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Nonsmooth Analysis is a relatively recent area of mathematical analysis. The literature about this subject consists mainly in research papers and books. The purpose of this book is to provide a handbook for undergraduate and graduate students of mathematics that introduce this interesting area in detail.Includes different kinds of sub and super differentials as well as generalized gradientsIncludes also the main tools of the theory, as Sum and Chain Rules or Mean Value theoremsContent is introduced in an elementary way, developing many examples, allowing the r

Nonsmooth optimization. --- Nonsmooth analysis --- Optimization, Nonsmooth --- Mathematical optimization

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Nonsmooth optimization. --- Calculus of variations. --- Isoperimetrical problems --- Variations, Calculus of --- Maxima and minima --- Nonsmooth analysis --- Optimization, Nonsmooth --- Mathematical optimization --- Calculus of variations --- Nonsmooth optimization

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This volume looks at the study of dynamical systems with discontinuities. Discontinuities arise when systems are subject to switches, decisions, or other abrupt changes in their underlying properties that require a ‘non-smooth’ definition. A review of current ideas and introduction to key methods is given, with a view to opening discussion of a major open problem in our fundamental understanding of what nonsmooth models are. What does a nonsmooth model represent: an approximation, a toy model, a sophisticated qualitative capturing of empirical law, or a mere abstraction? Tackling this question means confronting rarely discussed indeterminacies and ambiguities in how we define, simulate, and solve nonsmooth models. The author illustrates these with simple examples based on genetic regulation and investment games, and proposes precise mathematical tools to tackle them. The volume is aimed at students and researchers who have some experience of dynamical systems, whether as a modelling tool or studying theoretically. Pointing to a range of theoretical and applied literature, the author introduces the key ideas needed to tackle nonsmooth models, but also shows the gaps in understanding that all researchers should be bearing in mind. Mike Jeffrey is a researcher and lecturer at the University of Bristol with a background in mathematical physics, specializing in dynamics, singularities, and asymptotics.

Dynamics. --- Ergodic theory. --- Mathematical models. --- Dynamical Systems and Ergodic Theory. --- Mathematical Modeling and Industrial Mathematics. --- Nonsmooth optimization. --- Nonsmooth analysis --- Optimization, Nonsmooth --- Mathematical optimization --- Models, Mathematical --- Simulation methods --- Ergodic transformations --- Continuous groups --- Mathematical physics --- Measure theory --- Transformations (Mathematics) --- Dynamical systems --- Kinetics --- Mathematics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics

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Solving nonsmooth optimization (NSO) problems is critical in many practical applications and real-world modeling systems. The aim of this book is to survey various numerical methods for solving NSO problems and to provide an overview of the latest developments in the field. Experts from around the world share their perspectives on specific aspects of numerical NSO. The book is divided into four parts, the first of which considers general methods including subgradient, bundle and gradient sampling methods. In turn, the second focuses on methods that exploit the problem’s special structure, e.g. algorithms for nonsmooth DC programming, VU decomposition techniques, and algorithms for minimax and piecewise differentiable problems. The third part considers methods for special problems like multiobjective and mixed integer NSO, and problems involving inexact data, while the last part highlights the latest advancements in derivative-free NSO. Given its scope, the book is ideal for students attending courses on numerical nonsmooth optimization, for lecturers who teach optimization courses, and for practitioners who apply nonsmooth optimization methods in engineering, artificial intelligence, machine learning, and business. Furthermore, it can serve as a reference text for experts dealing with nonsmooth optimization.

Operations research. --- Management science. --- Decision making. --- Numerical analysis. --- Data mining. --- Economic theory. --- Operations Research, Management Science. --- Operations Research/Decision Theory. --- Numeric Computing. --- Data Mining and Knowledge Discovery. --- Economic Theory/Quantitative Economics/Mathematical Methods. --- Quantitative business analysis --- Management --- Problem solving --- Operations research --- Statistical decision --- Operational analysis --- Operational research --- Industrial engineering --- Management science --- Research --- System theory --- Economic theory --- Political economy --- Social sciences --- Economic man --- Algorithmic knowledge discovery --- Factual data analysis --- KDD (Information retrieval) --- Knowledge discovery in data --- Knowledge discovery in databases --- Mining, Data --- Database searching --- Mathematical analysis --- Deciding --- Decision (Psychology) --- Decision analysis --- Decision processes --- Making decisions --- Management decisions --- Choice (Psychology) --- Decision making --- Nonsmooth optimization. --- Nonsmooth analysis --- Optimization, Nonsmooth --- Mathematical optimization

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This book describes optimization models of clustering problems and clustering algorithms based on optimization techniques, including their implementation, evaluation, and applications. The book gives a comprehensive and detailed description of optimization approaches for solving clustering problems; the authors' emphasis on clustering algorithms is based on deterministic methods of optimization. The book also includes results on real-time clustering algorithms based on optimization techniques, addresses implementation issues of these clustering algorithms, and discusses new challenges arising from big data. The book is ideal for anyone teaching or learning clustering algorithms. It provides an accessible introduction to the field and it is well suited for practitioners already familiar with the basics of optimization. Provides a comprehensive description of clustering algorithms based on nonsmooth and global optimization techniques Addresses problems of real-time clustering in large data sets and challenges arising from big data Describes implementation and evaluation of optimization based clustering algorithms.

Electrical engineering. --- Pattern recognition. --- Signal processing. --- Image processing. --- Speech processing systems. --- Artificial intelligence. --- Data mining. --- Communications Engineering, Networks. --- Pattern Recognition. --- Signal, Image and Speech Processing. --- Artificial Intelligence. --- Data Mining and Knowledge Discovery. --- Nonsmooth optimization. --- Nonsmooth analysis --- Optimization, Nonsmooth --- Mathematical optimization --- Algorithmic knowledge discovery --- Factual data analysis --- KDD (Information retrieval) --- Knowledge discovery in data --- Knowledge discovery in databases --- Mining, Data --- Database searching --- AI (Artificial intelligence) --- Artificial thinking --- Electronic brains --- Intellectronics --- Intelligence, Artificial --- Intelligent machines --- Machine intelligence --- Thinking, Artificial --- Bionics --- Cognitive science --- Digital computer simulation --- Electronic data processing --- Logic machines --- Machine theory --- Self-organizing systems --- Simulation methods --- Fifth generation computers --- Neural computers --- Computational linguistics --- Electronic systems --- Information theory --- Modulation theory --- Oral communication --- Speech --- Telecommunication --- Singing voice synthesizers --- Pictorial data processing --- Picture processing --- Processing, Image --- Imaging systems --- Optical data processing --- Processing, Signal --- Information measurement --- Signal theory (Telecommunication) --- Design perception --- Pattern recognition --- Form perception --- Perception --- Figure-ground perception --- Electric engineering --- Engineering