zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
On stability and convergence of the population-dynamics in differential evolution. (English) Zbl 1200.68185

Summary: Theoretical analysis of the dynamics of evolutionary algorithms is believed to be very important to understand the search behavior of evolutionary algorithms and to develop more efficient algorithms. In this paper we investigate the dynamics of a canonical Differential Evolution (DE) algorithm with DE/rand/1 type mutation and binomial crossover. Differential Evolution (DE) is well known as a simple and efficient algorithm for global optimization over continuous spaces. Since its inception in 1995, DE has been finding many important applications in real-world optimization problems from diverse domains of science and engineering.

The paper proposes a simple mathematical model of the underlying evolutionary dynamics of a one-dimensional DE-population. The model shows that the fundamental dynamics of each search-agent (parameter vector) in DE employs the gradient-descent type search strategy (although it uses no analytical expression for the gradient itself), with a learning rate parameter that depends on control parameters like scale factor F and crossover rate CR of DE. The stability and convergence behavior of the proposed dynamics is analyzed in the light of Lyapunov’s stability theorems very near to the isolated equilibrium points during the final stages of the search. Empirical studies over simple objective functions are conducted in order to validate the theoretical analysis.

MSC:
68T05Learning and adaptive systems
65K05Mathematical programming (numerical methods)
68T20AI problem solving (heuristics, search strategies, etc.)
90C52Methods of reduced gradient type
90C59Approximation methods and heuristics