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On the convergence of the LJ search method

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Abstract

The convergence of the Luus-Jaakola search method for unconstrained optimization problems is established.

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Abbreviations

E n :

Euclideann-space

f :

Gradient off(x)

2 f :

Hessian matrix

(·)T :

Transpose of (·)

I :

Index set {1, 2, ...,n}

[x *(j) i1 ]:

Point around which search is made in the (j + 1)th iteration, i.e., [x *(j) 1l ,x *(j) 2l ,...,x *(j) n1 ]

r (i) i :

Range ofx *(i) il in the (j + 1)th iteration

l 1 :

min {r (0) i }

l 2 :

min {r (0) i }

A j :

Region of search in thejth iteration, i.e., {x εE n:x *(j-1) il −0.5r (j-1) i x ix *(j-1) il +0.5r (j-1) i ,i εI}

S αj :

Closed sphere with center origin and radiusα j

ɛ :

Reduction factor in each iteration

θ :

1−ɛ

Γ(·):

Gamma function

References

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Communicated by R. Jackson

Many discussions with Dr. S. N. Iyer, Professor of Electrical Engineering, College of Engineering, Trivandrum, India, are gratefully acknowledged. The author has great pleasure to thank Dr. K. Surendran, Professor, Department of Electrical Engineering, P.S.G. College of Technology, Coimbatore, India, for suggesting this work.

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Gopalakrishnan Nair, G. On the convergence of the LJ search method. J Optim Theory Appl 28, 429–434 (1979). https://doi.org/10.1007/BF00933384

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