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  • 1
    Electronic Resource
    Electronic Resource
    A quasi-Newton method can be obtained from a method of conjugate directions (1978)
    Springer
    Mathematical programming 15 (1978), S. 189-199 
    Details
    ISSN: 1436-4646
    Keywords: Algorithms ; Rate of Convergence ; Unconstrained Optimization ; Quasi-Newton ; Conjugate Direction Methods
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract In a recent paper McCormick and Ritter consider two classes of algorithms, namely methods of conjugate directions and quasi-Newton methods, for the problem of minimizing a function ofn variablesF(x). They show that the former methods possess ann-step superlinear rate of convergence while the latter are every step superlinear and therefore inherently superior. In this paper a simple and computationally inexpensive modification of a method of conjugate directions is presented. It is shown that the modified method is a quasi-Newton method and is thus every step superlinearly convergent. It is also shown that under certain assumptions on the second derivatives ofF the rate of convergence of the modified method isn-step quadratic.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01609017
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    Paper (German National Licenses)
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  • 2
    Electronic Resource
    Electronic Resource
    Active set algorithms for isotonic regression; A unifying framework (1990)
    Springer
    Mathematical programming 47 (1990), S. 425-439 
    Details
    ISSN: 1436-4646
    Keywords: Isotonic regression ; active sets
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract In this and subsequent papers we will show that several algorithms for the isotonic regression problem may be viewed as active set methods. The active set approach provides a unifying framework for studying algorithms for isotonic regression, simplifies the exposition of existing algorithms and leads to several new efficient algorithms. We also investigate the computational complexity of several algorithms. In this paper we consider the isotonic regression problem with respect to a complete order $$\begin{gathered} minimize\sum\limits_{i = 1}^n {w_i } (y_i - x_i )^2 \hfill \\ subject tox_1 \leqslant x_2 \leqslant \cdot \cdot \cdot \leqslant x_n \hfill \\ \end{gathered} $$ where eachw i is strictly positive and eachy i is an arbitrary real number. We show that the Pool Adjacent Violators algorithm (due to Ayer et al., 1955; Miles, 1959; Kruskal, 1964), is a dual feasible active set method and that the Minimum Lower Set algorithm (due to Brunk et al., 1957) is a primal feasible active set method of computational complexity O(n 2). We present a new O(n) primal feasible active set algorithm. Finally we discuss Van Eeden's method and show that it is of worst-case exponential time complexity.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01580873
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    Paper (German National Licenses)
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  • 3
    Electronic Resource
    Electronic Resource
    A method to accelerate the rate of convergence of a class of optimization algorithms (1975)
    Springer
    Mathematical programming 9 (1975), S. 139-160 
    Details
    ISSN: 1436-4646
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Supposez ∈ E n is a solution to the optimization problem minimizeF(x) s.t.x ∈ E n and an algorithm is available which iteratively constructs a sequence of search directions {s j } and points {x j } with the property thatx j →z. A method is presented to accelerate the rate of convergence of {x j } toz provided that n consecutive search directions are linearly independent. The accelerating method uses n iterations of the underlying optimization algorithm. This is followed by a special step and then another n iterations of the underlying algorithm followed by a second special step. This pattern is then repeated. It is shown that a superlinear rate of convergence applies to the points determined by the special step. The special step which uses only first derivative information consists of the computation of a search direction and a step size. After a certain number of iterations a step size of one will always be used. The acceleration method is applied to the projection method of conjugate directions and the resulting algorithm is shown to have an (n + 1)-step cubic rate of convergence. The acceleration method is based on the work of Best and Ritter [2].
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01681341
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  • 4
    Electronic Resource
    Electronic Resource
    The efficient frontier for bounded assets (2000)
    Springer
    Mathematical methods of operations research 52 (2000), S. 195-212 
    Details
    ISSN: 1432-5217
    Keywords: Key words: Parametric quadratic programming, mean-variance portfolio selection, efficient frontier, capital market line.
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract. This paper develops a closed form solution of the mean-variance portfolio selection problem for uncorrelated and bounded assets when an additional technical assumption is satisfied. Although the assumption of uncorrelated assets is unduly restrictive, the explicit determination of the efficient asset holdings in the presence of bound constraints gives insight into the nature of the efficient frontier. The mean-variance portfolio selection problem considered here deals with the budget constraint and lower bounds or the budget constraint and upper bounds. For the mean-variance portfolio selection problem dealing with lower bounds the closed form solution is derived for two cases: a universe of only risky assets and a universe of risky assets plus an additional asset which is risk free. For the mean-variance portfolio selection problem dealing with upper bounds, the results presented are for a universe consisting only of risky assets. In each case, the order in which the assets are driven to their bounds depends on the ordering of their expected returns.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s001860000073
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    Paper (German National Licenses)
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  • 5
    Book
    Book
    Linear programming: active set analysis and computer programs (1985)
    Englewood Cliffs, NJ :Prentice-Hall,
    Details
    Title: Linear programming: active set analysis and computer programs
    Author: Best, Michael J.
    Contributer: Ritter, Klaus
    Publisher: Englewood Cliffs, NJ :Prentice-Hall,
    Year of publication: 1985
    Pages: 369 S.
    Type of Medium: Book
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    Zuse Institute Berlin get availability status
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