Yaroslav Sergeyev

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• Ph.D., D.Sc., D.H.C., Editor-in-Chief, Journal “Operations Research Forum”, Springer, https://www.yaroslavsergeyev.com
• Distinguished Professor, Head of Numerical Calculus Laboratory at University of Calabria

The Turing machine is one of the simple abstract computational devices that can be used to investigate the limits of computability. In this paper, they are considered from several points of view that emphasize the importance and the relativity of mathematical languages used to describe the Turing machines. A deep investigation is performed on the i...

In this survey, a recent computational methodology paying a special attention to the separation of mathematical objects from numeral systems involved in their representation is described. It has been introduced with the intention to allow one to work with infinities and infinitesimals numerically in a unique computational framework in all the situa...

Global optimization is a field of mathematical programming dealing with finding global (absolute) minima of multi-dimensional multiextremal functions. Problems of this kind where the objective function is non-differentiable, satisfies the Lipschitz condition with an unknown Lipschitz constant, and is given as a “black-box” are very often encountere...

This paper considers hybrid systems – dynamical systems that exhibit both continuous and discrete behavior. Usually, in these systems, interactions between the continuous and discrete dynamics occur when a pre-defined function becomes equal to zero, i.e., in the system occurs a zero-crossing (the situation where the function only “touches” zero is...

In the paper, the global optimization problem of a multidimensional "black-box" function satisfying the Lipschitz condition over a hyperinterval with an unknown Lipschitz constant is considered. A new efficient algorithm for solving this problem is presented. At each iteration of the method a number of possible Lipschitz constants is chosen from a...

The paper presents PyGrossone, a general-purpose and domain-independent Python library for the Infinity Computer arithmetic allowing one to work numerically with different infinite and infinitesimal numbers. PyGrossone offers a set of arithmetic, elementary, trigonometric, and differentiation modules to perform computations with \documentclass[12pt...

In this article, we present a new numerical algorithm to detect the kernel shape parameter and the subdomain radius size within a partition of unity method for scattered data interpolation. Since an adaptive search of such hyperparameters is quite expensive from the computational point of view, we propose the use of a leave-one-out cross-validation...

In this article, we present a new numerical algorithm to detect the kernel shape parameter and the subdomain radius size within a partition of unity method for scattered data interpolation. Since an adaptive search of such hyperparameters is quite expensive from the computational point of view, we propose the use of a leave-one-out cross validation...

In this paper, the problem of approximating and visualizing the solution set of systems of nonlinear inequalities is considered. It is supposed that left-hand parts of the inequalities can be multiextremal and non-differentiable. Thus, traditional local methods using gradients cannot be applied in these circumstances. Problems of this kind arise in...

This work presents a new cutting plane method for lexicographic multi-objective integerlinear programming (LMOILP). The method uses Grossone Methodology to reformulate a LMOILP problem into one having a single non-Archimedean scalar objective function, asdone in [1] (but in that case in the absence of the integer constraints). The problem, without...

This is Book of Abstracts of the Fourth Triennial International Conference and Summer School NUMTA2023 “Numerical Computations: Theory and Algorithms”, 14-20 June 2023, Pizzo (VV), Italy. The Conference is organized by the Department of Computer Engineering, Modeling, Electronics and Systems Science of the University of Calabria, Italy in cooperati...

To capture the dynamics of modern Cyber-Physical Systems, hybrid system models are introduced to combine their continuous dynamics with the discrete ones. Unfortunately, one important negative issue can affect hybrid system models: the so-called Zeno phenomenon , which results in an infinite number of discrete transitions in a finite amount of time...

In this article, some classical paradoxes of infinity such as Galileo's paradox, Hilbert's paradox of the Grand Hotel, Thomson's lamp paradox, and the rectangle paradox of Torricelli are considered. In addition, three paradoxes regarding divergent series and a new paradox dealing with multiplication of elements of an infinite set are also described...

It is well known that the set of algebraic numbers (let us call it $A$) is countable. In this paper, instead of the usage of the classical terminology of cardinals proposed by Cantor, a recently introduced methodology using \G1-based infinite numbers is applied to measure the set $A$ (where the number \G1 is called \emph{grossone}). Our interest to...

It is well known that the set of algebraic numbers (let us call it A) is countable. In this paper, instead of the usage of the classical terminology of cardinals proposed by Cantor, a recently introduced methodology using 1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{...

In 1998, the paper Sergeyev (Math Program 81(1):127–146, 1998) has been published where a smooth piece-wise quadratic minorant has been proposed for multiextremal functions f ( x ) with the first derivative $$f'(x)$$ f ′ ( x ) satisfying the Lipschitz condition with a constant L , i.e., $$f'(x)$$ f ′ ( x ) cannot increase with the slope higher than...

In this paper, multi-dimensional global optimization problems are considered, where the objective function is supposed to be Lipschitz continuous, multiextremal, and without a known analytic expression. Two different approximations of Peano-Hilbert curve applied to reduce the problem to a univariate one satisfying the Hölder condition are discussed...

In this article, some classical paradoxes of infinity such as Galileo’s paradox, Hilbert’s paradox of the Grand Hotel, Thomson’s lamp paradox, and the rectangle paradox of Torricelli are considered. In addition, three paradoxes regarding divergent series and a new paradox dealing with multiplication of elements of an infinite set are also described...

This Chapter surveys a recent computational methodology allowing one to work with infinities and infinitesimals numerically on a supercomputer called the Infinity Computer that has been patented in several countries. This methodology applies the principle The whole is greater than the part to all numbers (finite, infinite, and infinitesimal) and to...

In this chapter we show how a lexicographic multi-objective linear programming problem (LMOLP) can be transformed into an equivalent, single-objective one, by using the Grossone Methodology. Then we provide a simplex-like algorithm, called GrossSimplex, able to solve the original LMOLP problem using a single run of the algorithm (its theoretical co...

There exist many applications where it is necessary to approximate numerically derivatives of a function f(x) which is given by a computer procedure. A novel way to efficiently compute exact derivatives (the word “exact” means here with respect to the accuracy of the implementation of f(x)) is presented in this Chapter. It uses a new kind of a supe...

A generator of classes of multidimensional test problems for benchmarking continuous constrained global optimization methods is proposed. It is based on the generator of test classes for global optimization proposed in 2003 by Gaviano, Kvasov, Lera, and Sergeyev and extends the previous generation procedure from the box-constrained case to the case...

This article describes a recently proposed methodology that allows one to work with infinitely large and infinitely small quantities on a computer. The approach uses a number of ideas that bring it closer to modern physics, in particular, the relativity of mathematical knowledge and its dependence on the tools used by mathematicians in their studie...

Numerical computing represents a critical aspect ofconventional computer architecture. Traditional computers adopt the IEEE 754-1985 binary floating-point standard to represent andwork with real numbers. Due to the architectural limitations of traditional computers, it is impossible to handle infinite and infinitesimal quanti- ties numerically. Thi...

In this paper we consider the problem of finding an optimal value of the shape parameter in radial basis function interpolation. In particular, we propose the use of a leave-one-out cross validation (LOOCV) technique combined with univariate global optimization methods, which involve strategies of global optimization with pessimistic improvement (G...

The problem of approximating and visualizing the solution set of systems of nonlinear inequalities can be frequently met in practice, in particular, when it is required to find the working space of some robots. In this paper, a method using Peano-Hilbert space-filling curves for the dimensionality reduction has been proposed for functions satisfyin...

This volume explores the connections between mathematical modeling, computational methods, and high performance computing, and how recent developments in these areas can help to solve complex problems in the natural sciences and engineering. The content of the book is based on talks and papers presented at the conference Modern Mathematical Methods...

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 gi...

Numerical computing is a key part of the traditional computer architecture. Almost all traditional computers implement the IEEE 754-1985 binary floating point standard to represent and work with numbers. The architectural limitations of traditional computers make impossible to work with infinite and infinitesimal quantities numerically. This paper...

In this contribution, a software simulator of the Infinity Computer is compared with MATLAB's symbolic computations in terms of execution times while solving two applied problems. The aim of this paper is to show advantages of the numerical nature of the Infinity Computer with respect to symbolic computations in solving difficult real-life problems...

The 14th International Workshop on Global Optimization was organized by Leiden University (Leiden Centre for Advanced Computer Science and Mathematical Institute) and the International Society of Global Optimization. One of the highlights of this workshop was a particular focus on the topic of multiobjective global optimization. LeGO 2018 is a work...

We consider an iterative computation of negative curvature directions, in large-scale unconstrained optimization frameworks, needed for ensuring the convergence toward stationary points which satisfy second-order necessary optimality conditions. We show that to the latter purpose, we can fruitfully couple the conjugate gradient (CG) method with a r...

In this paper, we deal with the computation of Lie derivatives, which are required, for example, in some numerical methods for the solution of differential equations. One common way for computing them is to use symbolic computation. Computer algebra software, however, might fail if the function is complicated, and cannot be even performed if an exp...

Lipschitz global optimization is an important research field with numerous applications in engineering, electronics, machine learning, optimal decision making, etc. In many of these applications, even in the univariate case, evaluations of the objective functions and derivatives are often time consuming and the number of function evaluations execut...

This paper is dedicated to numerical computation of higher order derivatives in Simulink. In this paper, a new module has been implemented to achieve this purpose within the Simulink-based Infinity Computer solution, recently introduced by the authors. This module offers several blocks to calculate higher order derivatives of a function given by th...

In the previous work (see [1]) the authors have shown how to solve a Lexicographic Multi-Objective Linear Programming (LMOLP) problem using the Grossone methodology described in [2]. That algorithm, called GrossSimplex, was a generalization of the well-known simplex algorithm, able to deal numerically with infinitesimal/infinite quantities. The aim...

This paper is dedicated to the Infinity Computer – a new type of a supercomputer allowing one to work numerically with finite, infinite, and infinitesimal numbers in one general framework. The existent software simulators of the Infinity Computer are used already for solving important real-world problems in applied mathematics. However, they are no...

Univariate box-constrained global optimization problems are considered, where the objective function is supposed to be Lipschitz continuous and multiextremal. It is assumed that its analytical representation is unknown (the function is given as a “black-box”) and even one its evaluation is a computationally expensive procedure. Geometric and inform...

In this work we have addressed lexicographic multi-objective linear programming problems where some of the variables are constrained to be integer. We have called this class of problems LMILP, which stands for Lexicographic Mixed Integer Linear Programming. Following one of the approach used to solve mixed integer linear programming problems, the b...

The two-volume set LNCS 11973 and 11974 constitute revised selected papers from the Third International Conference on Numerical Computations: Theory and Algorithms, NUMTA 2019, held in Crotone, Italy, in June 2019. This volume, LNCS 11973, consists of 34 full and 18 short papers chosen among papers presented at special streams and sessions of the...

The two-volume set LNCS 11973 and 11974 constitute revised selected papers from the Third International Conference on Numerical Computations: Theory and Algorithms, NUMTA 2019, held in Crotone, Italy, in June 2019. This volume, LNCS 11974, consists of 19 full and 32 short papers chosen among regular papers presented at the the Conference including...

In this paper, black-box global optimization problem with expensive function evaluations is considered. This problem is challenging for numerical methods due to the practical limits on computational budget often required by intelligent systems. For its efficient solution, a new DIRECT-type hybrid technique is proposed. The new algorithm incorporate...

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...

This paper is dedicated to the Infinity Computer – a new type of a supercomputer allowing one to work numerically with finite, infinite, and infinitesimal numbers in one general framework. The existent software simulators of the Infinity Computer are used already for solving important real-world problems in applied mathematics. However, they are no...

Multi-derivative one-step methods based upon Euler–Maclaurin integration formulae are considered for the solution of canonical Hamiltonian dynamical systems. Despite the negative result that simplecticity may not be attained by any multi-derivative Runge–Kutta methods, we show that the Euler–MacLaurin method of order p is conjugate-symplectic up to...

This commentary considers non-standard analysis and a recently introduced computational methodology based on the notion of \G1 (this symbol is called \emph{grossone}). The latter approach was developed with the intention to allow one to work with infinities and infinitesimals numerically in a unique computational framework and in all the situations...

Univariate Lipschitz global optimization problems are considered in this contribution. It is shown that in cases, where it is required to solve a scaled problem with very small or very large finite scaling constants, ill-conditioning can be provoked by scaling. It is established that this situation can be avoided using numerical infinities and infi...

In this paper, the global minimization problem of a multi-dimensional black-box Lipschitzian function is considered. In order to pass from the original Lipschitz multi-dimensional problem to a univariate one, an approach using space-filling curves to reduce the dimension is applied. The method does not use derivatives and, at each iteration, works...

We consider an iterative computation of negative curvature directions, in large scale optimization frameworks. We show that to the latter purpose, borrowing the ideas in [1, 3] and [4], we can fruitfully pair the Conjugate Gradient (CG) method with a recently introduced numerical approach involving the use of grossone [5]. In particular, though in...

This paper deals with an analysis of the Conjugate Gradient (CG) method (Hestenes and Stiefel in J Res Nat Bur Stand 49:409–436, 1952), in the presence of degenerates on indefinite linear systems. Several approaches have been proposed in the literature to issue the latter drawback in optimization frameworks, including reformulating the original lin...

Multi-derivative one-step methods based upon Euler-Maclaurin integration formulae are considered for the solution of canonical Hamiltonian dynamical systems. Despite the negative result that simplecticity may not be attained by any multi-derivative Runge-Kutta methods, we show that Euler-MacLaurin formulae are all topologically conjugate to a sympl...

A computational methodology called Grossone Infinity Computing introduced with the intention to allow one to work with infinities and infinitesimals numerically has been applied recently to a number of problems in numerical mathematics (optimization, numerical differentiation, numerical algorithms for solving ODEs, etc.). The possibility to use a s...

This commentary considers non-standard analysis and a recently introduced computational methodology based on the notion of \G1 (this symbol is called \emph{grossone}). The latter approach was developed with the intention to allow one to work with infinities and infinitesimals numerically in a unique computational framework and in all the situations...

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 independe...

Global optimization problems where evaluation of the objective function is an expensive operation arise frequently in engineering, decision making, optimal control, etc. There exist two huge but almost completely disjoint communities (they have different journals, different conferences, different test functions, etc.) solving these problems: a broa...

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 bo...

In this paper, multidimensional test problems for methods solving constrained Lipschitz global optimization problems are proposed. A new class of GKLS-based multidimensional test problems with continuously differentiable multiextremal objective functions and non-linear constraints is described. In these constrained problems, the global minimizer do...

Unconventional computing is about breaking boundaries in thinking, acting and computing. Typical topics of this non-typical field include, but are not limited to physics of computation, non-classical logics, new complexity measures, novel hardware, mechanical, chemical and quantum computing. Unconventional computing encourages a new style of thinki...

This paper is devoted to numerical global optimization algorithms applying several ideas to reduce the problem dimension. Two approaches to the dimensionality reduction are considered. The first one is based on the nested optimization scheme that reduces the multidimensional problem to a family of one-dimensional subproblems connected in a recursiv...

One-dimensional Lipschitz global optimization methods and novel accelerating techniques for these algorithms are described in detail in this Chapter.

The decision-maker (engineer, physicist, chemist, economist, etc.) frequently wants to find the best combination of a set of parameters (geometrical sizes, electrical and strength characteristics, etc.) describing a particular optimization model which provides the optimum (minimum or maximum) of a suitable objective function. Black-box multiextrema...

Two new diagonal methods based on the efficient partition strategy are studied in this Chapter. Both of them use multiple estimates of the Lipschitz constant as done in the popular DIRECT method. The methods are thoroughly explained and illustrated in this final Chapter; their numerical comparison is reported; the application of an important practi...

To extend one-dimensional algorithms to the multidimensional case, a class of diagonal algorithms thoroughly studied by the authors in the last years is considered in this Chapter. An efficient strategy, which is applied for partitioning the search domain, is described and compared with the existing diagonal partition strategies. This non-redundant...

This book begins with a concentrated introduction into deterministic global optimization and moves forward to present new original results from the authors who are well known experts in the field. Multiextremal continuous problems that have an unknown structure with Lipschitz objective functions and functions having the first Lipschitz derivatives...

Numerous problems arising in engineering applications can have several objectives to be satisfied. An important class of problems of this kind is lexicographic multi-objective problems where the first objective is incomparably more important than the second one which, in its turn, is incomparably more important than the third one, etc. In this pape...

In this chapter, a number of traditional models related to the percolation theory is taken into consideration: site percolation, gradient percolation, and forest-fire model. They are studied by means of a new computational methodology that gives a possibility to work with finite, infinite, and infinitesimal quantities numerically by using a new kin...

This book begins with a concentrated introduction into deterministic global optimization and moves forward to present new original results from the authors who are well known experts in the field. Multiextremal continuous problems that have an unknown structure with Lipschitz objective functions and functions having the first Lipschitz derivatives...

This book constitutes the thoroughly refereed post-conference proceedings of the 11th International Conference on Learning and Intelligent Optimization, LION 11, held in Nizhny,Novgorod, Russia, in June 2017. The 20 full papers (among these one GENOPT paper) and 15 short papers presented have been carefully reviewed and selected from 73 submission...

The sinusoidal parameter estimation problem is considered to fit a sum of damped sinusoids to a series of noisy observations. It is formulated as a nonlinear least-squares global optimization problem. A one-parametric case study is examined to determine an unknown frequency of a signal. Univariate Lipschitz-based deterministic methods are used for...

Lexicographic Multi-Objective Linear Programming (LMOLP) problems can be solved in two ways: preemptive and nonpreemptive. The preemptive approach requires the solution of a series of LP problems, with changing constraints (each time the next objective is added, a new constraint appears). The nonpreemptive approach is based on a scalarization of th...

New algorithms for the numerical solution of Ordinary Differential Equations (ODEs) with initial conditions are proposed. They are designed for working on a new kind of a supercomputer – the Infinity Computer – that is able to deal numerically with finite, infinite and infinitesimal numbers. Due to this fact, the Infinity Computer allows one to cal...