On strong homogeneity of two global optimization algorithms based on statistical models of multimodal objective functions

Žilinskas, Antanas

doi:10.1016/j.amc.2011.07.051

Cited by 42 publications

(33 citation statements)

References 15 publications

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“…(see [8,9,14,22,30,32,34,35,36,37,39,40,41,44,48,49]). It is important to emphasize that the new numeral system avoids situations of the type (5)- (7) providing results ensuring that if a is a numeral written in this system then for any a (i.e., a can be finite, infinite, or infinitesimal) it follows a + 1 > a.…”

Section: The Grossone Methodologymentioning

confidence: 99%

“…The new methodology has been successfully applied for studying a number of applications: percolation (see [14,44]), Euclidean and hyperbolic geometry (see [22,30]), fractals (see [32,34,41,44]), numerical differentiation and optimization (see [8,35,39,49]), infinite series (see [36,40,48]), the first Hilbert problem (see [37]), and cellular automata (see [9]). …”

Section: Introductionmentioning

confidence: 99%

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Single-tape and multi-tape Turing machines through the lens of the Grossone methodology

Sergeyev

Garro

2013

J Supercomput

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The paper investigates how the mathematical languages used to describe and to observe automatic computations influence the accuracy of the obtained results. In particular, we focus our attention on Single and Multi-tape Turing machines which are described and observed through the lens of a new mathematical language which is strongly based on three methodological ideas borrowed from Physics and applied to Mathematics, namely: the distinction between the object (we speak here about a mathematical object) of an observation and the instrument used for this observation; interrelations holding between the object and the tool used for the observation; the accuracy of the observation determined by the tool. Results of the observation executed by the traditional and new languages are compared and discussed.

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Section: The Grossone Methodologymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Single-tape and multi-tape Turing machines through the lens of the Grossone methodology

Sergeyev

Garro

2013

J Supercomput

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“…As shown in [41] DIRECT is not strongly homogeneous algorithm. In [22] the properties of DIRECT related to the scaling of the objective function is investigated once again.…”

Section: Modifications and Applications Of The Direct Algorithmmentioning

confidence: 99%

Application of Reduced-set Pareto-Lipschitzian Optimization to truss optimization

et al. 2015

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In this paper, a recently proposed global Lipschitz optimization algorithm PLOR (Pareto-Lipschitzian Optimization with Reduced-set) is further developed, investigated and applied to truss optimization problems. Partition patterns of the PLOR algorithm are similar to those of DIRECT (DIviding RECTangles), which was widely applied to different real-life problems. However here a set of all Lipschitz constants is reduced to just two: the maximal and the minimal ones. In such a way the PLOR approach is independent of any user-defined parameters and balances equally local and global search during the optimization process. An expanded list of other well-known DIRECT-type algorithms is used in investigation and experimental comparison using the standard test problems and truss optimization problems. The experimental investigation shows that the PLOR algorithm gives very competitive results to other DIRECT-type algorithms using standard test problems and performs pretty well on real truss optimization problems.

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“…One of the desirable properties of global optimization methods (see [7,35,41]) is their strong homogeneity meaning that a method produces the same sequences of trial points (i.e., points where the objective function f (x) is evaluated) independently of both shifting f (x) vertically and its multiplication by a scaling constant. In other words, it can be useful to optimize a scaled function g(x) = g(x; α, β) = αf (x) + β, α > 0,…”

Section: Introductionmentioning

confidence: 99%

“…Due to the presence of multiple local minima and non-differentiability of the objective function, classical local optimization techniques cannot be used for solving these problems and global optimization methods should be developed (see, e.g., [8,14,18,19,30,35,36,37,39,41]). …”

Section: Introductionmentioning

confidence: 99%

On strong homogeneity of a class of global optimization algorithms working with infinite and infinitesimal scales

Sergeyev

Kvasov

Mukhametzhanov

2018

Communications in Nonlinear Science and Numerical Simulation

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The necessity to find the global optimum of multiextremal functions arises in many applied problems where finding local solutions is insufficient. One of the desirable properties of global optimization methods is strong homogeneity meaning that a method produces the same sequences of points where the objective function is evaluated independently both of multiplication of the function by a scaling constant and of adding a shifting constant. In this paper, several aspects of global optimization using strongly homogeneous methods are considered. First, it is shown that even if a method possesses this property theoretically, numerically very small and large scaling constants can lead to illconditioning of the scaled problem. Second, a new class of global optimization problems where the objective function can have not only finite but also infinite or infinitesimal Lipschitz constants is introduced. Third, the strong homogeneity of several Lipschitz global optimization algorithms is studied in the framework of the Infinity Computing paradigm allowing one to work numerically with a variety of infinities and infinitesimals. Fourth, it is proved that a class of efficient univariate methods enjoys this property for finite, infinite and infinitesimal scaling and shifting constants. Finally, it is shown that in certain cases the usage of numerical infinities and infinitesimals can avoid ill-conditioning produced by scaling. Numerical experiments illustrating theoretical results are described.

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On strong homogeneity of two global optimization algorithms based on statistical models of multimodal objective functions

Cited by 42 publications

References 15 publications

Single-tape and multi-tape Turing machines through the lens of the Grossone methodology

Single-tape and multi-tape Turing machines through the lens of the Grossone methodology

Application of Reduced-set Pareto-Lipschitzian Optimization to truss optimization

On strong homogeneity of a class of global optimization algorithms working with infinite and infinitesimal scales

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