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.

There may be situations where there are no tasks on the BOINC project for a number of reasons. If a computing node does not receive a task as a response to a request, then it goes into an idle state for a while. This publication proposes a probabilistic model of the behavior of a single computing node in the absence of jobs on the server. The proposed model best describes the processes associated with computing in local infrastructures, but it is also possible to generalize the methodology for large-scale voluntary computing projects.

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 many scientific applications, in particular, in finding working spaces of robots where it is necessary to determine not one but all the solutions of the system of nonlinear inequalities. Global optimization algorithms can be taken as an inspiration for developing methods for solving this problem. In this article, two new methods using two different approximations of Peano–Hilbert space-filling curves actively used in global optimization are proposed. Convergence conditions of the new methods are established. Numerical experiments executed on problems regarding finding the working spaces of several robots show a promising performance of the new algorithms.

In this paper, we introduce smooth continuously differentiable upper and lower estimators for a univariate function with the first derivative satisfying an interval Lipschitz property being a generalization of the Lipschitz condition. The constructed estimators are piecewise quadratic and can be readily used for bounding ranges of a function, for reducing the search intervals in global optimization or for finding roots of non-linear equations. The new estimators are derived analytically and studied experimentally in the framework of branch-and-bound global optimization algorithms. The first experiments are very promising and show clearly a notable superiority of the proposed technique over the traditional approaches.

In the paper, we compute some lower bounds on time of parallel solving of the subset sum problem on a big number of processors by several versions of dynamic programming algorithm Balsub proposed before by Pisinger. Based on these lower bounds, we propose a version of Balsub which could be possibly effectively parallelized.

The paper considers the approximation of the robot workspace. The developed method, based on interval Newton operator relies on Baumann bicentered theorem. We used this method for approximation of the solution sets of undetermined non-linear equation. This problem refers to the one of the most important problems in robotics: workspace approximation, since the robot kinematic systems are set with undetermined (usually non-linear) systems. We perform experiments for the DexTar robotic system and visualize the obtained approximations. As expected the bicentered modification provides tight approximation of the workspace compared with classical Newton method.

Four parallel algorithms are considered that implement the branch-and-bound method (BnB) for solving problems of finding a global minimum. The algorithms are designed for computing systems with shared memory. The BnB is based on two basic operations: branching and eliminating. To implement the elimination operation, interval arithmetic is used, which for real intervals defines operations similar to ordinary arithmetic. The main difference between the algorithms lies in the different implementation of storing the list of subproblems. In the process of testing on a representative set of test problems, the speed of the algorithms, their scalability, and their resistance to search anomalies are investigated.

The paper aims at mitigating hot-spots during Offline Scheduling in IaaS (Infrastructure-as-a-Service) cloud systems. Unlike previous studies, the research focuses on identifying and resolving hot-spots not at servers, but at server racks. A two-phase algorithm for performing power-aware offline scheduling is proposed. The first phase aims at identifying and mitigating hot-spots at racks, while the second phase performs VM consolidation, i.e. minimization of the number of occupied servers while maintaining a feasible VM mapping and low migration costs. The proposed algorithm takes into account the dynamic nature of VM's resource consumption: it does not only resolve detected hot-spots, but also tries to avoid hot-spots in a reasonable future time period. The algorithm was tested with the data from a real IaaS cloud with different sets of algorithm's parameters. Experimental evaluation showed that the statistical estimates of the future VM's resource consumption provide the most reliable mapping, which is a result of minimization of the number of new hot-spot occurrences.

The article considers the application of numerical method based on bicentered Krawczyk operator for solving the problem of the robot workspace approximation with box constraints. We applied several modifications for approximation of the solution sets of the indeterminate system of nonlinear equations and compare it with basic method. All methods were tested on a passive orthosis robot, which is part of the lower limb rehabilitation system. A mathematical model of the mechanism kinematics is presented. We evaluate the efficiency of the considered approaches, compute and visualize the robot workspace for different parameters sets.

The research considered the motion control of a 2R planar robot consisting of two links connected in series through two revolute joints, suitable for the task of simulating the human arm and other similar mechanisms. The aim of the study was to apply Machine Learning (artificial neural networks, ANN) methods to control the position of the 2R planar robot's end-effector within the workspace along a given trajectory. The possibility of using ANNs to achieve a single solution of the Inverse Kinematics Task is shown.

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 $$f'(x)$$ satisfying the Lipschitz condition with a constant L, i.e., $$f'(x)$$ cannot increase with the slope higher than L and decrease with the slope smaller than $$-L$$ . 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 $$f'(x)$$ cannot increase with the slope higher than a constant $$\beta $$ and decrease with the slope smaller than $$\alpha $$ . The interval $$[\alpha ,\beta ]$$ is called the Lipschitz interval (clearly, in this case the Lipschitz constant $$L = \max \{|\alpha |, |\beta | \}$$ ). 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 $$ |\alpha | \ne |\beta |$$ 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.

Restoration of the 3D structure of a protein from the sequence of its amino acids (“folding”) is one of the most important and challenging problems in computational biology. The most accurate methods require enormous computational resources due to the large number of variables determining a protein’s shape. Coarse-grained models combining several protein atoms into one unified globule partially mitigate this issue. The paper studies one of these models where globules are located in the nodes of the two-dimensional triangular lattice. In this model, folding is reduced to the discrete optimization problem: find positions of protein’s globules to maximize the number of contacts between them. We consider a standard procedure that finds an exact solution to this problem. It first generates an H-core—a set of positions for hydrophobic globules, which is followed by mapping of protein’s hydrophobic globules to these positions by the constraint satisfaction techniques. We propose a way to avoid unnecessary enumeration by skipping infeasible H-cores prior to mapping. Another contribution of our paper is a procedure that automatically generates constraints to simplify finding the feasible mapping of proteins globules to the lattice nodes. Experiments show that the proposed techniques tremendously accelerate the problem’s solving process.

Parallel robots are mechanical systems with numerous applications in manufacturing, medicine, space industry, etc. Such robots are characterized by closed kinematic chains ensuring robustness and maneuverability. The design of a fully functional parallel robot is a time-consuming task as an engineer needs to enumerate and compare many variants in order to find one with the best characteristics. Typical characteristics are the area of the workspace, the number of singularity-free regions, dexterity index. The design of a parallel robot is thus a multi-objective constraint optimization problem. We propose to use interval analysis to reliably compute characteristics and multiobjective optimization methods to find the Pareto-optimal set of robot parameters. The proposed approach is illustrated on a set of several realistic parallel robotic systems.

The article considers approximation of solution sets to indeterminate systems for non-linear equations. We developed a method to obtain inner and outer approximations of such sets. The method uses interval analysis techniques and relies on the Krawczyk operator and its modification based on the Baumann bicentered interval extension. We developed software that efficiently constructs and visualizes the computed approximations of the solution sets for systems of non-linear equations. The software is implemented in Python programming language and is available for free download and use. Finding the solution set of indeterminate systems of equations finds at least one important application in practice: bounding the workspace of a robotic manipulator. We perform experiments for the 2-RPR robot and evaluate the tightness of the obtained approximations. As expected the Krawczyk bicentered method noticeably improves the quality of approximations as compared with the classical Krawczyk operator.

Reliable bounding of a function’s range is essential for deterministic global optimization, approximation, locating roots of nonlinear equations, and several other computational mathematics areas. Despite years of extensive research in this direction, there is still room for improvement. The traditional and compelling approach to this problem is interval analysis. We show that accounting convexity/concavity can significantly tighten the bounds computed by interval analysis. To make our approach applicable to a broad range of functions, we also develop the techniques for handling nondifferentiable composite functions. Traditional ways to ensure the convexity fail in such cases. Experimental evaluation showed the remarkable potential of the proposed methods.

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 satisfying the Lipschitz condition. Theoretical properties of the introduced algorithm showing advantages of this reduction in the context of the present problem have been established and convergence properties of this method have been studied. A number of experiments executed on test functions and problems regarding finding workspace of robots confirm theoretical results and show a promising character of the new methodology. (C) 2020 Elsevier Inc. All rights reserved.

The work is devoted to the development and study of a method for solving global optimization problems with interval constraints. The paper proposes a global optimization algorithm based on a deterministic method of selecting starting points for local search methods. The starting points are the extremum points of functions of one variable, obtained by restricting the objective function to straight, collinear coordinate vectors. The effectiveness of the proposed algorithm is demonstrated by the example of the problem of minimizing the energy of a fragment of a flat crystal lattice. The energy of interatomic interaction is calculated using the Tersoff potential. An experimental comparison is made of the developed algorithm with the classical version of the multi-start method, in which pseudo-random points uniformly distributed in the parallelepiped are used to select starting points. As a local search method, in both cases, one of the modifications of the coordinate wise descent method is used. The developed method can be applied to problems with an unknown analytical expression for an objective function that is often encountered in practice.

One of the most actual and complex problems of medicine and neurology is the rehabilitation of patients. Robotic systems currently occupy an important place in the comprehensive rehabilitation of neurological patients with severe motor impairment of various etiologies, as well as the most socially significant and common neurological diseases. The article discusses various architectures of robotic systems for the rehabilitation of the lower limbs. The article also analyzes the work of foreign scientists who deal with these issues. As a result, a classification of rehabilitation systems has been compiled and reviewed. The structure of the rehabilitation system based on a passive orthosis and active parallel 3-PRRR robot is proposed. A numerical algorithm has been developed to determine the workspace of a 3-PRRR robot, taking into account design limitations. The algorithm is based on the concept of non-uniform coverings using interval analysis methods and is implemented in the C++ programming language. The results of computational experiments are presented. The simulation results are visualized by converting a set of three-dimensional parallelepipeds into an STL file. However, many areas remain open to theoretical research. A further area of research is the optimization of the geometric parameters of the system for carrying out rehabilitation procedures.

This paper is devoted to the experimental and theoretical study of parallel versions of the dynamic programming algorithm for the Knapsack Problem. We considered the standard table-based dynamic programming algorithm of the Knapsack Problem for serial, shared memory parallel and distributed memory multiprocessors. Experimental comparison and the performance study of shared and distributed memory dynamic programming algorithms is carried out. The OpenMP and MPI parallel programming tools are used for experimental evaluation.

An easily implementable recursive parallelization strategy for solving the subset sum problem by the branch-and-bound method is proposed. Two different frontal and balanced variants of this strategy are compared. On an example of a particular case of the subset sum problem we show that the balanced variant is more effective than the frontal one. Moreover, we show that, for the considered particular case of the subset sum problem, the balanced variant is also time optimal.