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  • 1
    Electronic Resource
    Electronic Resource
    Hesitant adaptive search for global optimisation (1998)
    Springer
    Mathematical programming 81 (1998), S. 89-102 
    Details
    ISSN: 1436-4646
    Keywords: Global optimisation ; Adaptive search ; Simulated annealing ; Hesitation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Pure adaptive search is a stochastic algorithm which has been analysed in distinct ways for finite and continuous global optimisation. In this paper, motivated by the behaviour of practical algorithms such as simulated annealing, we extend these ideas. We present a unified theory which yields both the finite and continuous results for pure adaptive search. At the same time, we allow our extended algorithm to “hesitate” before improvement continues. Results are obtained for the expected number of iterations to convergence for such an algorithm. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01584846
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    Paper (German National Licenses)
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  • 2
    Electronic Resource
    Electronic Resource
    Pure adaptive search for finite global optimization (1995)
    Springer
    Mathematical programming 69 (1995), S. 443-448 
    Details
    ISSN: 1436-4646
    Keywords: Global optimization ; Discrete optimization ; Algorithm complexity ; Random search ; Markov chains
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Pure Adaptive Search is a stochastic algorithm which has been analyzed for continuous global optimization. When a uniform distribution is used in PAS, it has been shown to have complexity which is linear in dimension. We define strong and weak variations of PAS in the setting of finite global optimization and prove analogous results. In particular, for then-dimensional lattice {1,⋯,k} n , the expected number of iterations to find the global optimum is linear inn. Many discrete combinatorial optimization problems, although having intractably large domains, have quite small ranges. The strong version of PAS for all problems, and the weak version of PAS for a limited class of problems, has complexity the order of the size of the range.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01585570
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  • 3
    Electronic Resource
    Electronic Resource
    The bisection method in higher dimensions (1992)
    Springer
    Mathematical programming 55 (1992), S. 319-337 
    Details
    ISSN: 1436-4646
    Keywords: 49D37 ; Bisection ; simplex ; global optimisation ; linear convergence ; zonotope ; tiling
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Is the familiar bisection method part of some larger scheme? The aim of this paper is to present a natural and useful generalisation of the bisection method to higher dimensions. The algorithm preserves the salient features of the bisection method: it is simple, convergence is assured and linear, and it proceeds via a sequence of brackets whose infinite intersection is the set of points desired. Brackets are unions of similar simplexes. An immediate application is to the global minimisation of a Lipschitz continuous function defined on a compact subset of Euclidean space.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01581205
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  • 4
    Electronic Resource
    Electronic Resource
    Towards Pure Adaptive Search (1995)
    Springer
    Journal of global optimization 7 (1995), S. 93-110 
    Details
    ISSN: 1573-2916
    Keywords: Global optimisation ; stochastic ; random search ; localisation ; complexity
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The algorithm known as Pure Adaptive Search is a global optimisation ideal with desirable complexity. In this paper we temper it to a framework we term Somewhat Adaptive Search. This retains the desirable complexity, but allows scope for a practical realisation. We introduce a new algorithm termed Pure Localisation Search which attempts to reach the practical ideal. For a certain class of one variable functions the gap is bridged.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01100207
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  • 5
    Electronic Resource
    Electronic Resource
    Multidimensional bisection: The performance and the context (1993)
    Springer
    Journal of global optimization 3 (1993), S. 337-358 
    Details
    ISSN: 1573-2916
    Keywords: Bisection ; global optimisation ; branch and bound ; numerical performance ; simplex ; localisation ; deterministic ; mathematical programming ; 90C30 ; 65K05
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Two aspects of the multidimensional bisection algorithms for the global optimisation of Lipschitz continuous functions are investigated. Firstly, for several test functions we examine the numerical performance of the deepest point algorithm and two acceleration procedures. Secondly, we phrase the branch and bound framework of Horst and Tuy in terms of covers, and show the algorithms to be included in this framework. A result of Basso on the convergence of localisations is extended to higher dimensions.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01096775
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  • 6
    Electronic Resource
    Electronic Resource
    On computing the dispersion function (1988)
    Springer
    Journal of optimization theory and applications 58 (1988), S. 331-350 
    Details
    ISSN: 1573-2878
    Keywords: Distribution function ; dispersion function ; concentration function ; bisection ; interval minimization
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A two-dimensional analogue of the well-known bisection method for root finding is presented in order to solve the following problem, related to the dispersion function of a set of random variables: given distribution functionsF 1,...,F n and a probabilityp, find an interval [a,b] of minimum width such thatF i(b)−F i(a −)⩾p, fori=1,...,n.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00939689
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    Paper (German National Licenses)
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  • 7
    Electronic Resource
    Electronic Resource
    Estimation of the Lipschitz constant of a function (1996)
    Springer
    Journal of global optimization 8 (1996), S. 91-103 
    Details
    ISSN: 1573-2916
    Keywords: Global optimisation ; Lipschitz constant ; Reverse Weibull distribution ; Gnedenko condition ; asymptotic distribution
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A number of global optimisation algorithms rely on the value of the Lipschitz constant of the objective function. In this paper we present a stochastic method for estimating the Lipschitz constant. We show that the largest slope in a fixed size sample of slopes has an approximate Reverse Weibull distribution. Such a distribution is fitted to the largest slopes and the location parameter used as an estimator of the Lipschitz constant. Numerical results are presented.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00229304
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    Paper (German National Licenses)
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