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Positive Polynomials And Sums Of Squares
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Book Synopsis Positive Polynomials and Sums of Squares by : Murray Marshall
Download or read book Positive Polynomials and Sums of Squares written by Murray Marshall and published by American Mathematical Soc.. This book was released on 2008 with total page 201 pages. Available in PDF, EPUB and Kindle. Book excerpt: The study of positive polynomials brings together algebra, geometry and analysis. The subject is of fundamental importance in real algebraic geometry when studying the properties of objects defined by polynomial inequalities. Hilbert's 17th problem and its solution in the first half of the 20th century were landmarks in the early days of the subject. More recently, new connections to the moment problem and to polynomial optimization have been discovered. The moment problem relates linear maps on the multidimensional polynomial ring to positive Borel measures. This book provides an elementary introduction to positive polynomials and sums of squares, the relationship to the moment problem, and the application to polynomial optimization. The focus is on the exciting new developments that have taken place in the last 15 years, arising out of Schmudgen's solution to the moment problem in the compact case in 1991. The book is accessible to a well-motivated student at the beginning graduate level. The objects being dealt with are concrete and down-to-earth, namely polynomials in $n$ variables with real coefficients, and many examples are included. Proofs are presented as clearly and as simply as possible. Various new, simpler proofs appear in the book for the first time. Abstraction is employed only when it serves a useful purpose, but, at the same time, enough abstraction is included to allow the reader easy access to the literature. The book should be essential reading for any beginning student in the area.
Book Synopsis Certificates of Positivity for Real Polynomials by : Victoria Powers
Download or read book Certificates of Positivity for Real Polynomials written by Victoria Powers and published by Springer Nature. This book was released on 2021-11-26 with total page 161 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book collects and explains the many theorems concerning the existence of certificates of positivity for polynomials that are positive globally or on semialgebraic sets. A certificate of positivity for a real polynomial is an algebraic identity that gives an immediate proof of a positivity condition for the polynomial. Certificates of positivity have their roots in fundamental work of David Hilbert from the late 19th century on positive polynomials and sums of squares. Because of the numerous applications of certificates of positivity in mathematics, applied mathematics, engineering, and other fields, it is desirable to have methods for finding, describing, and characterizing them. For many of the topics covered in this book, appropriate algorithms, computational methods, and applications are discussed. This volume contains a comprehensive, accessible, up-to-date treatment of certificates of positivity, written by an expert in the field. It provides an overview of both the theory and computational aspects of the subject, and includes many of the recent and exciting developments in the area. Background information is given so that beginning graduate students and researchers who are not specialists can learn about this fascinating subject. Furthermore, researchers who work on certificates of positivity or use them in applications will find this a useful reference for their work.
Book Synopsis Moments, Positive Polynomials and Their Applications by : Jean-Bernard Lasserre
Download or read book Moments, Positive Polynomials and Their Applications written by Jean-Bernard Lasserre and published by World Scientific. This book was released on 2010 with total page 384 pages. Available in PDF, EPUB and Kindle. Book excerpt: 1. The generalized moment problem. 1.1. Formulations. 1.2. Duality theory. 1.3. Computational complexity. 1.4. Summary. 1.5. Exercises. 1.6. Notes and sources -- 2. Positive polynomials. 2.1. Sum of squares representations and semi-definite optimization. 2.2. Nonnegative versus s.o.s. polynomials. 2.3. Representation theorems : univariate case. 2.4. Representation theorems : mutivariate case. 2.5. Polynomials positive on a compact basic semi-algebraic set. 2.6. Polynomials nonnegative on real varieties. 2.7. Representations with sparsity properties. 2.8. Representation of convex polynomials. 2.9. Summary. 2.10. Exercises. 2.11. Notes and sources -- 3. Moments. 3.1. The one-dimensional moment problem. 3.2. The multi-dimensional moment problem. 3.3. The K-moment problem. 3.4. Moment conditions for bounded density. 3.5. Summary. 3.6. Exercises. 3.7. Notes and sources -- 4. Algorithms for moment problems. 4.1. The overall approach. 4.2. Semidefinite relaxations. 4.3. Extraction of solutions. 4.4. Linear relaxations. 4.5. Extensions. 4.6. Exploiting sparsity. 4.7. Summary. 4.8. Exercises. 4.9. Notes and sources. 4.10. Proofs -- 5. Global optimization over polynomials. 5.1. The primal and dual perspectives. 5.2. Unconstrained polynomial optimization. 5.3. Constrained polynomial optimization : semidefinite relaxations. 5.4. Linear programming relaxations. 5.5. Global optimality conditions. 5.6. Convex polynomial programs. 5.7. Discrete optimization. 5.8. Global minimization of a rational function. 5.9. Exploiting symmetry. 5.10. Summary. 5.11. Exercises. 5.12. Notes and sources -- 6. Systems of polynomial equations. 6.1. Introduction. 6.2. Finding a real solution to systems of polynomial equations. 6.3. Finding all complex and/or all real solutions : a unified treatment. 6.4. Summary. 6.5. Exercises. 6.6. Notes and sources -- 7. Applications in probability. 7.1. Upper bounds on measures with moment conditions. 7.2. Measuring basic semi-algebraic sets. 7.3. Measures with given marginals. 7.4. Summary. 7.5. Exercises. 7.6. Notes and sources -- 8. Markov chains applications. 8.1. Bounds on invariant measures. 8.2. Evaluation of ergodic criteria. 8.3. Summary. 8.4. Exercises. 8.5. Notes and sources -- 9. Application in mathematical finance. 9.1. Option pricing with moment information. 9.2. Option pricing with a dynamic model. 9.3. Summary. 9.4. Notes and sources -- 10. Application in control. 10.1. Introduction. 10.2. Weak formulation of optimal control problems. 10.3. Semidefinite relaxations for the OCP. 10.4. Summary. 10.5. Notes and sources -- 11. Convex envelope and representation of convex sets. 11.1. The convex envelope of a rational function. 11.2. Semidefinite representation of convex sets. 11.3. Algebraic certificates of convexity. 11.4. Summary. 11.5. Exercises. 11.6. Notes and sources -- 12. Multivariate integration 12.1. Integration of a rational function. 12.2. Integration of exponentials of polynomials. 12.3. Maximum entropy estimation. 12.4. Summary. 12.5. Exercises. 12.6. Notes and sources -- 13. Min-max problems and Nash equilibria. 13.1. Robust polynomial optimization. 13.2. Minimizing the sup of finitely many rational cunctions. 13.3. Application to Nash equilibria. 13.4. Exercises. 13.5. Notes and sources -- 14. Bounds on linear PDE. 14.1. Linear partial differential equations. 14.2. Notes and sources
Book Synopsis Positive Polynomials in Control by : Didier Henrion
Download or read book Positive Polynomials in Control written by Didier Henrion and published by Springer Science & Business Media. This book was released on 2005-01-14 with total page 332 pages. Available in PDF, EPUB and Kindle. Book excerpt: Positive Polynomials in Control originates from an invited session presented at the IEEE CDC 2003 and gives a comprehensive overview of existing results in this quickly emerging area. This carefully edited book collects important contributions from several fields of control, optimization, and mathematics, in order to show different views and approaches of polynomial positivity. The book is organized in three parts, reflecting the current trends in the area: 1. applications of positive polynomials and LMI optimization to solve various control problems, 2. a mathematical overview of different algebraic techniques used to cope with polynomial positivity, 3. numerical aspects of positivity of polynomials, and recently developed software tools which can be employed to solve the problems discussed in the book.
Book Synopsis Emerging Applications of Algebraic Geometry by : Mihai Putinar
Download or read book Emerging Applications of Algebraic Geometry written by Mihai Putinar and published by Springer Science & Business Media. This book was released on 2008-12-10 with total page 382 pages. Available in PDF, EPUB and Kindle. Book excerpt: Recent advances in both the theory and implementation of computational algebraic geometry have led to new, striking applications to a variety of fields of research. The articles in this volume highlight a range of these applications and provide introductory material for topics covered in the IMA workshops on "Optimization and Control" and "Applications in Biology, Dynamics, and Statistics" held during the IMA year on Applications of Algebraic Geometry. The articles related to optimization and control focus on burgeoning use of semidefinite programming and moment matrix techniques in computational real algebraic geometry. The new direction towards a systematic study of non-commutative real algebraic geometry is well represented in the volume. Other articles provide an overview of the way computational algebra is useful for analysis of contingency tables, reconstruction of phylogenetic trees, and in systems biology. The contributions collected in this volume are accessible to non-experts, self-contained and informative; they quickly move towards cutting edge research in these areas, and provide a wealth of open problems for future research.
Book Synopsis Positive Trigonometric Polynomials and Signal Processing Applications by : Bogdan Dumitrescu
Download or read book Positive Trigonometric Polynomials and Signal Processing Applications written by Bogdan Dumitrescu and published by Springer. This book was released on 2017-03-20 with total page 282 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book gathers the main recent results on positive trigonometric polynomials within a unitary framework. The book has two parts: theory and applications. The theory of sum-of-squares trigonometric polynomials is presented unitarily based on the concept of Gram matrix (extended to Gram pair or Gram set). The applications part is organized as a collection of related problems that use systematically the theoretical results.
Book Synopsis Positive Polynomials by : Alexander Prestel
Download or read book Positive Polynomials written by Alexander Prestel and published by Springer Science & Business Media. This book was released on 2013-04-17 with total page 269 pages. Available in PDF, EPUB and Kindle. Book excerpt: Positivity is one of the most basic mathematical concepts, involved in many areas of mathematics (analysis, real algebraic geometry, functional analysis, etc.). The main objective of the book is to give useful characterizations of polynomials. Beyond basic knowledge in algebra, only valuation theory as explained in the appendix is needed.
Book Synopsis From Polynomials to Sums of Squares by : T.H Jackson
Download or read book From Polynomials to Sums of Squares written by T.H Jackson and published by CRC Press. This book was released on 2023-05-09 with total page 200 pages. Available in PDF, EPUB and Kindle. Book excerpt: From Polynomials to Sums of Squares describes a journey through the foothills of algebra and number theory based around the central theme of factorization. The book begins by providing basic knowledge of rational polynomials, then gradually introduces other integral domains, and eventually arrives at sums of squares of integers. The text is complemented with illustrations that feature specific examples. Other than familiarity with complex numbers and some elementary number theory, very little mathematical prerequisites are needed. The accompanying disk enables readers to explore the subject further by removing the tedium of doing calculations by hand. Throughout the text there are practical activities involving the computer.
Book Synopsis Sum of Squares: Theory and Applications by : Pablo A. Parrilo
Download or read book Sum of Squares: Theory and Applications written by Pablo A. Parrilo and published by American Mathematical Soc.. This book was released on with total page 142 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume is based on lectures delivered at the 2019 AMS Short Course “Sum of Squares: Theory and Applications”, held January 14–15, 2019, in Baltimore, Maryland. This book provides a concise state-of-the-art overview of the theory and applications of polynomials that are sums of squares. This is an exciting and timely topic, with rich connections to many areas of mathematics, including polynomial and semidefinite optimization, real and convex algebraic geometry, and theoretical computer science. The six chapters introduce and survey recent developments in this area; specific topics include the algebraic and geometric aspects of sums of squares and spectrahedra, lifted representations of convex sets, and the algorithmic and computational implications of viewing sums of squares as a meta algorithm. The book also showcases practical applications of the techniques across a variety of areas, including control theory, statistics, finance and machine learning.
Book Synopsis Semidefinite Optimization and Convex Algebraic Geometry by : Grigoriy Blekherman
Download or read book Semidefinite Optimization and Convex Algebraic Geometry written by Grigoriy Blekherman and published by SIAM. This book was released on 2013-03-21 with total page 487 pages. Available in PDF, EPUB and Kindle. Book excerpt: An accessible introduction to convex algebraic geometry and semidefinite optimization. For graduate students and researchers in mathematics and computer science.
Author :Bogdan Alexandru Dumitrescu Publisher :Springer Science & Business Media ISBN 13 :1402051255 Total Pages :253 pages Book Rating :4.4/5 (2 download)
Book Synopsis Positive Trigonometric Polynomials and Signal Processing Applications by : Bogdan Alexandru Dumitrescu
Download or read book Positive Trigonometric Polynomials and Signal Processing Applications written by Bogdan Alexandru Dumitrescu and published by Springer Science & Business Media. This book was released on 2007-04-27 with total page 253 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book gathers the main recent results on positive trigonometric polynomials within a unitary framework. The book has two parts: theory and applications. The theory of sum-of-squares trigonometric polynomials is presented unitarily based on the concept of Gram matrix (extended to Gram pair or Gram set). The applications part is organized as a collection of related problems that use systematically the theoretical results.
Book Synopsis Recent Advances in Optimization and its Applications in Engineering by : Moritz Diehl
Download or read book Recent Advances in Optimization and its Applications in Engineering written by Moritz Diehl and published by Springer Science & Business Media. This book was released on 2010-09-21 with total page 535 pages. Available in PDF, EPUB and Kindle. Book excerpt: Mathematical optimization encompasses both a rich and rapidly evolving body of fundamental theory, and a variety of exciting applications in science and engineering. The present book contains a careful selection of articles on recent advances in optimization theory, numerical methods, and their applications in engineering. It features in particular new methods and applications in the fields of optimal control, PDE-constrained optimization, nonlinear optimization, and convex optimization. The authors of this volume took part in the 14th Belgian-French-German Conference on Optimization (BFG09) organized in Leuven, Belgium, on September 14-18, 2009. The volume contains a selection of reviewed articles contributed by the conference speakers as well as three survey articles by plenary speakers and two papers authored by the winners of the best talk and best poster prizes awarded at BFG09. Researchers and graduate students in applied mathematics, computer science, and many branches of engineering will find in this book an interesting and useful collection of recent ideas on the methods and applications of optimization.
Book Synopsis The Joint Spectral Radius by : Raphaël Jungers
Download or read book The Joint Spectral Radius written by Raphaël Jungers and published by Springer. This book was released on 2009-05-15 with total page 146 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph is based on the Ph.D. Thesis of the author [58]. Its goal is twofold: First, it presents most researchwork that has been done during his Ph.D., or at least the part of the work that is related with the joint spectral radius. This work was concerned with theoretical developments (part I) as well as the study of some applications (part II). As a second goal, it was the author’s feeling that a survey on the state of the art on the joint spectral radius was really missing in the literature, so that the ?rst two chapters of part I present such a survey. The other chapters mainly report personal research, except Chapter 5 which presents animportantapplicationofthejointspectralradius:thecontinuityofwavelet functions. The ?rst part of this monograph is dedicated to theoretical results. The ?rst two chapters present the above mentioned survey on the joint spectral radius. Its minimum-growth counterpart, the joint spectral subradius, is also considered. The next two chapters point out two speci?c theoretical topics, that are important in practical applications: the particular case of nonne- tive matrices, and the Finiteness Property. The second part considers applications involving the joint spectral radius.
Book Synopsis Fast Motions in Biomechanics and Robotics by : Moritz Diehl
Download or read book Fast Motions in Biomechanics and Robotics written by Moritz Diehl and published by Springer. This book was released on 2007-07-13 with total page 446 pages. Available in PDF, EPUB and Kindle. Book excerpt: In the past decades, much progress has been made in the field of walking robots. The current state of technology makes it possible to create humanoid robots that nearly walk like a human being, climb stairs, or avoid small - stacles. However, the dream of a robot running as fast and as elegantly as a human is still far from becoming reality. Control of such fast motions is still a big technological issue in robotics, and the maximum running speed of contemporary robots is still much smaller than that of human track runners. The conventional control approach that most of these robots are based on does not seem to be suitable to increase the running speeds up to a biological level. In order to address this challenge, we invited an interdisciplinary community of researchers from robotics, biomechanics, control engineering and applied mathematics to come together in Heidelberg at the Symposium “Fast Motions in Biomechanics and Robotics – Optimization & Feedback Control” which was held at the International Science Forum (IWH) on September 7–9, 2005. The number of participants in this symposium was kept small in order to promote discussions and enable a fruitful exchange of ideas.
Book Synopsis The Moment Problem by : Konrad Schmüdgen
Download or read book The Moment Problem written by Konrad Schmüdgen and published by Springer. This book was released on 2017-11-09 with total page 512 pages. Available in PDF, EPUB and Kindle. Book excerpt: This advanced textbook provides a comprehensive and unified account of the moment problem. It covers the classical one-dimensional theory and its multidimensional generalization, including modern methods and recent developments. In both the one-dimensional and multidimensional cases, the full and truncated moment problems are carefully treated separately. Fundamental concepts, results and methods are developed in detail and accompanied by numerous examples and exercises. Particular attention is given to powerful modern techniques such as real algebraic geometry and Hilbert space operators. A wide range of important aspects are covered, including the Nevanlinna parametrization for indeterminate moment problems, canonical and principal measures for truncated moment problems, the interplay between Positivstellensätze and moment problems on semi-algebraic sets, the fibre theorem, multidimensional determinacy theory, operator-theoretic approaches, and the existence theory and important special topics of multidimensional truncated moment problems. The Moment Problem will be particularly useful to graduate students and researchers working on moment problems, functional analysis, complex analysis, harmonic analysis, real algebraic geometry, polynomial optimization, or systems theory. With notes providing useful background information and exercises of varying difficulty illustrating the theory, this book will also serve as a reference on the subject and can be used for self-study.
Book Synopsis Mathematical Systems Theory in Biology, Communications, Computation and Finance by : Joachim Rosenthal
Download or read book Mathematical Systems Theory in Biology, Communications, Computation and Finance written by Joachim Rosenthal and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 508 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains survey and research articles by some of the leading researchers in mathematical systems theory - a vibrant research area in its own right. Many authors have taken special care that their articles are self-contained and accessible also to non-specialists.
Book Synopsis Real Algebraic Geometry and Optimization by : Thorsten Theobald
Download or read book Real Algebraic Geometry and Optimization written by Thorsten Theobald and published by American Mathematical Society. This book was released on 2024-04-17 with total page 312 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides a comprehensive and user-friendly exploration of the tremendous recent developments that reveal the connections between real algebraic geometry and optimization, two subjects that were usually taught separately until the beginning of the 21st century. Real algebraic geometry studies the solutions of polynomial equations and polynomial inequalities over the real numbers. Real algebraic problems arise in many applications, including science and engineering, computer vision, robotics, and game theory. Optimization is concerned with minimizing or maximizing a given objective function over a feasible set. Presenting key ideas from classical and modern concepts in real algebraic geometry, this book develops related convex optimization techniques for polynomial optimization. The connection to optimization invites a computational view on real algebraic geometry and opens doors to applications. Intended as an introduction for students of mathematics or related fields at an advanced undergraduate or graduate level, this book serves as a valuable resource for researchers and practitioners. Each chapter is complemented by a collection of beneficial exercises, notes on references, and further reading. As a prerequisite, only some undergraduate algebra is required.
