Positive Polynomials And Sums Of Squares

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Positive Polynomials and Sums of Squares

Murray Marshall
Positive Polynomials and Sums of Squares Book PDF

Author : Murray Marshall
Publisher : American Mathematical Soc.
Page : 201 pages
File Size : 51,8 Mb
Release : 2008
Category : Geometry, Algebraic
ISBN : 9780821844021

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Positive Polynomials and Sums of Squares by Murray Marshall Pdf

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.

Moments, Positive Polynomials and Their Applications

Jean-Bernard Lasserre
Moments Positive Polynomials and Their Applications Book PDF

Author : Jean-Bernard Lasserre
Publisher : World Scientific
Page : 384 pages
File Size : 55,5 Mb
Release : 2010
Category : Mathematics
ISBN : 9781848164468

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Moments, Positive Polynomials and Their Applications by Jean-Bernard Lasserre Pdf

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

Positive Polynomials in Control

Didier Henrion,Andrea Garulli
Positive Polynomials in Control Book PDF

Author : Didier Henrion,Andrea Garulli
Publisher : Springer Science & Business Media
Page : 332 pages
File Size : 46,7 Mb
Release : 2005-01-14
Category : Technology & Engineering
ISBN : 3540239480

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Positive Polynomials in Control by Didier Henrion,Andrea Garulli Pdf

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.

Certificates of Positivity for Real Polynomials

Victoria Powers
Certificates of Positivity for Real Polynomials Book PDF

Author : Victoria Powers
Publisher : Springer Nature
Page : 161 pages
File Size : 46,6 Mb
Release : 2021-11-26
Category : Mathematics
ISBN : 9783030855475

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Certificates of Positivity for Real Polynomials by Victoria Powers Pdf

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.

Positive Trigonometric Polynomials and Signal Processing Applications

Bogdan Dumitrescu
Positive Trigonometric Polynomials and Signal Processing Applications Book PDF

Author : Bogdan Dumitrescu
Publisher : Springer
Page : 282 pages
File Size : 54,6 Mb
Release : 2017-03-20
Category : Technology & Engineering
ISBN : 9783319536880

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Positive Trigonometric Polynomials and Signal Processing Applications by Bogdan Dumitrescu Pdf

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.

Emerging Applications of Algebraic Geometry

Mihai Putinar,Seth Sullivant
Emerging Applications of Algebraic Geometry Book PDF

Author : Mihai Putinar,Seth Sullivant
Publisher : Springer Science & Business Media
Page : 382 pages
File Size : 42,9 Mb
Release : 2008-12-10
Category : Mathematics
ISBN : 9780387096865

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Emerging Applications of Algebraic Geometry by Mihai Putinar,Seth Sullivant Pdf

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.

Positive Polynomials

Alexander Prestel,Charles Delzell
Positive Polynomials Book PDF

Author : Alexander Prestel,Charles Delzell
Publisher : Springer Science & Business Media
Page : 269 pages
File Size : 43,7 Mb
Release : 2013-04-17
Category : Mathematics
ISBN : 9783662046487

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Positive Polynomials by Alexander Prestel,Charles Delzell Pdf

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.

From Polynomials to Sums of Squares

T.H Jackson
From Polynomials to Sums of Squares Book PDF

Author : T.H Jackson
Publisher : CRC Press
Page : 200 pages
File Size : 46,6 Mb
Release : 2023-05-09
Category : Mathematics
ISBN : 9781000948783

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From Polynomials to Sums of Squares by T.H Jackson Pdf

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.

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 : 51,5 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.

Positive Trigonometric Polynomials and Signal Processing Applications

Bogdan Alexandru Dumitrescu
Positive Trigonometric Polynomials and Signal Processing Applications Book PDF

Author : Bogdan Alexandru Dumitrescu
Publisher : Springer Science & Business Media
Page : 253 pages
File Size : 43,9 Mb
Release : 2007-04-27
Category : Mathematics
ISBN : 9781402051258

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Positive Trigonometric Polynomials and Signal Processing Applications by Bogdan Alexandru Dumitrescu Pdf

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.

Recent Advances in Optimization and its Applications in Engineering

Moritz Diehl,Francois Glineur,Elias Jarlebring,Wim Michiels
Recent Advances in Optimization and its Applications in Engineering Book PDF

Author : Moritz Diehl,Francois Glineur,Elias Jarlebring,Wim Michiels
Publisher : Springer Science & Business Media
Page : 535 pages
File Size : 41,7 Mb
Release : 2010-09-21
Category : Technology & Engineering
ISBN : 9783642125980

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Recent Advances in Optimization and its Applications in Engineering by Moritz Diehl,Francois Glineur,Elias Jarlebring,Wim Michiels Pdf

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.

The Joint Spectral Radius

Raphaël Jungers
The Joint Spectral Radius Book PDF

Author : Raphaël Jungers
Publisher : Springer
Page : 146 pages
File Size : 51,7 Mb
Release : 2009-05-15
Category : Technology & Engineering
ISBN : 9783540959809

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The Joint Spectral Radius by Raphaël Jungers Pdf

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.

Sum of Squares: Theory and Applications

Pablo A. Parrilo,Rekha R. Thomas
Sum of Squares Theory and Applications Book PDF

Author : Pablo A. Parrilo,Rekha R. Thomas
Publisher : American Mathematical Soc.
Page : 142 pages
File Size : 45,8 Mb
Release : 2024-06-30
Category : Education
ISBN : 9781470450250

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Sum of Squares: Theory and Applications by Pablo A. Parrilo,Rekha R. Thomas Pdf

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.

Fast Motions in Biomechanics and Robotics

Moritz Diehl,Katja Mombaur
Fast Motions in Biomechanics and Robotics Book PDF

Author : Moritz Diehl,Katja Mombaur
Publisher : Springer
Page : 446 pages
File Size : 45,9 Mb
Release : 2007-07-13
Category : Technology & Engineering
ISBN : 9783540361190

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Fast Motions in Biomechanics and Robotics by Moritz Diehl,Katja Mombaur Pdf

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.

The Moment Problem

Konrad Schmüdgen
The Moment Problem Book PDF

Author : Konrad Schmüdgen
Publisher : Springer
Page : 512 pages
File Size : 49,7 Mb
Release : 2017-11-09
Category : Mathematics
ISBN : 9783319645469

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The Moment Problem by Konrad Schmüdgen Pdf

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.