Understanding and Using Linear Programming

Author: Jiri Matousek,Bernd Gärtner

Publisher: Springer Science & Business Media

ISBN: 3540307176

Category: Mathematics

Page: 226

View: 8111

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The book is an introductory textbook mainly for students of computer science and mathematics. Our guiding phrase is "what every theoretical computer scientist should know about linear programming". A major focus is on applications of linear programming, both in practice and in theory. The book is concise, but at the same time, the main results are covered with complete proofs and in sufficient detail, ready for presentation in class. The book does not require more prerequisites than basic linear algebra, which is summarized in an appendix. One of its main goals is to help the reader to see linear programming "behind the scenes".

Understanding and Using Linear Programming

Author: Jiri Matousek,Bernd Gärtner

Publisher: Springer Science & Business Media

ISBN: 3540306978

Category: Mathematics

Page: 226

View: 7597

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The book is an introductory textbook mainly for students of computer science and mathematics. Our guiding phrase is "what every theoretical computer scientist should know about linear programming". A major focus is on applications of linear programming, both in practice and in theory. The book is concise, but at the same time, the main results are covered with complete proofs and in sufficient detail, ready for presentation in class. The book does not require more prerequisites than basic linear algebra, which is summarized in an appendix. One of its main goals is to help the reader to see linear programming "behind the scenes".

Understanding and Using Linear Programming

Author: Jiri Matousek,Bernd Gärtner

Publisher: Springer

ISBN: 9783540818403

Category: Mathematics

Page: 226

View: 1607

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The book is an introductory textbook mainly for students of computer science and mathematics. Our guiding phrase is "what every theoretical computer scientist should know about linear programming". A major focus is on applications of linear programming, both in practice and in theory. The book is concise, but at the same time, the main results are covered with complete proofs and in sufficient detail, ready for presentation in class. The book does not require more prerequisites than basic linear algebra, which is summarized in an appendix. One of its main goals is to help the reader to see linear programming "behind the scenes".

Approximation Algorithms and Semidefinite Programming

Author: Bernd Gärtner,Jiri Matousek

Publisher: Springer Science & Business Media

ISBN: 3642220150

Category: Mathematics

Page: 251

View: 2626

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Semidefinite programs constitute one of the largest classes of optimization problems that can be solved with reasonable efficiency - both in theory and practice. They play a key role in a variety of research areas, such as combinatorial optimization, approximation algorithms, computational complexity, graph theory, geometry, real algebraic geometry and quantum computing. This book is an introduction to selected aspects of semidefinite programming and its use in approximation algorithms. It covers the basics but also a significant amount of recent and more advanced material. There are many computational problems, such as MAXCUT, for which one cannot reasonably expect to obtain an exact solution efficiently, and in such case, one has to settle for approximate solutions. For MAXCUT and its relatives, exciting recent results suggest that semidefinite programming is probably the ultimate tool. Indeed, assuming the Unique Games Conjecture, a plausible but as yet unproven hypothesis, it was shown that for these problems, known algorithms based on semidefinite programming deliver the best possible approximation ratios among all polynomial-time algorithms. This book follows the “semidefinite side” of these developments, presenting some of the main ideas behind approximation algorithms based on semidefinite programming. It develops the basic theory of semidefinite programming, presents one of the known efficient algorithms in detail, and describes the principles of some others. It also includes applications, focusing on approximation algorithms.

Linear Programming

Methods and Applications

Author: Saul I. Gass

Publisher: Courier Corporation

ISBN: 048643284X

Category: Mathematics

Page: 532

View: 7125

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Comprehensive, well-organized volume, suitable for undergraduates, covers theoretical, computational, and applied areas in linear programming. Expanded, updated edition; useful both as a text and as a reference book. 1995 edition.

Linear Programming

Author: Vasek Chvatal

Publisher: Macmillan

ISBN: 9780716715870

Category: Mathematics

Page: 478

View: 6835

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For upper-division/graduate courses in operations research/management science, mathematics, and computer science, this text covers basic theory, selected applications, network flow problems, and advanced techniques.

Numerical Optimization

Theoretical and Practical Aspects

Author: Joseph-Frédéric Bonnans,Jean Charles Gilbert,Claude Lemarechal,Claudia A. Sagastizábal

Publisher: Springer Science & Business Media

ISBN: 3662050781

Category: Mathematics

Page: 423

View: 3596

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This book starts with illustrations of the ubiquitous character of optimization, and describes numerical algorithms in a tutorial way. It covers fundamental algorithms as well as more specialized and advanced topics for unconstrained and constrained problems. This new edition contains computational exercises in the form of case studies which help understanding optimization methods beyond their theoretical description when coming to actual implementation.

Linear and Integer Optimization

Theory and Practice, Third Edition

Author: Gerard Sierksma,Yori Zwols

Publisher: CRC Press

ISBN: 1498743129

Category: Business & Economics

Page: 686

View: 9072

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Presenting a strong and clear relationship between theory and practice, Linear and Integer Optimization: Theory and Practice is divided into two main parts. The first covers the theory of linear and integer optimization, including both basic and advanced topics. Dantzig’s simplex algorithm, duality, sensitivity analysis, integer optimization models, and network models are introduced. More advanced topics also are presented including interior point algorithms, the branch-and-bound algorithm, cutting planes, complexity, standard combinatorial optimization models, the assignment problem, minimum cost flow, and the maximum flow/minimum cut theorem. The second part applies theory through real-world case studies. The authors discuss advanced techniques such as column generation, multiobjective optimization, dynamic optimization, machine learning (support vector machines), combinatorial optimization, approximation algorithms, and game theory. Besides the fresh new layout and completely redesigned figures, this new edition incorporates modern examples and applications of linear optimization. The book now includes computer code in the form of models in the GNU Mathematical Programming Language (GMPL). The models and corresponding data files are available for download and can be readily solved using the provided online solver. This new edition also contains appendices covering mathematical proofs, linear algebra, graph theory, convexity, and nonlinear optimization. All chapters contain extensive examples and exercises. This textbook is ideal for courses for advanced undergraduate and graduate students in various fields including mathematics, computer science, industrial engineering, operations research, and management science.

An Illustrated Guide to Linear Programming

Author: Saul I. Gass

Publisher: Courier Corporation

ISBN: 0486319601

Category: Computers

Page: 192

View: 3577

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Entertaining, nontechnical introduction covers basic concepts of linear programming and its relationship to operations research; geometric interpretation and problem solving, solution techniques, network problems, much more. Only high-school algebra needed.

Programming for Mathematicians

Author: Raymond Seroul

Publisher: Springer Science & Business Media

ISBN: 3642571298

Category: Mathematics

Page: 432

View: 4947

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Aimed at teaching mathematics students how to program using their knowledge of mathematics, the entire books emphasis is on "how to think" when programming. Three methods for constructing an algorithm or a program are used: manipulation and enrichment of existing code; use of recurrent sequences; deferral of code writing, in order to deal with one difficulty at a time. Many theorems are mathematically proved and programmed, and the text concludes with an explanation of how a compiler works and how to compile "by hand" little programs. Intended for anyone who thinks mathematically and wants to program and play with mathematics.

Linear Algebra

Author: Harold M. Edwards

Publisher: Springer Science & Business Media

ISBN: 9780817637316

Category: Mathematics

Page: 184

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* Proposes a radically new and thoroughly algorithmic approach to linear algebra * Each proof is an algorithm described in English that can be translated into the computer language the class is using and put to work solving problems and generating new examples * Designed for a one-semester course, this text gives the student many examples to work through and copious exercises to test their skills and extend their knowledge of the subject

Thirty-three Miniatures

Mathematical and Algorithmic Applications of Linear Algebra

Author: Jiří Matoušek

Publisher: American Mathematical Soc.

ISBN: 0821849778

Category: Mathematics

Page: 182

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This volume contains a collection of clever mathematical applications of linear algebra, mainly in combinatorics, geometry, and algorithms. Each chapter covers a single main result with motivation and full proof in at most ten pages and can be read independently of all other chapters (with minor exceptions), assuming only a modest background in linear algebra. The topics include a number of well-known mathematical gems, such as Hamming codes, the matrix-tree theorem, the Lovasz bound on the Shannon capacity, and a counterexample to Borsuk's conjecture, as well as other, perhaps less popular but similarly beautiful results, e.g., fast associativity testing, a lemma of Steinitz on ordering vectors, a monotonicity result for integer partitions, or a bound for set pairs via exterior products. The simpler results in the first part of the book provide ample material to liven up an undergraduate course of linear algebra. The more advanced parts can be used for a graduate course of linear-algebraic methods or for seminar presentations. Table of Contents: Fibonacci numbers, quickly; Fibonacci numbers, the formula; The clubs of Oddtown; Same-size intersections; Error-correcting codes; Odd distances; Are these distances Euclidean?; Packing complete bipartite graphs; Equiangular lines; Where is the triangle?; Checking matrix multiplication; Tiling a rectangle by squares; Three Petersens are not enough; Petersen, Hoffman-Singleton, and maybe 57; Only two distances; Covering a cube minus one vertex; Medium-size intersection is hard to avoid; On the difficulty of reducing the diameter; The end of the small coins; Walking in the yard; Counting spanning trees; In how many ways can a man tile a board?; More bricks--more walls?; Perfect matchings and determinants; Turning a ladder over a finite field; Counting compositions; Is it associative?; The secret agent and umbrella; Shannon capacity of the union: a tale of two fields; Equilateral sets; Cutting cheaply using eigenvectors; Rotating the cube; Set pairs and exterior products; Index. (STML/53)

Linear Programming: An Introduction to Finite Improvement Algorithms

Second Edition

Author: Daniel Solow

Publisher: Courier Corporation

ISBN: 0486493768

Category: Mathematics

Page: 432

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This text covers the basic theory and computation for a first course in linear programming, including substantial material on mathematical proof techniques and sophisticated computation methods. Includes Appendix on using Excel. 1984 edition.

Higher-Dimensional Algebraic Geometry

Author: Olivier Debarre

Publisher: Springer Science & Business Media

ISBN: 147575406X

Category: Mathematics

Page: 234

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The classification theory of algebraic varieties is the focus of this book. This very active area of research is still developing, but an amazing quantity of knowledge has accumulated over the past twenty years. The authors goal is to provide an easily accessible introduction to the subject. The book starts with preparatory and standard definitions and results, then moves on to discuss various aspects of the geometry of smooth projective varieties with many rational curves, and finishes in taking the first steps towards Moris minimal model program of classification of algebraic varieties by proving the cone and contraction theorems. The book is well-organized and the author has kept the number of concepts that are used but not proved to a minimum to provide a mostly self-contained introduction.

Using the Borsuk-Ulam Theorem

Lectures on Topological Methods in Combinatorics and Geometry

Author: Jiri Matousek

Publisher: Springer Science & Business Media

ISBN: 3540766499

Category: Mathematics

Page: 214

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To the uninitiated, algebraic topology might seem fiendishly complex, but its utility is beyond doubt. This brilliant exposition goes back to basics to explain how the subject has been used to further our understanding in some key areas. A number of important results in combinatorics, discrete geometry, and theoretical computer science have been proved using algebraic topology. While the results are quite famous, their proofs are not so widely understood. This book is the first textbook treatment of a significant part of these results. It focuses on so-called equivariant methods, based on the Borsuk-Ulam theorem and its generalizations. The topological tools are intentionally kept on a very elementary level. No prior knowledge of algebraic topology is assumed, only a background in undergraduate mathematics, and the required topological notions and results are gradually explained.

Convex Analysis and Variational Problems

Author: Ivar Ekeland,Roger Témam

Publisher: SIAM

ISBN: 9781611971088

Category: Calculus of variations

Page: 402

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This book contains different developments of infinite dimensional convex programming in the context of convex analysis, including duality, minmax and Lagrangians, and convexification of nonconvex optimization problems in the calculus of variations (infinite dimension). It also includes the theory of convex duality applied to partial differential equations; no other reference presents this in a systematic way. The minmax theorems contained in this book have many useful applications, in particular the robust control of partial differential equations in finite time horizon. First published in English in 1976, this SIAM Classics in Applied Mathematics edition contains the original text along with a new preface and some additional references.

Algorithmics for Hard Problems

Introduction to Combinatorial Optimization, Randomization, Approximation, and Heuristics

Author: Juraj Hromkovič

Publisher: Springer Science & Business Media

ISBN: 3662052695

Category: Computers

Page: 538

View: 8398

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Algorithmic design, especially for hard problems, is more essential for success in solving them than any standard improvement of current computer tech nologies. Because of this, the design of algorithms for solving hard problems is the core of current algorithmic research from the theoretical point of view as well as from the practical point of view. There are many general text books on algorithmics, and several specialized books devoted to particular approaches such as local search, randomization, approximation algorithms, or heuristics. But there is no textbook that focuses on the design of algorithms for hard computing tasks, and that systematically explains, combines, and compares the main possibilities for attacking hard algorithmic problems. As this topic is fundamental for computer science, this book tries to close this gap. Another motivation, and probably the main reason for writing this book, is connected to education. The considered area has developed very dynami cally in recent years and the research on this topic discovered several profound results, new concepts, and new methods. Some of the achieved contributions are so fundamental that one can speak about paradigms which should be in cluded in the education of every computer science student. Unfortunately, this is very far from reality. This is because these paradigms are not sufficiently known in the computer science community, and so they are insufficiently com municated to students and practitioners.

Understanding Machine Learning

From Theory to Algorithms

Author: Shai Shalev-Shwartz,Shai Ben-David

Publisher: Cambridge University Press

ISBN: 1107057132

Category: Computers

Page: 409

View: 2594

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Introduces machine learning and its algorithmic paradigms, explaining the principles behind automated learning approaches and the considerations underlying their usage.

Variational Calculus and Optimal Control

Optimization with Elementary Convexity

Author: John L. Troutman

Publisher: Springer Science & Business Media

ISBN: 1461207371

Category: Mathematics

Page: 462

View: 1964

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An introduction to the variational methods used to formulate and solve mathematical and physical problems, allowing the reader an insight into the systematic use of elementary (partial) convexity of differentiable functions in Euclidian space. By helping students directly characterize the solutions for many minimization problems, the text serves as a prelude to the field theory for sufficiency, laying as it does the groundwork for further explorations in mathematics, physics, mechanical and electrical engineering, as well as computer science.

Measure Theory and Probability Theory

Author: Krishna B. Athreya,Soumendra N. Lahiri

Publisher: Springer Science & Business Media

ISBN: 038732903X

Category: Business & Economics

Page: 618

View: 3386

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This is a graduate level textbook on measure theory and probability theory. The book can be used as a text for a two semester sequence of courses in measure theory and probability theory, with an option to include supplemental material on stochastic processes and special topics. It is intended primarily for first year Ph.D. students in mathematics and statistics although mathematically advanced students from engineering and economics would also find the book useful. Prerequisites are kept to the minimal level of an understanding of basic real analysis concepts such as limits, continuity, differentiability, Riemann integration, and convergence of sequences and series. A review of this material is included in the appendix. The book starts with an informal introduction that provides some heuristics into the abstract concepts of measure and integration theory, which are then rigorously developed. The first part of the book can be used for a standard real analysis course for both mathematics and statistics Ph.D. students as it provides full coverage of topics such as the construction of Lebesgue-Stieltjes measures on real line and Euclidean spaces, the basic convergence theorems, L^p spaces, signed measures, Radon-Nikodym theorem, Lebesgue's decomposition theorem and the fundamental theorem of Lebesgue integration on R, product spaces and product measures, and Fubini-Tonelli theorems. It also provides an elementary introduction to Banach and Hilbert spaces, convolutions, Fourier series and Fourier and Plancherel transforms. Thus part I would be particularly useful for students in a typical Statistics Ph.D. program if a separate course on real analysis is not a standard requirement. Part II (chapters 6-13) provides full coverage of standard graduate level probability theory. It starts with Kolmogorov's probability model and Kolmogorov's existence theorem. It then treats thoroughly the laws of large numbers including renewal theory and ergodic theorems with applications and then weak convergence of probability distributions, characteristic functions, the Levy-Cramer continuity theorem and the central limit theorem as well as stable laws. It ends with conditional expectations and conditional probability, and an introduction to the theory of discrete time martingales. Part III (chapters 14-18) provides a modest coverage of discrete time Markov chains with countable and general state spaces, MCMC, continuous time discrete space jump Markov processes, Brownian motion, mixing sequences, bootstrap methods, and branching processes. It could be used for a topics/seminar course or as an introduction to stochastic processes. Krishna B. Athreya is a professor at the departments of mathematics and statistics and a Distinguished Professor in the College of Liberal Arts and Sciences at the Iowa State University. He has been a faculty member at University of Wisconsin, Madison; Indian Institute of Science, Bangalore; Cornell University; and has held visiting appointments in Scandinavia and Australia. He is a fellow of the Institute of Mathematical Statistics USA; a fellow of the Indian Academy of Sciences, Bangalore; an elected member of the International Statistical Institute; and serves on the editorial board of several journals in probability and statistics. Soumendra N. Lahiri is a professor at the department of statistics at the Iowa State University. He is a fellow of the Institute of Mathematical Statistics, a fellow of the American Statistical Association, and an elected member of the International Statistical Institute.