Evaluating the CMA Evolution Strategy on Multimodal Test Functions

             Nikolaus Hansen and Stefan Kern 

Abstract: 
  In this paper the performance of the CMA evolution strategy with
  rank-$\mu$-update and weighted recombination is empirically
  investigated on eight multimodal test functions.  In particular the
  effect of the population size $\lambda$ on the performance is
  investigated.  Increasing the population size remarkably improves
  the performance on six of the eight test functions. The optimal
  population size takes a wide range of values, but, with one
  exception, scales sub-linearly with the problem dimension. The
  global optimum can be located in all but one function. The
  performance for locating the global optimum scales between linear
  and cubic with the problem dimension. In a comparison to
  state-of-the-art global search strategies the CMA evolution strategy
  achieves superior performance on multimodal, non-separable test
  functions without intricate parameter tuning.

Errata: 

  Section 3.1, "Additional bounds are implemented...E.g. 
     f_Schwefel(x) + 10^4 sum_{i=1}^n \theta(|x_i|-500) x_i^2 is minimized..."
  the formula must read
     f_Schwefel(x) + 10^4 sum_{i=1}^n \theta(|x_i|-500)*(|x_i|-500)^2 is minimized...

  Section 3.2: "The starting point x^(0) is sampled uniformly..."
    must read "The starting point <x>_w^(0) is sampled uniformly..."


In: Parallel Problem Solving from Nature PPSN VIII, Proceedings, 2004.   
    http://www.springer.de/comp/lncs/index.html