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A restricted trust region algorithm for unconstrained optimization. (English) Zbl 0556.90075
This paper proposes an efficient implementation of a trust-region-like algorithm. The trust region is restricted to an appropriately chosen two- dimensional subspace. Convergence properties are discussed and numerical results are reported.

90C30 Nonlinear programming
49M37 Numerical methods based on nonlinear programming
65K05 Numerical mathematical programming methods
GQTPAR; minpack
Full Text: DOI
[1] Levenberg, K.,A Method for the Solution of Certain Nonlinear Problems in Least Squares, Quarterly of Applied Mathematics, Vol. 2, pp. 164-168, 1944. · Zbl 0063.03501
[2] Marquardt, D. W.,An Algorithm for Least Squares Estimation of Nonlinear Parameters, SIAM Journal on Applied Mathematics, Vol. 11, pp. 431-441, 1963. · Zbl 0112.10505
[3] Goldfeld, S. M., Quandt, R. E., andTrotter, H. F.,Maximization by Quadratic Hill Climbing, Econometrica, Vol. 34, pp. 541-551, 1966. · Zbl 0145.40901
[4] Hebden, M. D.,An Algorithm for Minimization Using Exact Second Derivatives, Atomic Energy Research Establishment, Harwell, England, Report TP-515, 1973.
[5] Moré, J. J.,The Levenberg-Marquardt Algorithm: Implementation and Theory, Proceedings of the Dundee Conference on Numerical Analysis, Edited by G. A. Watson, Springer-Verlag, Berlin, Germany, pp. 105-116, 1978.
[6] Gay, D. M.,Computing Optimal Locally Constrained Steps, SIAM Journal on Sciences and Statistical Computations, Vol. 2, pp. 186-197, 1981. · Zbl 0467.65027
[7] Sorensen, D. C.,Newton’s Method with a Model Trust Region Modification, SIAM Journal on Numerical Analysis, Vol. 19, pp. 409-426, 1982. · Zbl 0483.65039
[8] Moré, J. J., andSorensen, D. C.,Computing a Trust Region Step, SIAM Journal on Sciences and Statistical Computations, Vol. 4, pp. 553-572, 1983. · Zbl 0551.65042
[9] Bulteau, J. P., andVial, J. P.,Curvilinear Path and Trust Region in Unconstrained Optimization: A Convergence Analysis, Université Catholique de Louvain, Louvain-la-Neuve, Belgium, CORE Discussion Paper No. 8337, 1983.
[10] Bulteau, J. P., andVial, J. P.,Unconstrained Optimization by Approximation of a Projected Gradient Path, Université Catholique de Louvain, Louvain-la-Neuve, Belgium, CORE Discussion Paper No. 8352, 1983.
[11] Moré, J. J.,Recent developments in Algorithms and Software for Trust Region Methods, Mathematical Programming?The State of the Art?Bonn 1982, Edited by A. Backen, M. Grötschel, and B. Korte, Springer-Verlag, Berlin, Germany, pp. 258-287, 1983.
[12] Powell, M. J. D.,A Hybrid Method for Nonlinear Equations, Numerical Methods for Nonlinear Algebraic Equations, Edited by P. Rabinowitz, Gordon and Breach, London, England, pp. 87-114, 1970.
[13] Powell, M. J. D.,Convergence Properties of a Class of Minimization Algorithms, Nonlinear Programming 2, Edited by O. L. Mangasarian, R. R. Meyer, and S. M. Robinson, Academic Press, New York, New York, pp. 1-27, 1975.
[14] Fletcher, R., andFreeman, T. L.,A Modified Newton Method for Minimization, Journal of Optimization Theory and Applications, Vol. 23, pp. 357-372, 1977. · Zbl 0348.65058
[15] Moré, J. J., andSorensen, D. C.,On the Use of Directions of Negative Curvature in a Modified Newton Method, Mathematical Programming, Vol. 16, pp. 1-20, 1979. · Zbl 0394.90093
[16] Goldfarb, D.,Curvilinear Path Steplength Algorithms for Minimization Which Use Directions of Negative Curvature, Mathematical Programming, Vol. 18, pp. 31-40, 1980. · Zbl 0428.90068
[17] Bunch, J. R., andParlett, B. N.,Direct Methods for Solving Symmetric Indefinite Systems of Linear Equations, SIAM Journal of Numerical Analysis, Vol. 8, pp. 639-655, 1971. · Zbl 0222.65038
[18] Fletcher, R.,Factorizing Symmetric Indefinite Matrices, Linear Algebra and Its Applications, Vol. 14, pp. 257-272, 1976. · Zbl 0336.65022
[19] Fletcher, R., andPowell, M. J. D.,On the Modifications of LDL T Factorizations, Mathematics of Computation, Vol. 28, pp. 1067-1087, 1974. · Zbl 0293.65018
[20] Gill, P. E., andMurray, W.,Modification of Matrix Factorizations after a Rank-One Change, The State of the Art in Numerical Analysis, Edited by D. A. H. Jacobs, Academic Press, London, England, pp. 55-83, 1977.
[21] Zang, I.,A New Arc Algorithm for Unconstrained Optimization, Mathematical Programming, Vol. 15, pp. 36-52, 1978. · Zbl 0392.90076
[22] Moré, J. J., Garbow, B. S., andHillstrom, K. E.,Testing Unconstrained Optimization Software, ACM Transactions on Mathematical Software, Vol. 7, pp. 17-41, 1981. · Zbl 0454.65049
[23] Sorensen, D.,Updating the Symmetric Indefinite Factorization with Applications in a Modified Newton’s Method, Argonne National Laboratory, Argonne, Illinois, Report No. ANL-77-49, 1977.
[24] Engvall, J. L.,Numerical Algorithm for Solving Overdetermined Systems of Nonlinear Equations, NASA, Document No. N70-35600, 1970.
[25] Zangwill, W. J.,Nonlinear Programming via Penalty Functions, Management Science, Vol. 13, pp. 344-358, 1967. · Zbl 0171.18202
[26] Davidon, W. C.,New Least Square Algorithms, Journal of Optimization Theory and Applications, Vol. 18, pp. 187-188, 1976. · Zbl 0299.65037
[27] Hock, W., andSchittkowski, K.,Test Examples for Nonlinear Programming Codes, Springer-Verlag, Berlin, Germany, 1981. · Zbl 0452.90038
[28] Dixon, L. C. W.,Conjugate Directions without Linear Searches, Journal of the Institute of Mathematics and Its Applications, Vol. 11, pp. 317-328, 1973. · Zbl 0259.65060
[29] Polak, E.,A Modified Secant Method for Unconstrained Minimization, Mathematical Programming, Vol. 6, pp. 264-280, 1974. · Zbl 0287.90025
[30] Spedicato, E.,Computational Experience with Quasi-Newton Algorithms for Minimization Problems of Moderately Large Size, CISE, Segrate, Italy, Report No. CISE-N-175, 1975. · Zbl 0397.90088
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