The Rosenbrock function is a well-known benchmark for numerical optimization problems, which is frequently used to assess the performance of Evolutionary Algorithms. The classical Rosenbrock function, which is a two-dimensional unimodal function, has been extended to higher dimensions in recent years. Many researchers take the high-dimensional Rosenbrock function as a unimodal function by instinct. In 2001 and 2002, Hansen and Deb found that the Rosenbrock function is not a unimodal function for higher dimensions although no theoretical analysis was provided. This paper shows that the n-dimensional (n = 4 30) Rosenbrock function has 2 minima, and analysis is proposed to verify this. The local minima in some cases are presented. In addition, this paper demonstrates that one of the local minima for the 20-variable Rosenbrock function found by Deb might not in fact be a local minimum.
[1]
K. Dejong,et al.
An analysis of the behavior of a class of genetic adaptive systems
,
1975
.
[2]
Xin Yao,et al.
Evolutionary programming made faster
,
1999,
IEEE Trans. Evol. Comput..
[3]
E. Kreyszig,et al.
Advanced Engineering Mathematics.
,
1974
.
[4]
Ioannis B. Theocharis,et al.
Microgenetic algorithms as generalized hill-climbing operators for GA optimization
,
2001,
IEEE Trans. Evol. Comput..
[5]
Kalyanmoy Deb,et al.
A Computationally Efficient Evolutionary Algorithm for Real-Parameter Optimization
,
2002,
Evolutionary Computation.
[6]
Nikolaus Hansen,et al.
Completely Derandomized Self-Adaptation in Evolution Strategies
,
2001,
Evolutionary Computation.
[7]
David E. Goldberg,et al.
Genetic Algorithms in Search Optimization and Machine Learning
,
1988
.