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Lamarckian memetic algorithms: local optimum and connectivity structure analysis

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Abstract

Memetic algorithms (MAs) represent an emerging field that has attracted increasing research interest in recent times. Despite the popularity of the field, we remain to know rather little of the search mechanisms of MAs. Given the limited progress made on revealing the intrinsic properties of some commonly used complex benchmark problems and working mechanisms of Lamarckian memetic algorithms in general non-linear programming, we introduce in this work for the first time the concepts of local optimum structure and generalize the notion of neighborhood to connectivity structure for analysis of MAs. Based on the two proposed concepts, we analyze the solution quality and computational efficiency of the core search operators in Lamarckian memetic algorithms. Subsequently, the structure of local optimums of a few representative and complex benchmark problems is studied to reveal the effects of individual learning on fitness landscape and to gain clues into the success or failure of MAs. The connectivity structure of local optimum for different memes or individual learning procedures in Lamarckian MAs on the benchmark problems is also investigated to understand the effects of choice of memes in MA design.

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Abbreviations

f (x):

Objective or fitness function

x*:

Global optimum

x (i) :

ith element of vector x

r (t)(y|x):

Conditional probability density function of having offspring y given parent x at generation t

d(x, y):

Euclidean distance \({\|{\bf x}-{\bf y}\|=\sqrt{\sum_{i=1}^{n}{(x_i - y_i)^2}}}\) between x and y

Ψ :

A set of local optimums

B v :

Basin of attraction of local optimum v

p v (x):

Probability of converging to local optimum v from x by means of individual learning

T (x, y):

Probability of converging to local optimum y from x by means of reproduction and individual learning

C(x′, x′′):

Computational effort incurred to arrive at x′′ from x′ by means of individual learning

E[.|P]:

Expectation of a measure conditioned to population P

C v :

Maximum computational effort required to converge to local optimum starting from any point within the basin of attraction B v

n :

Number of dimensions

N :

Population size

S(.):

Selection operator

R(.):

Reproduction operator

I L(.):

Individual learning operator

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Authors and Affiliations

  1. Centre for Computational Intelligence, Division of Information System, School of Computer Engineering, Nanyang Technological University, Blk N4, B1a-02, Nanyang Avenue, Singapore, 639798, Singapore

    Minh Nghia Le & Yew-Soon Ong

  2. Honda Research Institute Europe GmbH, Carl-Legien-Strasse 30, 63073, Offenbach/Main, Germany

    Yaochu Jin & Bernhard Sendhoff

Authors
  1. Minh Nghia Le
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  2. Yew-Soon Ong
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  3. Yaochu Jin
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  4. Bernhard Sendhoff
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Correspondence to Yew-Soon Ong.

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Le, M.N., Ong, YS., Jin, Y. et al. Lamarckian memetic algorithms: local optimum and connectivity structure analysis. Memetic Comp. 1, 175–190 (2009). https://doi.org/10.1007/s12293-009-0016-9

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