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  • 1
    Electronic Resource
    Electronic Resource
    SCAN-98: an IMACS/GAMM International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics (1999)
    Springer
    Reliable computing 5 (1999), S. 101-102 
    Details
    ISSN: 1573-1340
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1026402010437
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    Paper (German National Licenses)
    Fulltext
  • 2
    Electronic Resource
    Electronic Resource
    Nonlinear coordinate transformations for unconstrained optimization II. Theoretical background (1993)
    Springer
    Journal of global optimization 3 (1993), S. 359-375 
    Details
    ISSN: 1573-2916
    Keywords: Global optimization ; smooth unconstrained optimization ; geodesic convexity ; unimodal function ; nonlinear coordinate transformation ; 65K05 ; 90C30 ; 54H00
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this two-part article, nonlinear coordinate transformations are discussed in order to simplify global unconstrained optimization problems and to test their unimodality on the basis of the analytical structure of the objective functions. If the transformed problems can be quadratic in some or all the variables, then the optimum can be calculated directly, without an iterative procedure, or the number of variables to be optimized can be reduced. Otherwise, the analysis of the structure can serve as a first phase for solving global unconstrained optimization problems. The first part treats real-life problems where the presented technique is applied and the transformation steps are constructed. The second part of the article deals with the differential geometrical background and the conditions of the existence of such transformations.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01096776
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    Paper (German National Licenses)
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  • 3
    Electronic Resource
    Electronic Resource
    Constructing large feasible suboptimal intervals for constrained nonlinear optimization (1995)
    Springer
    Annals of operations research 58 (1995), S. 279-293 
    Details
    ISSN: 1572-9338
    Keywords: Constrained nonlinear optimization ; sensitivity analysis ; inclusion function ; interval arithmetic ; 90C30 ; 65K05
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract An algorithm for finding a large feasiblen-dimensional interval for constrained global optimization is presented. Then-dimensional interval is iteratively enlarged about a seed point while maintaining feasibility. An interval subdivision method may be used to check feasibility of the growing box. The resultant feasible interval is constrained to lie within a given level set, thus ensuring it is close to the optimum. The ability to determine such a feasible interval is useful for exploring the neighbourhood of the optimum, and can be practically used in manufacturing considerations. The numerical properties of the algorithm are tested and demonstrated by an example problem.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02096403
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    Paper (German National Licenses)
    Fulltext
  • 4
    Electronic Resource
    Electronic Resource
    Interval method for bounding level sets: Revisited and tested with global optimization problems (1990)
    Springer
    BIT 30 (1990), S. 650-657 
    Details
    ISSN: 1572-9125
    Keywords: 90C30 ; 65K05 ; Inclusion function ; interval arithmetic ; level set ; global optimization
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract An interval method for bounding level sets, modified to increase its efficiency and to get sharper bounding boxes, is presented. The new algorithm was tested with standard global optimization test problems. The test results show that, while the modified method gives a more valuable, guaranteed reliability result set, it is competitive with non-interval methods in terms of CPU time and number of function evaluations.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01933213
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    Paper (German National Licenses)
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  • 5
    Electronic Resource
    Electronic Resource
    Multisection in Interval Branch-and-Bound Methods for Global Optimization – I. Theoretical Results (2000)
    Springer
    Journal of global optimization 16 (2000), S. 371-392 
    Details
    ISSN: 1573-2916
    Keywords: Branch-and-bound method ; Global optimization ; Interval arithmetic ; Multisection ; Accelerating devices
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We have investigated variants of interval branch-and-bound algorithms for global optimization where the bisection step was substituted by the subdivision of the current, actual interval into many subintervals in a single iteration step. The convergence properties of the multisplitting methods, an important class of multisection procedures are investigated in detail. We also studied theoretically the convergence improvements caused by multisection on algorithms which involve the accelerating tests (like e.g. the monotonicity test). The results are published in two papers, the second one contains the numerical test result.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1008354711345
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    Paper (German National Licenses)
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  • 6
    Electronic Resource
    Electronic Resource
    Multisection in Interval Branch-and-Bound Methods for Global Optimization II. Numerical Tests (2000)
    Springer
    Journal of global optimization 16 (2000), S. 219-228 
    Details
    ISSN: 1573-2916
    Keywords: Branch-and-bound method ; Global optimization ; Interval arithmetic ; Multisection ; Accelerating devices
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We have investigated variants of interval branch-and-bound algorithms for global optimization where the bisection step was substituted by the subdivision of the current, actual interval into many subintervals in a single iteration step. The results are published in two papers, the first one contains the theoretical investigations on the convergence properties. An extensive numerical study indicates that multisection can substantially improve the efficiency of interval global optimization procedures, and multisection seems to be indispensable in solving hard global optimization problems in a reliable way.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1008359223042
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    Paper (German National Licenses)
    Fulltext
  • 7
    Electronic Resource
    Electronic Resource
    Nonlinear coordinate transformations for unconstrained optimization I. Basic transformations (1993)
    Springer
    Journal of global optimization 3 (1993), S. 213-221 
    Details
    ISSN: 1573-2916
    Keywords: Global optimization ; nonlinear parameter transformation ; unconstrained optimization ; unimodal function
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this two-part article, nonlinear coordinate transformations are discussed to simplify unconstrained global optimization problems and to test their unimodality on the basis of the analytical structure of the objective functions. If the transformed problems are quadratic in some or all the variables, then the optimum can be calculated directly, without an iterative procedure, or the number of variables to be optimized can be reduced. Otherwise the analysis of the structure can serve as a first phase for solving unconstrained global optimization problems. The first part treats real-life problems where the presented technique is applied and the transformation steps are constructed. The second part of the article deals with the differential geometrical background and the conditions of the existence of such transformations.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01096739
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    Paper (German National Licenses)
    Fulltext
  • 8
    Electronic Resource
    Electronic Resource
    On the selection of subdivision directions in interval branch-and-bound methods for global optimization (1995)
    Springer
    Journal of global optimization 7 (1995), S. 183-207 
    Details
    ISSN: 1573-2916
    Keywords: Global optimization ; interval arithmetic ; branch-and-bound ; interval subdivision
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper investigates the influence of the interval subdivision selection rule on the convergence of interval branch-and-bound algorithms for global optimization. For the class of rules that allows convergence, we study the effects of the rules on a model algorithm with special list ordering. Four different rules are investigated in theory and in practice. A wide spectrum of test problems is used for numerical tests indicating that there are substantial differences between the rules with respect to the required CPU time, the number of function and derivative evaluations, and the necessary storage space. Two rules can provide considerable improvements in efficiency for our model algorithm.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01097060
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    Paper (German National Licenses)
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