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  • 1
    Electronic Resource
    Electronic Resource
    Trust region dogleg path algorithms for unconstrained minimization (1999)
    Springer
    Annals of operations research 87 (1999), S. 407-418 
    Details
    ISSN: 1572-9338
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: 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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1018957708498
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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
    Hybrid method for nonlinear least-square problems without calculating derivatives (1990)
    Springer
    Journal of optimization theory and applications 65 (1990), S. 555-574 
    Details
    ISSN: 1573-2878
    Keywords: Nonlinear least squares ; hybrid method ; Gauss-Newton method ; BFGS method ; finite-termination property
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper presents a no-derivative modification of the hybrid Gauss-Newton-BFGS method for nonlinear least-square problems suggested initially by Al-Baali and Fletcher and modified later by Fletcher and Xu. The modification is made in such a way that, in a Gauss-Newton step, the Broyden's rank-one updating formula is used to obtain an approximate Jacobian and, in a BFGS step, the Jacobian is estimated using difference formulas. A set of numerical comparisons among the new hybrid method, the Gauss-Newton-Broyden method, and the finite-difference BFGS method is made and shows that the new hybrid method combines the better features of the Gauss-Newton-Broyden method and the finite-difference BFGS method. This paper also extends to the least-square problem the finite-termination property of the Broyden method, proved for a nonsingular system of equations by Gay and for the full-rank rectangular system of equations by Gerber and Luk.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00939566
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    Paper (German National Licenses)
    Fulltext
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