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Digitale Medien

Scaled Optimal Path Trust-Region Algorithm (1999)

Xu, C. X. ; Zhang, J. Z.

Springer

Journal of optimization theory and applications 102 (1999), S. 127-146

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ISSN: 1573-2878

Schlagwort(e): Trust-region methods ; unconstrained optimization ; Bunch–Parlett factorization ; optimal paths ; global convergence ; local convergence rates

Quelle: Springer Online Journal Archives 1860-2000

Thema: Mathematik

Notizen: Abstract Trust-region algorithms solve a trust-region subproblem at each iteration. Among the methods solving the subproblem, the optimal path algorithm obtains the solution to the subproblem in full-dimensional space by using the eigenvalues and eigenvectors of the system. Although the idea is attractive, the existing optimal path method seems impractical because, in addition to factorization, it requires either the calculation of the full eigensystem of a matrix or repeated factorizations of matrices at each iteration. In this paper, we propose a scaled optimal path trust-region algorithm. The algorithm finds a solution of the subproblem in full-dimensional space by just one Bunch–Parlett factorization for symmetric matrices at each iteration and by using the resulting unit lower triangular factor to scale the variables in the problem. A scaled optimal path can then be formed easily. The algorithm has good convergence properties under commonly used conditions. Computational results for small-scale and large-scale optimization problems are presented which show that the algorithm is robust and effective.

Materialart: Digitale Medien

URL: http://dx.doi.org/10.1023/A:1021846613163

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Digitale Medien

Projected quasi-Newton algorithm with trust region for constrained optimization (1990)

Zhang, J. Z. ; Zhu, D. T.

Springer

Journal of optimization theory and applications 67 (1990), S. 369-393

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ISSN: 1573-2878

Schlagwort(e): Two-sided projected Hessians ; trust regions ; differentiable penalty functions ; global convergence ; two-step Q-superlinear rate ; constrained optimization

Quelle: Springer Online Journal Archives 1860-2000

Thema: Mathematik

Notizen: Abstract In Ref. 1, Nocedal and Overton proposed a two-sided projected Hessian updating technique for equality constrained optimization problems. Although local two-step Q-superlinear rate was proved, its global convergence is not assured. In this paper, we suggest a trust-region-type, two-sided, projected quasi-Newton method, which preserves the local two-step superlinear convergence of the original algorithm and also ensures global convergence. The subproblem that we propose is as simple as the one often used when solving unconstrained optimization problems by trust-region strategies and therefore is easy to implement.

Materialart: Digitale Medien

URL: http://dx.doi.org/10.1007/BF00940481

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Globally Convergent Inexact Generalized Newton Method for First-Order Differentiable Optimization Problems (2000)

Pu, D. ; Zhang, J.

Springer

Journal of optimization theory and applications 106 (2000), S. 551-568

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ISSN: 1573-2878

Schlagwort(e): nonsmooth optimization ; inexact Newton methods ; generalized Newton methods ; global convergence ; superlinear rate

Quelle: Springer Online Journal Archives 1860-2000

Thema: Mathematik

Notizen: Abstract Motivated by the method of Martinez and Qi (Ref. 1), we propose in this paper a globally convergent inexact generalized Newton method to solve unconstrained optimization problems in which the objective functions have Lipschitz continuous gradient functions, but are not twice differentiable. This method is implementable, globally convergent, and produces monotonically decreasing function values. We prove that the method has locally superlinear convergence or even quadratic convergence rate under some mild conditions, which do not assume the convexity of the functions.

Materialart: Digitale Medien

URL: http://dx.doi.org/10.1023/A:1004605428858

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