A parallel method for finding the global minimum of univariate functions

Sergeyev, Yaroslav D.; Grishagin, Vladimir A.

doi:10.1007/bf02207778

“…In the framework of this approach many multidimensional global search algorithms have been worked out (Sergeyev, Y.D., Grishagin V.A., 1994), (Strongin, R.G., Sergeyev, Ya.D. , 2000), (Gergel, V.P., Strongin, R.G., 2003), (Hime, A. E., Oliveira Jr., H.A., Petraglia, A., 2011), (Sergeyev, Ya.…”

Section: Introductionmentioning

confidence: 99%

Local Tuning in Nested Scheme of Global Optimization

Gergel

¹

,

Grishagin

²

,

Israfilov

³

2015

Procedia Computer Science

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Numerical methods for global optimization of the multidimensional multiextremal functions in the framework of the approach oriented at dimensionality reduction by means of the nested optimization scheme are considered. This scheme reduces initial multidimensional problem to a set of univariate subproblems connected recursively. That enables to apply efficient univariate algorithms for solving the multidimensional problems. The nested optimization scheme served as the source of many methods for optimization of Lipschitzian function. However, in all of them there is the problem of estimating the Lipschitz constant as the parameter of the function optimized and, as a consequence, of tuning to it the optimization method. In the methods proposed earlier, as a rule, a global estimate (related to whole search domain) is used whereas local Lipschitz constants in some subdomains can differ significantly from the global constant. It can slow down the optimization process considerably. To overcome this drawback in the article the finer estimates of a priori unknown Lipschitz constants taking into account local properties of the objective function are considered and used in the nested optimization scheme. The results of numerical experiments presented demonstrate the advantages of methods with mixed (local and global) estimates of Lipschitz constants in comparison with the use of the global ones only.

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“…A vast literature is dedicated to the problem (2), (3) and algorithms for its solving (see, e.g., [8,10,11,15,18,29,30,31,34,36,39]). In particular, in practice it can be useful to optimize a scaled function g(x) from (1) instead of the original objective function f (x) (see, e.g., [7,35,41]).…”

Section: Introductionmentioning

confidence: 99%

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

Sergeyev

¹

,

Kvasov

²

,

Mukhametzhanov

³

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.

show abstract

“…A vast literature is dedicated to the problem (2), (3) and algorithms for its solving (see, e.g., [8,10,11,15,18,29,30,31,34,36,39]). In particular, in practice it can be useful to optimize a scaled function g(x) from (1) instead of the original objective function f (x) (see, e.g., [7,35,41]).…”

Section: Introductionmentioning

confidence: 99%

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

Sergeyev,

Kvasov,

Mukhametzhanov

2018

Preprint

0

View full text Add to dashboard Cite

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.

show abstract

A parallel method for finding the global minimum of univariate functions

Cited by 42 publications

References 13 publications

Local Tuning in Nested Scheme of Global Optimization

Local Tuning in Nested Scheme of Global Optimization

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

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

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