Yaroslav D. Sergeyev

Yaroslav D. Sergeyev

A new computational methodology for executing calculations with infinite and infinitesimal quantities is described in this paper. It is based on the principle `The part is less than the whole' introduced by Ancient Greeks and applied to all numbers (finite, infinite, and infinitesimal) and to all sets and processes (finite and infinite). It is shown that it becomes possible to write down finite, infinite, and infinitesimal numbers by a finite number of symbols as particular cases of a unique framework. The new methodology has allowed us to introduce the Infinity Computer working with such numbers (its simulator has already been realized). Examples dealing with divergent series, infinite sets, and limits are given.

Yaroslav D. Sergeyev

There exist many applications where it is necessary to approximate numerically derivatives of a function which is given by a computer procedure. In particular, all the fields of optimization have a special interest in such a kind of information. In this paper, a new way to do this is presented for a new kind of a computer -- the Infinity Computer -- able to work numerically with finite, infinite, and infinitesimal numbers. It is proved that the Infinity Computer is able to calculate values of derivatives of a higher order for a wide class of functions represented by computer procedures. It is shown that the ability to compute derivatives of arbitrary order automatically and accurate to working precision is an intrinsic property of the Infinity Computer related to its way of functioning. Numerical examples illustrating the new concepts and numerical tools are given.

Yaroslav D. Sergeyev

Traditional computers work with finite numbers. Situations where the usage of infinite or infinitesimal quantities is required are studied mainly theoretically. In this paper, a recently introduced computational methodology (that is not related to the non-standard analysis) is used to work with finite, infinite, and infinitesimal numbers \textit{numerically}. This can be done on a new kind of a computer - the Infinity Computer - able to work with all these types of numbers. The new computational tools both give possibilities to execute computations of a new type and open new horizons for creating new mathematical models where a computational usage of infinite and/or infinitesimal numbers can be useful. A number of numerical examples showing the potential of the new approach and dealing with divergent series, limits, probability theory, linear algebra, and calculation of volumes of objects consisting of parts of different dimensions are given.

Yaroslav D. Sergeyev

A recently developed computational methodology for executing numerical calculations with infinities and infinitesimals is described in this paper. The developed approach has a pronounced applied character and is based on the principle `The part is less than the whole' introduced by Ancient Greeks. This principle is used with respect to all numbers (finite, infinite, and infinitesimal) and to all sets and processes (finite and infinite). The point of view on infinities and infinitesimals (and in general, on Mathematics) presented in this paper uses strongly physical ideas emphasizing interrelations holding between a mathematical object under the observation and tools used for this observation. It is shown how a new numeral system allowing one to express different infinite and infinitesimal quantities in a unique framework can be used for theoretical and computational purposes. Numerous examples dealing with infinite sets, divergent series, limits, and probability theory are given.

Yaroslav D. Sergeyev, Dmitri E. Kvasov, Marat S. Mukhametzhanov

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 \emph{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 ill-conditioning 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 \emph{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.

Yaroslav D. Sergeyev, Antonio Candelieri, Dmitri E. Kvasov et al.

In this paper, the problem of safe global maximization (it should not be confused with robust optimization) of expensive noisy black-box functions satisfying the Lipschitz condition is considered. The notion "safe" means that the objective function $f(x)$ during optimization should not violate a "safety" threshold, for instance, a certain a priori given value $h$ in a maximization problem. Thus, any new function evaluation (possibly corrupted by noise) must be performed at "safe points" only, namely, at points $y$ for which it is known that the objective function $f(y) > h$. The main difficulty here consists in the fact that the used optimization algorithm should ensure that the safety constraint will be satisfied at a point $y$ before evaluation of $f(y)$ will be executed. Thus, it is required both to determine the safe region $Ω$ within the search domain~$D$ and to find the global maximum within $Ω$. An additional difficulty consists in the fact that these problems should be solved in the presence of the noise. This paper starts with a theoretical study of the problem and it is shown that even though the objective function $f(x)$ satisfies the Lipschitz condition, traditional Lipschitz minorants and majorants cannot be used due to the presence of the noise. Then, a $δ$-Lipschitz framework and two algorithms using it are proposed to solve the safe global maximization problem. The first method determines the safe area within the search domain and the second one executes the global maximization over the found safe region. For both methods a number of theoretical results related to their functioning and convergence is established. Finally, numerical experiments confirming the reliability of the proposed procedures are performed.

Yaroslav D. Sergeyev, Dmitri E. Kvasov

In many practical decision-making problems it happens that functions involved in optimization process are black-box with unknown analytical representations and hard to evaluate. In this paper, a global optimization problem is considered where both the goal function~$f(x)$ and its gradient $f'(x)$ are black-box functions. It is supposed that $f'(x)$ satisfies the Lipschitz condition over the search hyperinterval with an unknown Lipschitz constant~$K$. A new deterministic `Divide-the-Best' algorithm based on efficient diagonal partitions and smooth auxiliary functions is proposed in its basic version, its convergence conditions are studied and numerical experiments executed on eight hundred test functions are presented.

Dmitri E. Kvasov, Yaroslav D. Sergeyev

In many important design problems, some decisions should be made by finding the global optimum of a multiextremal objective function subject to a set of constrains. Frequently, especially in engineering applications, the functions involved in optimization process are black-box with unknown analytical representations and hard to evaluate. Such computationally challenging decision-making problems often cannot be solved by traditional optimization techniques based on strong suppositions about the problem (convexity, differentiability, etc.). Nature and evolutionary inspired metaheuristics are also not always successful in finding global solutions to these problems due to their multiextremal character. In this paper, some innovative and powerful deterministic approaches developed by the authors to construct numerical methods for solving the mentioned problems are surveyed. Their efficiency is shown on solving both the classes of random test problems and some practical engineering tasks.