### Working paper

## Packing of Circles on Square Flat Torus as Global Optimization of Mixed Integer Nonlinear problem

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 proceedings publication is a compilation of selected contributions from the “Third International Conference on the Dynamics of Information Systems” which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents state-of-the art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study.

Inspired by human learning mechanisms, a novel meta-heuristic algorithm named human learning optimization (HLO) is presented in this paper in which the individual learning operator, social learning operator, random exploration learning operator and re-learning operator are developed to generate new solutions and search for the optima by mimicking the human learning process. Then HLO is applied to solve the well-known 5.100 and 10.100 multi-dimensional knapsack problems from the OR-library and the performance of HLO is compared with that of other meta-heuristics collected from the recent literature. The experimental results show that the presented HLO achieves the best performance in comparison with other meta-heuristics, which demonstrates that HLO is a promising optimization tool.

This book constitutes revised selected papers from the First International Workshop on Machine Learning, Optimization, and Big Data, MOD 2015, held in Taormina, Sicily, Italy, in July 2015. The 32 papers presented in this volume were carefully reviewed and selected from 73 submissions. They deal with the algorithms, methods and theories relevant in data science, optimization and machine learning.

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.

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 form for an unbiased estimate of the coefficient of determination of a linear regression model is obtained. It is calculated by using a sample from a multivariate normal distribution. This estimate is proposed as an alternative criterion for a choice of regression factors.