Evolutionary programming based on non-uniform mutation

doi:10.1016/j.amc.2006.06.107

Applied Mathematics and Computation

Volume 192, Issue 1, 1 September 2007, Pages 1-11

https://doi.org/10.1016/j.amc.2006.06.107 Get rights and content

Abstract

A new evolutionary programming using non-uniform mutation instead of Gaussian, Cauchy and Lévy mutations is proposed. Evolutionary programming with non-uniform mutation (NEP) has the merits of searching the space uniformly at the early stage and very locally at the later stage during the programming. For a suite of 14 benchmark problems, NEP outperforms the improved evolutionary programming using mutation based on Lévy probability distribution (ILEP) for multimodal functions with many local minima while being comparable to ILEP in performance for unimodal and multimodal functions with only a few minima. The detailed theoretical analysis of the executing process of NEP and the expected step size on non-uniform mutation are given.

Introduction

Inspired by biological evolution and natural selection, intelligent computation algorithms are proposed to provide powerful tools for solving many difficult problems. Genetic algorithms (GAs) [1], [2], [3], [4], [5], evolution strategies (ESs) [6], [7], and evolutionary programming (EP) [8], [9] are especially noticeable among them. In GAs, the crossover operator plays the major role and mutation is always seen as an assistant operator. In ESs and EP, however, mutation has been considered as the main operator. GAs usually adopt a high crossover probability and a low mutation probability, while ESs and EP apply mutation to every individual. One, two, multipoint, or uniform crossover and uniform mutation [4], [10] are often used in GAs. Besides Gaussian mutation [8], [9], self-adaptation mutations [11], [12], self-adaptation rules from ESs [13], Cauchy [14] and Lévy probability distribution [15] mutations have also been incorporated into EP. These new operators can greatly enhance the performance of the algorithms on some specific problems. About some recent progress in algorithms, please refer to [16], [17], [18], [19], [20].

In this paper, we focus on EP. In classical EP (CEP), each parent generates an offspring via Gaussian mutation, and better individuals among parents and offspring become the parents of the next generation. In [14], a “fast evolutionary programming (FEP)” using mutation based on Cauchy distribution was proposed. FEP converges faster to a better solution than CEP for a number of multimodal functions with many local minima [14]. We know that Cauchy probability distribution is a special case of Lévy probability distribution. Lee and Yao [15] generalized FEP and proposed an EP by using mutation based on Lévy probability distribution (LEP).

Yao and co-workers [14], [15] claimed that “higher probability of making longer jumps” is a key point that makes FEP and LEP perform better than CEP in certain cases, and that “longer jumps” are detrimental if the current point is already very close to the global optimum. Inspired by this comment, we borrow non-uniform mutation operators from [10] in order to consider a new EP based on this kind of operators (NEP). Non-uniform mutation operators have the feature of searching the space uniformly at the early stage and very locally at the later stage.

Theoretical analysis on how NEP works is given based on probability theory. First, the expected step size of non-uniform mutation is calculated. Its derivative with respect to the generation variable t is less than zero, which implies the monotonously decreasing exploring region with the progress of the algorithm. Second, we obtain a quantitative description of the step size through analyzing the step size equation. Theoretical analysis also supports the fact that NEP is not sensitive to the search space size [21].

The rest of the paper is organized as follows: In Section 2, NEP algorithm is presented. In Section 3, comprehensive experiments are done to compare NEP with the recently proposed ILEP [15]. In Section 4, theoretical analysis on the executing process of NEP is given. An adaptive NEP algorithm is proposed in Section 5 and conclusions are reached in the last section.

Section snippets

Non-uniform evolutionary programming (NEP)

In this section, we introduce an evolutionary programming algorithm based on a non-uniform mutation operator.

Experiments and analysis

In our algorithm we let the sum of the crossover probability (pc) and the mutation probability (pm) be 1 to ensure that NEP generates the same number of offspring compared with other EPs in terms of probabilistic sense. Precisely, we let pc be 0.4 and pm 0.6.

Theoretical analysis on the executing process of NEP

In Section 3, experiments show that NEP is fast and robust for most benchmark functions tested. In this section, we try to give the underling reasons by conducting theoretical analysis.

Suppose that X = {x₁, … , x_k, … , x_m} is a real-coded chromosome in the population and the component x_k (whose lower and upper bounds are L_k, U_k) is selected for variation through some mutation operation and $x_{k}^{'}$ is obtained. A decimal fraction r is uniformly and randomly generated in [0, 1].

The Influence of the parameter b

Besides t in Eq. (5), there is another parameter b that we have been treating it as a constant. However, it is better to treat b as a parameter. In fact, b determines the degree of non-uniformity [10]. Next we will analyze this parameter to show how it affects the search step. For a given t in Eq. (5), let $g (b) ≔ \frac{U_{k} - L_{k}}{2} \times (1 - \frac{1}{1 + (1 - \frac{t}{T})^{b}}) = \frac{U_{k} - L_{k}}{2} \times \frac{1}{1 + {(1 - \frac{t}{T})}^{- b}},$ then we have $\frac{\partial g (b)}{\partial b} = \frac{U_{k} - L_{k}}{2} \times \frac{- 1}{(1 + (1 - \frac{t}{T})^{- b})^{2}} \times {(1 - \frac{t}{T})}^{- b} \times \ln (1 - \frac{t}{T}) = \frac{U_{k} - L_{k}}{2} \times \frac{\ln (\frac{T}{T - t}) \times {(1 - \frac{t}{T})}^{- b}}{{(1 + {(1 - \frac{t}{T})}^{- b})}^{2}} > 0 .$

It follows from Eq. (12) that the expected step size of the

Conclusion

In this paper, a new EP algorithm is proposed based on a non-uniform mutation operator. Comparisons with the newly proposed ILEP based on Lévy probability mutation, NEP is faster and more robust for most multimodal benchmark functions tested.

Detailed theoretical analysis on its working schemes of NEP is presented. NEP searches the space uniformly at the early stage of the algorithm and very locally at the later stage. The probabilistic gradual decreasing jump length makes the algorithm

Acknowledgements

The first author thanks Prof. Xin Yao, Dr. Jun He, Jin Li, Xiaoli Li, Yong Xu and Dan Chen in the School of Computer Science, the University of Birmingham, UK, for their helps and suggestions. Partially supported by a National Key Basic Research Project of China and by a USA NSF grant CCR-0201253.

References (25)

Y. Liang et al.
Evolution strategies with exclusion-based selection operators and a Fourier series auxiliary function
Appl. Math. Comput.
(2006)
L. Wang et al.
Hybrid genetic algorithm based on quantum computing for numerical optimization and parameter estimation
Appl. Math. Comput.
(2005)
D. Whitley et al.
Evaluating evolutionary algorithms
Artif. Intell.
(1996)
A.S. Fraser
Simulation of genetic system by automatic digital computers. I. Introduction
Austral. J. Biol. Sci.
(1957)
H.J. Bremerman
Optimization through evolution and recombination
J.H. Holland
Adaptation in Natural and Artificial Systems
(1975)
D.E. Goldberg
Genetic Algorithms in Search, Optimization and Machine Learning
(1989)
T. Kuo et al.
A genetic algorithm with disruptive selection
IEEE Trans. Syst. Man Cybern.
(1996)
H.P. Schwefel
Numerical Optimization of Computer Models
(1977)
X. Yao et al.
Fast evolution strategies
Contr. Cybern.
(1997)

L.J. Fogel et al.

Artificial Intelligence through Simulated Evolution

(1966)

D.B. Fogel

Evolutionary Computation: Toward a New Philosophy of Machine Intelligence

(1995)

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View full text

Evolutionary programming based on non-uniform mutation

Abstract

Introduction

Section snippets

Non-uniform evolutionary programming (NEP)

Experiments and analysis

Theoretical analysis on the executing process of NEP

The Influence of the parameter b

Conclusion

Acknowledgements

Appl. Math. Comput.

Appl. Math. Comput.

Artif. Intell.

Simulation of genetic system by automatic digital computers. I. Introduction

Austral. J. Biol. Sci.

Optimization through evolution and recombination

Adaptation in Natural and Artificial Systems

Genetic Algorithms in Search, Optimization and Machine Learning

A genetic algorithm with disruptive selection

IEEE Trans. Syst. Man Cybern.

Numerical Optimization of Computer Models

Fast evolution strategies

Contr. Cybern.

Artificial Intelligence through Simulated Evolution

Evolutionary Computation: Toward a New Philosophy of Machine Intelligence