Abstract
In this paper, we propose a class of convenient curvilinear search algorithms to solve trust region problems arising from unconstrained optimization. The curvilinear paths we set are dogleg paths, generated mainly by employing Bunch‐Parlett factorization for general symmetric matrices which may be indefinite. These algorithms are easy to use and globally convergent. It is proved that these algorithms satisfy the first‐ and second‐order stationary point convergence properties and that the convergence rate is quadratic under commonly used conditions.
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J.P. Bulteau and J.Ph. Vial, Unconstrained optimization by approximation of a projected gradient path, Research Report 8352, Center for Operations Research and Econometrics, Université Catholique de Louvain, Belgium, 1983.
J.P. Bulteau and J.Ph. Vial, A restricted trust region algorithm for unconstrained optimization, Journal of Optimization Theory and Applications 47(1985)413-434.
J.P. Bulteau and J.Ph. Vial, Curvilinear path and trust region in unconstrained optimization: A convergence analysis, Mathematical Programming Study 30(1987)82-101.
J.R. Bunch and B.N. Parlett, Direct method for solving symmetric indefinite systems of linear equations, SIAM Journal on Numerical Analysis 8(1971) 639-655.
J.R. Bunch, L. Kaufman and B.N. Parlett, Decomposition of a symmetric matrix, Numerische Mathematik 27(1976)95-109.
R.H. Byrd, R.B. Schnabel and G.A. Shultz, Approximate solution of the trust region problem by minimization over two-dimensional subspaces, Mathematical Programming 40(1988) 247-263.
J.E. Dennis and H.H.W Mei, Two new unconstrained optimization algorithms which use function and gradient values, Journal of Optimization Theory and Applications 28(1979)453-482.
J.J. Moré, B.S. Garbow and K.E. Hillstrom, Testing unconstrained optimization software, ACM Transactions on Mathematical Software 7(1981)17-41.
S.G. Nash, Newton-type minimization via the Lanczos method, SIAM J. on Numerical Analysis 21(1984)770-788.
M.J.D. Powell, A hybrid method for nonlinear equations, in: Numerical Methods for Nonlinear Algebraic Equations, ed. Ph. Rabinowitz, Gordon and Breach, 1970, pp. 87-114.
G.A. Shultz, R.B. Schnabel and R.H. Byrd, A family of trust-region-based algorithms for unconstrained minimization with strong global convergence properties, SIAM Journal on Numerical Analysis 22(1985)47-67.
J.-Z. Zhang and C.-X. Xu, A class of indefinite dogleg path methods for unconstrained minimization, Research Report MA-94-09, City University of Hong Kong, Hong Kong, 1994.
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Zhang, JZ., Xu, CX. Trust region dogleg path algorithms for unconstrained minimization. Annals of Operations Research 87, 407–418 (1999). https://doi.org/10.1023/A:1018957708498
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DOI: https://doi.org/10.1023/A:1018957708498