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## Approximating a solution set of nonlinear inequalities

In this paper we propose a method for solving systems of nonlinear inequalities with predefined accuracy based on nonuniform covering concept formerly adopted for global optimization. The method generates inner and outer approximations of the solution set. We describe the general concept and three ways of numerical implementation of the method. The first one is applicable only in a few cases when a minimum and a maximum of the constraints convolution function can be found analytically. The second implementation uses a global optimization method to find extrema of the constraints convolution function numerically. The third one is based on extrema approximation with Lipschitz under- and overestimations. We obtain theoretical bounds on the complexity and the accuracy of the generated approximations as well as compare proposed approaches theoretically and experimentally.

This paper argues that modeling granularity and approximation (Krifka 2007; Lewis 1979) is crucial for capturing important aspects of the distribution and interpretation of adjectives and their modifiers, modulo certain differences between modified adjectives and numerals. In addition, the paper presents supporting experimental results with minimizers like *slightly* and maximizers like *completely*.

Consideration was given to the omega square Cramer–von Mises tests intended to verify the goodness hypothesis about the distribution of the observed multivariable random vector with the distribution in the unit cube. The limit distribution of the statistics of these tests was defined by the distribution of an infinite quadratic form in the normal random variables. For convenience of computing its distribution, the residue of the quadratic form was approximated by a finite linear combination of the χ2-distributed random variables. Formulas for determination of the residue parameters were established.

In this paper the technique of parametric and structural synthesis of systems of maintenance of thermal modes of electronic equipment, as optimality criterion uses the ratio of the price system - the quality (the degree of approximation of the temperature to the desired values). Describes the developed software which allows to obtain practical results through effective selection systems for ensuring the thermal regime at the stage of preliminary design and the detailed design of electronic equipment.

Nowadays decision tree learning is one of the most popular classification and regression techniques. Though decision trees are not accurate on their own, they make very good base learners for advanced tree-based methods such as random forests and gradient boosted trees. However, applying ensembles of trees deteriorates interpretability of the final model. Another problem is that decision tree learning can be seen as a greedy search for a good classification hypothesis in terms of some information-based criterion such as Gini impurity or information gain. But in case of small data sets the global search might be possible. In this paper, we propose an FCA-based lazy classification technique where each test instance is classified with a set of the best (in terms of some information-based criterion) rules. In a set of benchmarking experiments, the proposed strategy is compared with decision tree and nearest neighbor learning.

This volume contains a collection of papers based on lectures and presentations delivered at the International Conference on Constructive Nonsmooth Analysis (CNSA) held in St. Petersburg (Russia) from June 18-23, 2012. This conference was organized to mark the 50th anniversary of the birth of nonsmooth analysis and nondifferentiable optimization and was dedicated to J.-J. Moreau and the late B.N. Pshenichnyi, A.M. Rubinov, and N.Z. Shor, whose contributions to NSA and NDO remain invaluable.

The first four chapters of the book are devoted to the theory of nonsmooth analysis. Chapters 5-8 contain new results in nonsmooth mechanics and calculus of variations. Chapters 9-13 are related to nondifferentiable optimization, and the volume concludes with four chapters containing interesting and important historical chapters, including tributes to three giants of nonsmooth analysis, convexity, and optimization: Alexandr Alexandrov, Leonid Kantorovich, and Alex Rubinov. The last chapter provides an overview and important snapshots of the 50-year history of convex analysis and optimization.

The increasing of the efficiency of technological modes of steel products manufacturing requires simulation of metal forming during hot deformation. To obtain correct results, one should set the correct initial and boundary conditions, including the mechanical properties of materials, which represent the dependence of the stress-strain and strain rate at maintained temperature. In the experiments one must reveal the mechanical properties and constants of the steels according to strain rate, predetermined temperature and chemical composition. So, the type of test is usually dependents on the technology process, which simulation will be using the obtained information. One can identify four main types of tests used in the hot deformation: compression, tension, torsion and rupture tests. The simplest tests are considered as uniaxial compression or tension tests. The results of these tests are the curves of <<flow stress -- strain>>. The present study describes an approximation method of test results for uniaxial compression of cylindrical samples made from AISI304 steel. During this work a mathematical model of the <<stress -- strain>> relation has been described. An algorithm that determines the necessary numerical coefficients for this model was developed. As a result, the equation of the material state, which is characterized by the stress relation on the strain, strain rate (0.15, 0.5, 1.5, 5 and 15 inverse seconds) and temperature (800, 950, 1080 and 1200 degree Celsius) was found. Also the approximation comparison with the experimental results were obtained.

Most of the existing books on optimization focus on the problem of computing locally optimal solutions. Global optimization is concerned with the computation and characterization of global optima of nonlinear functions. Global optimization problems are widespread in the mathematical modeling of real world systems for a very broad range of applications. During the past three decades many new theoretical, algorithmic, and computational contributions have helped to solve globally multi-extreme problems arising from important practical applications. Introduction to Global Optimization is the first comprehensive textbook that covers the fundamentals in global optimization. The second edition includes algorithms, applications, and complexity results for quadratic programming, concave minimization, DC and Lipshitz problems, decomposition algorithms for nonconvex optimization, and nonlinear network flow problems. Each chapter contains illustrative examples and ends with carefully selected exercises, which are designed to help the student to get a grasp of the material and enhance their knowledge of global optimization methods. Audience: This textbook is addressed not only to students of mathematical programming, but to all scientists in various disciplines who need global optimization methods to model and solve problems.

A new method was proposed to solve the global minimization problems of the Hölder functions on compact sets obeying continuous functions. The method relies on the Monte Carlo batch processing intended for constructing the sequences of values of the “quasi-global” minima and their decrements. A numerical procedure was proposed to generate a probabilistic stopping rule whose operability was corroborated by numerous tests and benchmarks with algorithmically defined functions.

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.

Event logs collected by modern information and technical systems usually contain enough data for automated process models discovery. A variety of algorithms was developed for process models discovery, conformance checking, log to model alignment, comparison of process models, etc., nevertheless a quick analysis of ad-hoc selected parts of a journal still have not get a full-fledged implementation. This paper describes an ROLAP-based method of multidimensional event logs storage for process mining. The result of the analysis of the journal is visualized as directed graph representing the union of all possible event sequences, ranked by their occurrence probability. Our implementation allows the analyst to discover process models for sublogs defined by ad-hoc selection of criteria and value of occurrence probability

The Handbook of CO₂ in Power Systems' objective is to include the state-of-the-art developments that occurred in power systems taking CO₂ emission into account. The book includes power systems operation modeling with CO₂ emissions considerations, CO₂ market mechanism modeling, CO₂ regulation policy modeling, carbon price forecasting, and carbon capture modeling. For each of the subjects, at least one article authored by a world specialist on the specific domain is included.