Algebraic And Geometric Ideas In The Theory Of Discrete Optimization

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Algebraic and Geometric Ideas in the Theory of Discrete Optimization

Jesus A. De Loera,Raymond Hemmecke,Matthias K?ppe
Algebraic and Geometric Ideas in the Theory of Discrete Optimization Book PDF

Author : Jesus A. De Loera,Raymond Hemmecke,Matthias K?ppe
Publisher : SIAM
Page : 320 pages
File Size : 46,5 Mb
Release : 2013-01-31
Category : Mathematics
ISBN : 9781611972436

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Algebraic and Geometric Ideas in the Theory of Discrete Optimization by Jesus A. De Loera,Raymond Hemmecke,Matthias K?ppe Pdf

In recent years, many new techniques have emerged in the mathematical theory of discrete optimization that have proven to be effective in solving a number of hard problems. This book presents these recent advances, particularly those that arise from algebraic geometry, commutative algebra, convex and discrete geometry, generating functions, and other tools normally considered outside of the standard curriculum in optimization. These new techniques, all of which are presented with minimal prerequisites, provide a transition from linear to nonlinear discrete optimization. This book can be used as a textbook for advanced undergraduates or first-year graduate students in mathematics, computer science or operations research. It is also appropriate for mathematicians, engineers, and scientists engaged in computation who wish to gain a deeper understanding of how and why algorithms work.

Semidefinite Optimization and Convex Algebraic Geometry

Grigoriy Blekherman,Pablo A. Parrilo,Rekha R. Thomas
Semidefinite Optimization and Convex Algebraic Geometry Book PDF

Author : Grigoriy Blekherman,Pablo A. Parrilo,Rekha R. Thomas
Publisher : SIAM
Page : 487 pages
File Size : 50,7 Mb
Release : 2013-03-21
Category : Mathematics
ISBN : 9781611972283

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Semidefinite Optimization and Convex Algebraic Geometry by Grigoriy Blekherman,Pablo A. Parrilo,Rekha R. Thomas Pdf

An accessible introduction to convex algebraic geometry and semidefinite optimization. For graduate students and researchers in mathematics and computer science.

Discrete Geometry and Optimization

Károly Bezdek,Antoine Deza,Yinyu Ye
Discrete Geometry and Optimization Book PDF

Author : Károly Bezdek,Antoine Deza,Yinyu Ye
Publisher : Springer Science & Business Media
Page : 341 pages
File Size : 46,5 Mb
Release : 2013-07-09
Category : Mathematics
ISBN : 9783319002002

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Discrete Geometry and Optimization by Károly Bezdek,Antoine Deza,Yinyu Ye Pdf

​Optimization has long been a source of both inspiration and applications for geometers, and conversely, discrete and convex geometry have provided the foundations for many optimization techniques, leading to a rich interplay between these subjects. The purpose of the Workshop on Discrete Geometry, the Conference on Discrete Geometry and Optimization, and the Workshop on Optimization, held in September 2011 at the Fields Institute, Toronto, was to further stimulate the interaction between geometers and optimizers. This volume reflects the interplay between these areas. The inspiring Fejes Tóth Lecture Series, delivered by Thomas Hales of the University of Pittsburgh, exemplified this approach. While these fields have recently witnessed a lot of activity and successes, many questions remain open. For example, Fields medalist Stephen Smale stated that the question of the existence of a strongly polynomial time algorithm for linear optimization is one of the most important unsolved problems at the beginning of the 21st century. The broad range of topics covered in this volume demonstrates the many recent and fruitful connections between different approaches, and features novel results and state-of-the-art surveys as well as open problems.

Geometric Methods and Optimization Problems

Vladimir Boltyanski,Horst Martini,V. Soltan
Geometric Methods and Optimization Problems Book PDF

Author : Vladimir Boltyanski,Horst Martini,V. Soltan
Publisher : Springer Science & Business Media
Page : 438 pages
File Size : 50,6 Mb
Release : 2013-12-11
Category : Mathematics
ISBN : 9781461553199

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Geometric Methods and Optimization Problems by Vladimir Boltyanski,Horst Martini,V. Soltan Pdf

VII Preface In many fields of mathematics, geometry has established itself as a fruitful method and common language for describing basic phenomena and problems as well as suggesting ways of solutions. Especially in pure mathematics this is ob vious and well-known (examples are the much discussed interplay between lin ear algebra and analytical geometry and several problems in multidimensional analysis). On the other hand, many specialists from applied mathematics seem to prefer more formal analytical and numerical methods and representations. Nevertheless, very often the internal development of disciplines from applied mathematics led to geometric models, and occasionally breakthroughs were b~ed on geometric insights. An excellent example is the Klee-Minty cube, solving a problem of linear programming by transforming it into a geomet ric problem. Also the development of convex programming in recent decades demonstrated the power of methods that evolved within the field of convex geometry. The present book focuses on three applied disciplines: control theory, location science and computational geometry. It is our aim to demonstrate how methods and topics from convex geometry in a wider sense (separation theory of convex cones, Minkowski geometry, convex partitionings, etc.) can help to solve various problems from these disciplines.

Discrete Geometry and Algebraic Combinatorics

Alexander Barg,Oleg R. Musin
Discrete Geometry and Algebraic Combinatorics Book PDF

Author : Alexander Barg,Oleg R. Musin
Publisher : American Mathematical Society
Page : 202 pages
File Size : 52,5 Mb
Release : 2014-08-28
Category : Mathematics
ISBN : 9781470409050

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Discrete Geometry and Algebraic Combinatorics by Alexander Barg,Oleg R. Musin Pdf

This volume contains the proceedings of the AMS Special Session on Discrete Geometry and Algebraic Combinatorics held on January 11, 2013, in San Diego, California. The collection of articles in this volume is devoted to packings of metric spaces and related questions, and contains new results as well as surveys of some areas of discrete geometry. This volume consists of papers on combinatorics of transportation polytopes, including results on the diameter of graphs of such polytopes; the generalized Steiner problem and related topics of the minimal fillings theory; a survey of distance graphs and graphs of diameters, and a group of papers on applications of algebraic combinatorics to packings of metric spaces including sphere packings and topics in coding theory. In particular, this volume presents a new approach to duality in sphere packing based on the Poisson summation formula, applications of semidefinite programming to spherical codes and equiangular lines, new results in list decoding of a family of algebraic codes, and constructions of bent and semi-bent functions.

Problems and Solutions for Integer and Combinatorial Optimization

Mustafa Ç. Pınar,Deniz Akkaya
Problems and Solutions for Integer and Combinatorial Optimization Book PDF

Author : Mustafa Ç. Pınar,Deniz Akkaya
Publisher : SIAM
Page : 148 pages
File Size : 51,8 Mb
Release : 2023-11-10
Category : Mathematics
ISBN : 9781611977769

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Problems and Solutions for Integer and Combinatorial Optimization by Mustafa Ç. Pınar,Deniz Akkaya Pdf

The only book offering solved exercises for integer and combinatorial optimization, this book contains 102 classroom tested problems of varying scope and difficulty chosen from a plethora of topics and applications. It has an associated website containing additional problems, lecture notes, and suggested readings. Topics covered include modeling capabilities of integer variables, the Branch-and-Bound method, cutting planes, network optimization models, shortest path problems, optimum tree problems, maximal cardinality matching problems, matching-covering duality, symmetric and asymmetric TSP, 2-matching and 1-tree relaxations, VRP formulations, and dynamic programming. Problems and Solutions for Integer and Combinatorial Optimization: Building Skills in Discrete Optimization is meant for undergraduate and beginning graduate students in mathematics, computer science, and engineering to use for self-study and for instructors to use in conjunction with other course material and when teaching courses in discrete optimization.

An Introduction to Convexity, Optimization, and Algorithms

Heinz H. Bauschke,Walaa M. Moursi
An Introduction to Convexity Optimization and Algorithms Book PDF

Author : Heinz H. Bauschke,Walaa M. Moursi
Publisher : SIAM
Page : 192 pages
File Size : 49,6 Mb
Release : 2023-12-20
Category : Mathematics
ISBN : 9781611977806

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An Introduction to Convexity, Optimization, and Algorithms by Heinz H. Bauschke,Walaa M. Moursi Pdf

This concise, self-contained volume introduces convex analysis and optimization algorithms, with an emphasis on bridging the two areas. It explores cutting-edge algorithms—such as the proximal gradient, Douglas–Rachford, Peaceman–Rachford, and FISTA—that have applications in machine learning, signal processing, image reconstruction, and other fields. An Introduction to Convexity, Optimization, and Algorithms contains algorithms illustrated by Julia examples and more than 200 exercises that enhance the reader’s understanding of the topic. Clear explanations and step-by-step algorithmic descriptions facilitate self-study for individuals looking to enhance their expertise in convex analysis and optimization. Designed for courses in convex analysis, numerical optimization, and related subjects, this volume is intended for undergraduate and graduate students in mathematics, computer science, and engineering. Its concise length makes it ideal for a one-semester course. Researchers and professionals in applied areas, such as data science and machine learning, will find insights relevant to their work.

Integer Programming and Combinatorial Optimization

Jon Lee,Jens Vygen
Integer Programming and Combinatorial Optimization Book PDF

Author : Jon Lee,Jens Vygen
Publisher : Springer
Page : 429 pages
File Size : 44,6 Mb
Release : 2014-05-17
Category : Computers
ISBN : 9783319075570

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Integer Programming and Combinatorial Optimization by Jon Lee,Jens Vygen Pdf

This book constitutes the refereed proceedings of the 17th International Conference on Integer Programming and Combinatorial Optimization, IPCO 2014, held in Bonn, Germany, in June 2014. The 34 full papers presented were carefully reviewed and selected from 143 submissions. The conference is a forum for researchers and practitioners working on various aspects of integer programming and combinatorial optimization. The aim is to present recent developments in theory, computation, and applications in these areas. The scope of IPCO is viewed in a broad sense, to include algorithmic and structural results in integer programming and combinatorial optimization as well as revealing computational studies and novel applications of discrete optimization to practical problems.

Introduction to Nonlinear Optimization

Amir Beck
Introduction to Nonlinear Optimization Book PDF

Author : Amir Beck
Publisher : SIAM
Page : 364 pages
File Size : 45,7 Mb
Release : 2023-06-29
Category : Mathematics
ISBN : 9781611977622

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Introduction to Nonlinear Optimization by Amir Beck Pdf

Built on the framework of the successful first edition, this book serves as a modern introduction to the field of optimization. The author’s objective is to provide the foundations of theory and algorithms of nonlinear optimization as well as to present a variety of applications from diverse areas of applied sciences. Introduction to Nonlinear Optimization gradually yet rigorously builds connections between theory, algorithms, applications, and actual implementation. The book contains several topics not typically included in optimization books, such as optimality conditions in sparsity constrained optimization, hidden convexity, and total least squares. Readers will discover a wide array of applications such as circle fitting, Chebyshev center, the Fermat–Weber problem, denoising, clustering, total least squares, and orthogonal regression. These applications are studied both theoretically and algorithmically, illustrating concepts such as duality. Python and MATLAB programs are used to show how the theory can be implemented. The extremely popular CVX toolbox (MATLAB) and CVXPY module (Python) are described and used. More than 250 theoretical, algorithmic, and numerical exercises enhance the reader's understanding of the topics. (More than 70 of the exercises provide detailed solutions, and many others are provided with final answers.) The theoretical and algorithmic topics are illustrated by Python and MATLAB examples. This book is intended for graduate or advanced undergraduate students in mathematics, computer science, electrical engineering, and potentially other engineering disciplines.

Lectures on Stochastic Programming: Modeling and Theory, Third Edition

Alexander Shapiro,Darinka Dentcheva,Andrzej Ruszczyński
Lectures on Stochastic Programming Modeling and Theory Third Edition Book PDF

Author : Alexander Shapiro,Darinka Dentcheva,Andrzej Ruszczyński
Publisher : SIAM
Page : 540 pages
File Size : 55,9 Mb
Release : 2021-08-19
Category : Mathematics
ISBN : 9781611976595

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Lectures on Stochastic Programming: Modeling and Theory, Third Edition by Alexander Shapiro,Darinka Dentcheva,Andrzej Ruszczyński Pdf

An accessible and rigorous presentation of contemporary models and ideas of stochastic programming, this book focuses on optimization problems involving uncertain parameters for which stochastic models are available. Since these problems occur in vast, diverse areas of science and engineering, there is much interest in rigorous ways of formulating, analyzing, and solving them. This substantially revised edition presents a modern theory of stochastic programming, including expanded and detailed coverage of sample complexity, risk measures, and distributionally robust optimization. It adds two new chapters that provide readers with a solid understanding of emerging topics; updates Chapter 6 to now include a detailed discussion of the interchangeability principle for risk measures; and presents new material on formulation and numerical approaches to solving periodical multistage stochastic programs. Lectures on Stochastic Programming: Modeling and Theory, Third Edition is written for researchers and graduate students working on theory and applications of optimization, with the hope that it will encourage them to apply stochastic programming models and undertake further studies of this fascinating and rapidly developing area.

Evaluation Complexity of Algorithms for Nonconvex Optimization

Coralia Cartis,Nicholas I. M. Gould,Philippe L. Toint
Evaluation Complexity of Algorithms for Nonconvex Optimization Book PDF

Author : Coralia Cartis,Nicholas I. M. Gould,Philippe L. Toint
Publisher : SIAM
Page : 549 pages
File Size : 42,8 Mb
Release : 2022-07-06
Category : Mathematics
ISBN : 9781611976991

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Evaluation Complexity of Algorithms for Nonconvex Optimization by Coralia Cartis,Nicholas I. M. Gould,Philippe L. Toint Pdf

A popular way to assess the “effort” needed to solve a problem is to count how many evaluations of the problem functions (and their derivatives) are required. In many cases, this is often the dominating computational cost. Given an optimization problem satisfying reasonable assumptions—and given access to problem-function values and derivatives of various degrees—how many evaluations might be required to approximately solve the problem? Evaluation Complexity of Algorithms for Nonconvex Optimization: Theory, Computation, and Perspectives addresses this question for nonconvex optimization problems, those that may have local minimizers and appear most often in practice. This is the first book on complexity to cover topics such as composite and constrained optimization, derivative-free optimization, subproblem solution, and optimal (lower and sharpness) bounds for nonconvex problems. It is also the first to address the disadvantages of traditional optimality measures and propose useful surrogates leading to algorithms that compute approximate high-order critical points, and to compare traditional and new methods, highlighting the advantages of the latter from a complexity point of view. This is the go-to book for those interested in solving nonconvex optimization problems. It is suitable for advanced undergraduate and graduate students in courses on advanced numerical analysis, data science, numerical optimization, and approximation theory.

Moment and Polynomial Optimization

Jiawang Nie
Moment and Polynomial Optimization Book PDF

Author : Jiawang Nie
Publisher : SIAM
Page : 484 pages
File Size : 40,9 Mb
Release : 2023-06-15
Category : Mathematics
ISBN : 9781611977608

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Moment and Polynomial Optimization by Jiawang Nie Pdf

Moment and polynomial optimization is an active research field used to solve difficult questions in many areas, including global optimization, tensor computation, saddle points, Nash equilibrium, and bilevel programs, and it has many applications. The author synthesizes current research and applications, providing a systematic introduction to theory and methods, a comprehensive approach for extracting optimizers and solving truncated moment problems, and a creative methodology for using optimality conditions to construct tight Moment-SOS relaxations. This book is intended for applied mathematicians, engineers, and researchers entering the field. It can be used as a textbook for graduate students in courses on convex optimization, polynomial optimization, and matrix and tensor optimization.

Modern Nonconvex Nondifferentiable Optimization

Ying Cui,Jong-Shi Pang
Modern Nonconvex Nondifferentiable Optimization Book PDF

Author : Ying Cui,Jong-Shi Pang
Publisher : SIAM
Page : 792 pages
File Size : 48,5 Mb
Release : 2021-12-02
Category : Mathematics
ISBN : 9781611976748

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Modern Nonconvex Nondifferentiable Optimization by Ying Cui,Jong-Shi Pang Pdf

Starting with the fundamentals of classical smooth optimization and building on established convex programming techniques, this research monograph presents a foundation and methodology for modern nonconvex nondifferentiable optimization. It provides readers with theory, methods, and applications of nonconvex and nondifferentiable optimization in statistical estimation, operations research, machine learning, and decision making. A comprehensive and rigorous treatment of this emergent mathematical topic is urgently needed in today’s complex world of big data and machine learning. This book takes a thorough approach to the subject and includes examples and exercises to enrich the main themes, making it suitable for classroom instruction. Modern Nonconvex Nondifferentiable Optimization is intended for applied and computational mathematicians, optimizers, operations researchers, statisticians, computer scientists, engineers, economists, and machine learners. It could be used in advanced courses on optimization/operations research and nonconvex and nonsmooth optimization.

Algebraic Statistics

Seth Sullivant
Algebraic Statistics Book PDF

Author : Seth Sullivant
Publisher : American Mathematical Soc.
Page : 490 pages
File Size : 52,5 Mb
Release : 2018-11-19
Category : Geometry, Algebraic
ISBN : 9781470435172

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Algebraic Statistics by Seth Sullivant Pdf

Algebraic statistics uses tools from algebraic geometry, commutative algebra, combinatorics, and their computational sides to address problems in statistics and its applications. The starting point for this connection is the observation that many statistical models are semialgebraic sets. The algebra/statistics connection is now over twenty years old, and this book presents the first broad introductory treatment of the subject. Along with background material in probability, algebra, and statistics, this book covers a range of topics in algebraic statistics including algebraic exponential families, likelihood inference, Fisher's exact test, bounds on entries of contingency tables, design of experiments, identifiability of hidden variable models, phylogenetic models, and model selection. With numerous examples, references, and over 150 exercises, this book is suitable for both classroom use and independent study.

First-Order Methods in Optimization

Amir Beck
First Order Methods in Optimization Book PDF

Author : Amir Beck
Publisher : SIAM
Page : 487 pages
File Size : 54,7 Mb
Release : 2017-10-02
Category : Mathematics
ISBN : 9781611974997

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First-Order Methods in Optimization by Amir Beck Pdf

The primary goal of this book is to provide a self-contained, comprehensive study of the main ?rst-order methods that are frequently used in solving large-scale problems. First-order methods exploit information on values and gradients/subgradients (but not Hessians) of the functions composing the model under consideration. With the increase in the number of applications that can be modeled as large or even huge-scale optimization problems, there has been a revived interest in using simple methods that require low iteration cost as well as low memory storage. The author has gathered, reorganized, and synthesized (in a unified manner) many results that are currently scattered throughout the literature, many of which cannot be typically found in optimization books. First-Order Methods in Optimization offers comprehensive study of first-order methods with the theoretical foundations; provides plentiful examples and illustrations; emphasizes rates of convergence and complexity analysis of the main first-order methods used to solve large-scale problems; and covers both variables and functional decomposition methods.