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  • 1
    Electronic Resource
    Electronic Resource
    Globally Convergent Inexact Generalized Newton Method for First-Order Differentiable Optimization Problems (2000)
    Springer
    Journal of optimization theory and applications 106 (2000), S. 551-568 
    Details
    ISSN: 1573-2878
    Keywords: nonsmooth optimization ; inexact Newton methods ; generalized Newton methods ; global convergence ; superlinear rate
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: 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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1004605428858
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    Paper (German National Licenses)
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  • 2
    Electronic Resource
    Electronic Resource
    Scaled Optimal Path Trust-Region Algorithm (1999)
    Springer
    Journal of optimization theory and applications 102 (1999), S. 127-146 
    Details
    ISSN: 1573-2878
    Keywords: Trust-region methods ; unconstrained optimization ; Bunch–Parlett factorization ; optimal paths ; global convergence ; local convergence rates
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: 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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1021846613163
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    Paper (German National Licenses)
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  • 3
    Electronic Resource
    Electronic Resource
    Projected quasi-Newton algorithm with trust region for constrained optimization (1990)
    Springer
    Journal of optimization theory and applications 67 (1990), S. 369-393 
    Details
    ISSN: 1573-2878
    Keywords: Two-sided projected Hessians ; trust regions ; differentiable penalty functions ; global convergence ; two-step Q-superlinear rate ; constrained optimization
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: 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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00940481
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    Paper (German National Licenses)
    Fulltext
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