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 :
-
miniɛ {r (0) i }
- l 2 :
-
miniɛ {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 i⩽x *(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
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.
Luus, R.,Optimal Control by Direct Search on Feedback Gain Matrix, Chemical Engineering Science, Vol. 29, pp. 1013–1017, 1974.
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.
Oh, S. H., andLuus, R.,Optimal Feedback Control of Time-Delay Systems, AIChE Journal, Vol. 22, pp. 140–147, 1976.
Gopalakrishnan Nair, G.,Optimal Control Using LJ Search Method, Automatic Control Theory and Applications, Vol. 4, pp. 1–6, 1976.
Gopalakrishnan Nair, G.,Suboptimal Control of Nonlinear Systems, Automatica, Vol. 14, pp. 517–519, 1978.
Gopalakrishnan Nair, G.,Suboptimal Control of Nonlinear Time-Delay Systems, Journal of Optimization Theory and Applications (to appear).
Wolfe, P.,Methods of Nonlinear Programming, Nonlinear Programming, Edited by J. Abadie, John Wiley and Sons, New York, New York, 1967.
Simmons, G. F.,Introduction to Topology and Modern Analysis, McGraw-Hill Book Company, New York, New York, 1963.
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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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DOI: https://doi.org/10.1007/BF00933384