On the convergence of the LJ search method

  • G. Gopalakrishnan Nair
Technical Note

Abstract

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

Key Words

Optimal search techniques convergence proofs sufficient conditions Cantor's intersection theorem 

Notation

En

Euclideann-space

f

Gradient off(x)

2f

Hessian matrix

(·)T

Transpose of (·)

I

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

[xi1*(j)]

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

ri(i)

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

l1

min {r i (0) }

l2

min {r i (0) }

Aj

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

Sαj

Closed sphere with center origin and radiusαj

ɛ

Reduction factor in each iteration

θ

1−ɛ

Γ(·)

Gamma function

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References

  1. 1.
    Luus, R., andJaakola, T. H. I.,Optimization by Direct Search and Systematic Reduction of the Size of Search Region, AIChE Journal, Vol. 19, pp. 760–766, 1973.Google Scholar
  2. 2.
    Luus, R.,Optimal Control by Direct Search on Feedback Gain Matrix, Chemical Engineering Science, Vol. 29, pp. 1013–1017, 1974.Google Scholar
  3. 3.
    Luus, R.,Practical Approach to Time-Optimal Control of Nonlinear Systems, Industrial and Engineering Chemistry, Process Design and Development, Vol. 13, pp. 405–408, 1974.Google Scholar
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    Oh, S. H., andLuus, R.,Optimal Feedback Control of Time-Delay Systems, AIChE Journal, Vol. 22, pp. 140–147, 1976.Google Scholar
  5. 5.
    Gopalakrishnan Nair, G.,Optimal Control Using LJ Search Method, Automatic Control Theory and Applications, Vol. 4, pp. 1–6, 1976.Google Scholar
  6. 6.
    Gopalakrishnan Nair, G.,Suboptimal Control of Nonlinear Systems, Automatica, Vol. 14, pp. 517–519, 1978.Google Scholar
  7. 7.
    Gopalakrishnan Nair, G.,Suboptimal Control of Nonlinear Time-Delay Systems, Journal of Optimization Theory and Applications (to appear).Google Scholar
  8. 8.
    Wolfe, P.,Methods of Nonlinear Programming, Nonlinear Programming, Edited by J. Abadie, John Wiley and Sons, New York, New York, 1967.Google Scholar
  9. 9.
    Simmons, G. F.,Introduction to Topology and Modern Analysis, McGraw-Hill Book Company, New York, New York, 1963.Google Scholar

Copyright information

© Plenum Publishing Corporation 1979

Authors and Affiliations

  • G. Gopalakrishnan Nair
    • 1
  1. 1.Department of MathematicsCollege of EngineeringTrivandrumIndia

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