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  • 1
    Book
    Book
    Numerical linear algebra and optimization (1991)
    Reading, MA u.a. :Addison-Wesley,
    Details
    Title: Numerical linear algebra and optimization
    Author: Gill, Philip E.
    Contributer: Murray, Walter , Wright, Margaret H.
    Publisher: Reading, MA u.a. :Addison-Wesley,
    Year of publication: 1991
    Pages: 426 S.
    Type of Medium: Book
    URL: http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0714.65063&format=complete
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  • 2
    Book
    Book
    Practical optimization (1981)
    London u.a. :Academic Press,
    Details
    Title: Practical optimization
    Author: Gill, Philip E.
    Contributer: Murray, Walter , Wright Margaret H.
    Publisher: London u.a. :Academic Press,
    Year of publication: 1981
    Pages: 401 S.
    Type of Medium: Book
    URL: http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0503.90062&format=complete
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  • 3
    Electronic Resource
    Electronic Resource
    Properties of a representation of a basis for the null space (1985)
    Springer
    Mathematical programming 33 (1985), S. 172-186 
    Details
    ISSN: 1436-4646
    Keywords: Nonlinear Optimization ; Matrix Factorization ; Null-Space Continuity
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Given a rectangular matrixA(x) that depends on the independent variablesx, many constrained optimization methods involve computations withZ(x), a matrix whose columns form a basis for the null space ofA T(x). WhenA is evaluated at a given point, it is well known that a suitableZ (satisfyingA T Z = 0) can be obtained from standard matrix factorizations. However, Coleman and Sorensen have recently shown that standard orthogonal factorization methods may produce orthogonal bases that do not vary continuously withx; they also suggest several techniques for adapting these schemes so as to ensure continuity ofZ in the neighborhood of a given point. This paper is an extension of an earlier note that defines the procedure for computingZ. Here, we first describe howZ can be obtained byupdating an explicit QR factorization with Householder transformations. The properties of this representation ofZ with respect to perturbations inA are discussed, including explicit bounds on the change inZ. We then introduceregularized Householder transformations, and show that their use implies continuity of the full matrixQ. The convergence ofZ andQ under appropriate assumptions is then proved. Finally, we indicate why the chosen form ofZ is convenient in certain methods for nonlinearly constrained optimization.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01582244
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  • 4
    Electronic Resource
    Electronic Resource
    A practical anti-cycling procedure for linearly constrained optimization (1989)
    Springer
    Mathematical programming 45 (1989), S. 437-474 
    Details
    ISSN: 1436-4646
    Keywords: Linear programming ; simplex method ; active-set methods ; degeneracy ; cycling
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract A procedure is described for preventing cycling in active-set methods for linearly constrained optimization, including the simplex method. The key ideas are a limited acceptance of infeasibilities in all variables, and maintenance of a “working” feasibility tolerance that increases over a long sequence of iterations. The additional work per iteration is nominal, and “stalling” cannot occur with exact arithmetic. The method appears to be reliable, based on computational results for the first 53 linear programming problems in theNetlib set.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01589114
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  • 5
    Electronic Resource
    Electronic Resource
    Implementation issues in solving nonlinear equations for two-point boundary value problems (1990)
    Springer
    Computing 45 (1990), S. 17-37 
    Details
    ISSN: 1436-5057
    Keywords: 65L10 (boundary value problems) ; 68N99 (mathematical software) ; Boundary value problems ; nonlinear algebraic equations ; merit functions ; mathematical software ; watchdog strategy
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Komplizierte numerische Methoden enthalten oft Teilprobleme, die sich leicht mathematisch formulieren lassen, die aber schwierig in Computerprogramme umgewandelt werden können. Mehrere solche Fragestellungen traten bei der Erstellung eines Newton-Verfahrens als Bestandteil eines Differenzenverfahrens für 2-Punkt Randwert-probleme auf. Wir beschreiben die praktischen und theoretischen Überlegungen, welche den Entscheidungen zugrunde liegen, die schließlich zum Computerprogramm führten. Insbesondere betonen wir dabei zwei “Watchdog”-Strategien, welche die Zuverlässigkeit verbessern und ein frühes Abbrechen der Newton-Iteration ermöglichen.
    Notes: Abstract Complex numerical methods often contain subproblems that are easy to state in mathematical form, but difficult to translate into software. Several algorithmic isues of this nature arise in implementing a Newton iteration scheme as part of a finite-difference method for two-point boundary value problems. We describe the practical as well as theoretical considerations behind the decisions included in the final code, with special emphasis on two “watchdog” strategies designed to improve reliability and allow early termination of the Newton iterates.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02250582
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  • 6
    Electronic Resource
    Electronic Resource
    A note on a sufficient-decrease criterion for a non-derivative step-length procedure (1982)
    Springer
    Mathematical programming 23 (1982), S. 349-352 
    Details
    ISSN: 1436-4646
    Keywords: Step-Length Selection ; Linesearch ; Non-Derivative Methods
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract A step-length algorithm is an essential part of many descent methods for unconstrained and constrained optimization. In this note we present a criterion that defines an acceptable step length when only function values are available at trial step lengths.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01583799
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    Paper (German National Licenses)
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  • 7
    Electronic Resource
    Electronic Resource
    Some properties of the Hessian of the logarithmic barrier function (1994)
    Springer
    Mathematical programming 67 (1994), S. 265-295 
    Details
    ISSN: 1436-4646
    Keywords: Barrier methods ; Interior methods ; Barrier Hessian ; Ill-conditioning ; Rank-revealing Cholesky factorization
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract More than twenty years ago, Murray and Lootsma showed that Hessian matrices of the logarithmic barrier function become increasingly ill-conditioned at points on the barrier trajectory as the solution is approached. This paper explores some further characteristics of the barrier Hessian. We first show that, except in two special cases, the barrier Hessian is ill-conditioned in an entire region near the solution. At points in a more restricted region (including the barrier trajectory itself), this ill-conditioning displays a special structure connected with subspaces defined by the Jacobian of the active constraints. We then indicate how a Cholesky factorization with diagonal pivoting can be used to detect numerical rank-deficiency in the barrier Hessian, and to provide information about the underlying subspaces without making an explicit prediction of the active constraints. Using this subspace information, a close approximation to the Newton direction can be calculated by solving linear systems whose condition reflects that of the original problem.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01582224
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