doc.txt

GENERAL PURPOSE: 
The CMA-ES (Evolution Strategy with Covariance Matrix Adaptation) is a
robust search/optimization method. The goal is to minimize a given
objective function, f: R^N -> R.  The CMA-ES should be applied, if
e.g. BFGS and/or conjugate gradient methods fail due to a rugged
search landscape (e.g. discontinuities, outliers, noise, local optima,
etc.). Learning the covariance matrix in the CMA-ES is similar to
learning the inverse Hessian matrix in a quasi-Newton method. On
smooth landscapes the CMA-ES is roughly ten times slower than BFGS,
assuming derivatives are not directly available. For up to N=10
parameters the simplex direct search method (Nelder & Mead) is
sometimes faster, but less robust than CMA-ES.  On considerably hard
problems the search (a single run) is expected to take between 100*N
and 300*N^2 function evaluations. But you might be lucky...

APPLICATION REMARK: 

The adaptation of the covariance matrix (e.g. by the CMA) is
equivalent to a general linear transformation of the problem
variables. Nevertheless, every problem specific knowledge about the
best problem transformation should be exploited before starting the
search procedure and an appropriate a priori transformation should be
applied to the problem. In particular a decision should be taken
whether variables, which are positive by nature, should be taken in
the log scale. A hard lower variable bound can also be realized by
taking the square. All variables should be re-scaled such that they
"live" in a similar search range width (for example, but not
necessarily between zero and one), such that the initial standard
deviation can be chosen the same for all variables.


LINKS
  http://www.lri.fr/~hansen/cmaesintro.html
  http://www.lri.fr/~hansen/publications.html

TUTORIAL:
	http://www.lri.fr/~hansen/cmatutorial.pdf

REFERENCES:

Hansen, N, and S. Kern (2004).  Evaluating the CMA Evolution
  Strategy on Multimodal Test Functions. In: Eighth International
  Conference on Parallel Problem Solving from Nature PPSN VIII,
  Proceedings, pp. 282-291, Berlin: Springer

Hansen, N., S.D. Müller and P. Koumoutsakos (2003): Reducing the
  Time Complexity of the Derandomized Evolution Strategy with
  Covariance Matrix Adaptation (CMA-ES). Evolutionary Computation,
  11(1).

Hansen, N. and A. Ostermeier (2001). Completely Derandomized
  Self-Adaptation in Evolution Strategies. Evolutionary Computation,
  9(2), pp. 159-195.

Hansen, N. and A. Ostermeier (1996). Adapting arbitrary normal
  mutation distributions in evolution strategies: The covariance
  matrix adaptation. In Proceedings of the 1996 IEEE International
  Conference on Evolutionary Computation, pp. 312-317.