Abstract 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 L and decrease with the slope smaller than $$-L$$ - L . This minorant has been successfully applied in several efficient global optimization algorithms and used in engineering applications. In the present paper, it is supposed that the first derivative $$f'(x)$$ f ′ ( x ) cannot increase with the slope higher than a constant $$\beta $$ β and decrease with the slope smaller than $$\alpha $$ α . The interval $$[\alpha ,\beta ]$$ [ α , β ] is called the Lipschitz interval (clearly, in this case the Lipschitz constant $$L = \max \{|\alpha |, |\beta | \}$$ L = max { | α | , | β | } ). For this class of functions, smooth piece-wise estimators (minorants and majorants) have been proposed and applied in global optimization. Both theoretically and experimentally (on 200 randomly generated test problems) it has been shown that in cases where $$ |\alpha | \ne |\beta |$$ | α | ≠ | β | the new estimators can give a significant improvement w.r.t. those proposed in Sergeyev (Math Program 81(1):127–146, 1998), for example, in the framework of branch-and-bound global optimization methods.

Abstract In this paper, we propose a deterministic approach for finding the first zero-crossing point of a differentiable, possibly multiextremal univariate function, that combines a Branch-and-Bound framework with piece-wise linear under- and overestimators derived from Lipschitz intervals. Two algorithms are presented: a baseline algorithm and a version enhanced with interval reduction techniques. Theoretical guarantees of correctness and finite termination of the introduced methods are established. Extensive numerical experiments on 27 benchmark problems demonstrate that the proposed methods outperform the existing interval Branch-and-Bound approach both in terms of running time and accuracy.

Abstract 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 (LOOCV) technique combined with univariate global optimization tools from the class of Lipschitz derivative-free methods. Conventional LOOCV methods often suffer from ill-conditioning, particularly in high-dimensional settings, leading to computational inefficiencies. To address these issues, we consider efficient global optimization strategies characterized by optimistic and pessimistic improvements. The resulting algorithm allows us to improve the performance of the standard approach in terms of both accuracy and efficiency. Numerical results deriving from the study of some test cases and an application to real-world data support our analysis.

Abstract 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. The first of them, piecewise-linear approximation, is broadly used in global optimization and not only whereas the second one, non-univalent approximation, is less known. Multi-dimensional geometric algorithms employing these Peano curve approximations are introduced and their convergence conditions are established. Numerical experiments executed on 800 randomly generated test functions taken from the literature show a promising performance of algorithms employing Peano curve approximations w.r.t. their direct competitors.

Abstract 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 occurring during the model’s simulation that leads to inconsistent results. In this context, the paper investigates the use of a recently proposed numerical algorithm, based on the Infinity Computer methodology, to handle the Zeno phenomenon and evaluate it with respect to standard numerical methods by considering the hybrid system models of two exemplary Cyber-Physical Systems: the Water tanks and the Thermostat.

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 of nonlinear constraints. The user has the possibility to fix the difficulty of tests in an intuitive way by choosing several types of constraints. A detailed information (including the global solution) for each of 100 problems in each generated class is provided to the user. The generator is particularly suited for testing black-box optimization algorithms that normally address low or medium dimensional problems with hard to evaluate objective functions.

Abstract 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 is dedicated to the Infinity Computer, a new kind of a supercomputer that allows one to perform numerical computations with finite, infinite, and infinitesimal numbers. The already available software simulator of the Infinity Computer is used in different research domains for solving important real-world problems, where precision represents a key aspect. However, the software simulator is not suitable for solving problems in control theory and dynamics, where visual programming tools like Simulink are used frequently. In this context, the paper presents an innovative solution that allows one to use the Infinity Computer arithmetic within the Simulink environment. It is shown that the proposed solution is user-friendly, general purpose, and domain independent.

In this lecture, a recent computational methodology is described. It has been introduced with the intention to allow one to work with infinities and infinitesimals numerically in a unique computational framework. It is based on the principle ‘The part is less than the whole’ applied to all quantities (finite, infinite, and infinitesimal) and to all sets and processes (finite and infinite). The methodology uses as a computational device the Infinity Computer (patented in USA and EU) working numerically with infinite and infinitesimal numbers that can be written in a positional system with an infinite radix. On a number of examples (numerical differentiation, divergent series, ordinary differential equations, fractals, set theory, etc.) it is shown that the new approach can be useful from both theoretical and computational points of view. The main attention is dedicated to applications in optimization (local, global, and multi-objective). The accuracy of the obtained results is continuously compared with results obtained by traditional tools used to work with mathematical objects involving infinity. The Infinity Calculator working with infinities and infinitesimals numerically is shown during the lecture. For more information see http://www.theinfinitycomputer.com and this survey: Sergeyev Ya.D. Numerical infinities and infinitesimals: Methodology, applications, and repercussions on two Hilbert problems, EMS Surveys in Mathematical Sciences, 2017, 4(2), 219–320. For more information see http://www.theinfinitycomputer.com and this survey: Sergeyev Ya.D. Numerical infinities and infinitesimals: Methodology, applications, and repercussions on two Hilbert problems, EMS Surveys in Mathematical Sciences, 2017, 4(2), 219–320.

A procedure for generating non-differentiable, continuously differentiable, and twice continuously differentiable classes of test functions for multiextremal multidimensional box-constrained global optimization is presented. Each test class consists of 100 functions. Test functions are generated by defining a convex quadratic function systematically distorted by polynomials in order to introduce local minima. To determine a class, the user defines the following parameters: (i) problem dimension, (ii) number of local minima, (iii) value of the global minimum, (iv) radius of the attraction region of the global minimizer, (v) distance from the global minimizer to the vertex of the quadratic function. Then, all other necessary parameters are generated randomly for all 100 functions of the class. Full information about each test function including locations and values of all local minima is supplied to the user. Partial derivatives are also generated where possible.