmaes, parma::cmaes

Examples

if (FALSE) {
##  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
}