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  • 1
    Electronic Resource
    Electronic Resource
    An analytical approach to global optimization (1991)
    Springer
    Mathematical programming 52 (1991), S. 227-254 
    Details
    ISSN: 1436-4646
    Keywords: Global optimization ; analytical methods ; computer algebra
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Global optimization problems with a few variables and constraints arise in numerous applications but are seldom solved exactly. Most often only a local optimum is found, or if a global optimum is detected no proof is provided that it is one. We study here the extent to which such global optimization problems can be solved exactly using analytical methods. To this effect, we propose a series of tests, similar to those of combinatorial optimization, organized in a branch-and-bound framework. The first complete solution of two difficult test problems illustrates the efficiency of the resulting algorithm. Computational experience with the programbagop, which uses the computer algebra systemmacsyma, is reported on. Many test problems from the compendiums of Hock and Schittkowski and others sources have been solved.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01582889
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  • 2
    Electronic Resource
    Electronic Resource
    Global optimization of univariate Lipschitz functions: II. New algorithms and computational comparison (1992)
    Springer
    Mathematical programming 55 (1992), S. 273-292 
    Details
    ISSN: 1436-4646
    Keywords: Global optimization ; algorithm ; univariate function ; Lipschitz function ; computation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract We consider the following global optimization problems for a Lipschitz functionf implicitly defined on an interval [a, b]. Problem P′: find a globallyε-optimal value off and a corresponding point; Problem Q″: find a set of disjoint subintervals of [a, b] containing only points with a globallyε-optimal value and the union of which contains all globally optimal points. A two-phase algorithm is proposed for Problem P′. In phase I, this algorithm obtains rapidly a solution which is often globallyε-optimal. Moreover, a sufficient condition onf for this to be the case is given. In phase II, the algorithm proves theε-optimality of the solution obtained in phase I or finds a sequence of points of increasing value containing one with a globallyε-optimal value. The new algorithm is empirically compared (on twenty problems from the literature) with a best possible algorithm (for which the optimal value is assumed to be known), with a passive algorithm and with the algorithms of Evtushenko, Galperin, Shen and Zhu, Piyavskii, Timonov and Schoen. For smallε, the new algorithm requires only a few percent more function evaluations than the best possible one. An extended version of Piyavskii's algorithm is proposed for problem Q″. A sufficient condition onf is given for the globally optimal points to be in one-to-one correspondance with the obtained intervals. This result is achieved for all twenty test problems.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01581203
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    Paper (German National Licenses)
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  • 3
    Electronic Resource
    Electronic Resource
    Decomposition and interval arithmetic applied to global minimization of polynomial and rational functions (1993)
    Springer
    Journal of global optimization 3 (1993), S. 421-437 
    Details
    ISSN: 1573-2916
    Keywords: Global optimization ; decomposition ; interval arithmetic ; Optimisation globale ; décomposition ; arithmétique d'intervalles
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Description / Table of Contents: Résumé On montre que l'algorithme récent d'optimisation globale basé sur la décomposition du à Floudas et Visweswaran, lorsqu'on le spécialise au cas de fonctions polynômiales, est équivalent à une méthode d'optimisation globale basée sur l'arithmétique d'intervalles, qui applique l'extension naturelle à la forme de la pente de la corde du développement de Taylor. Plusieurs variantes plus efficaces utilisant d'autres formes de l'arithmétique d'intervalles sont explorées. On propose des extensions au cas des fonctions fractionnaires. On présente des résultats de calcul comparatifs.
    Notes: Abstract A recent global optimization algorithm using decomposition (GOP), due to Floudas and Visweswaran, when specialized to the case of polynomial functions is shown to be equivalent to an interval arithmetic global optimization algorithm which applies natural extension to the cord-slope form of Taylor's expansion. Several more efficient variants using other forms of interval arithmetic are explored. Extensions to rational functions are presented. Comparative computational experiences are reported.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01096413
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    Paper (German National Licenses)
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  • 4
    Electronic Resource
    Electronic Resource
    Finding maximum likelihood estimators for the three-parameter Weibull distribution (1994)
    Springer
    Journal of global optimization 5 (1994), S. 373-397 
    Details
    ISSN: 1573-2916
    Keywords: Global optimization ; decomposition ; maximum likelihood estimation ; Weibull distribution
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Much work has been devoted to the problem of finding maximum likelihood estimators for the three-parameter Weibull distribution. This problem has not been clearly recognized as a global optimization one and most methods from the literature occasionally fail to find a global optimum. We develop a global optimization algorithm which uses first order conditions and projection to reduce the problem to a univariate optimization one. Bounds on the resulting function and its first order derivative are obtained and used in a branch-and-bound scheme. Computational experience is reported. It is also shown that the solution method we propose can be extended to the case of right censored samples.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01096687
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    Paper (German National Licenses)
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  • 5
    Electronic Resource
    Electronic Resource
    On Timonov's algorithm for global optimization of univariate Lipschitz functions (1991)
    Springer
    Journal of global optimization 1 (1991), S. 37-46 
    Details
    ISSN: 1573-2916
    Keywords: Global optimization ; univariate function ; Lipschitz function
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Timonov proposes an algorithm for global maximization of univariate Lipschitz functions in which successive evaluation points are chosen in order to ensure at each iteration a maximal expected reduction of the “region of indeterminacy”, which contains all globally optimal points. It is shown that such an algorithm does not necessarily converge to a global optimum.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00120664
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    Paper (German National Licenses)
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  • 6
    Electronic Resource
    Electronic Resource
    Global optimization of univariate Lipschitz functions: I. Survey and properties (1992)
    Springer
    Mathematical programming 55 (1992), S. 251-272 
    Details
    ISSN: 1436-4646
    Keywords: Global optimization ; algorithm ; univariate function ; Lipschitz function ; convergence
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract We consider the following global optimization problems for a univariate Lipschitz functionf defined on an interval [a, b]: Problem P: find a globally optimal value off and a corresponding point; Problem P′: find a globallyε-optimal value off and a corresponding point; Problem Q: localize all globally optimal points; Problem Q′: find a set of disjoint subintervals of small length whose union contains all globally optimal points; Problem Q″: find a set of disjoint subintervals containing only points with a globallyε-optimal value and whose union contains all globally optimal points. We present necessary conditions onf for finite convergence in Problem P and Problem Q, recall the concepts necessary for a worst-case and an empirical study of algorithms (i.e., those ofpassive and ofbest possible algorithms), summarize and discuss algorithms of Evtushenko, Piyavskii-Shubert, Timonov, Schoen, Galperin, Shen and Zhu, presenting them in a simplified and uniform way, in a high-level computer language. We address in particular the problems of using an approximation for the Lipschitz constant, reducing as much as possible the expected length of the region of indeterminacy which contains all globally optimal points and avoiding remaining subintervals without points with a globallyε-optimal value. New algorithms for Problems P′ and Q″ and an extensive computational comparison of algorithms are presented in a companion paper.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01581202
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    Paper (German National Licenses)
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