Certificates Of Positivity For Real Polynomials

Author : Victoria Powers
Publisher : Springer Nature
Page : 161 pages
File Size : 44,7 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.

Author : Alexander Prestel,Charles Delzell
Publisher : Springer Science & Business Media
Page : 269 pages
File Size : 44,6 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.

Author : Dia Zeidan,Juan C. Cortés,Aliaa Burqan,Ahmad Qazza,Jochen Merker,Gharib Gharib
Publisher : Springer Nature
Page : 476 pages
File Size : 42,6 Mb
Release : 2023-05-29
Category : Mathematics
ISBN : 9789819904471

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Mathematics and Computation by Dia Zeidan,Juan C. Cortés,Aliaa Burqan,Ahmad Qazza,Jochen Merker,Gharib Gharib Pdf

This book collects select papers presented at the 7th International Arab Conference on Mathematics and Computations (IACMC 2022), held from 11–13 May 2022, at Zarqa University, Zarqa, Jordan. These papers discuss a new direction for mathematical sciences. Researchers, professionals and educators will be exposed to research results contributed by worldwide scholars in fundamental and advanced interdisciplinary mathematical research such as differential equations, dynamical systems, matrix analysis, numerical methods and mathematical modelling. The vision of this book is to establish prototypes in completed, current and future mathematical and applied sciences research from advanced and developing countries. The book is intended to make an intellectual contribution to the theory and practice of mathematics. This proceedings would connect scientists in this part of the world to the international level.

Author : Thorsten Theobald
Publisher : American Mathematical Society
Page : 312 pages
File Size : 55,9 Mb
Release : 2024-04-18
Category : Mathematics
ISBN : 9781470476366

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Real Algebraic Geometry and Optimization by Thorsten Theobald Pdf

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.

Author : Michal Kočvara,Bernard Mourrain,Cordian Riener
Publisher : Springer Nature
Page : 274 pages
File Size : 47,5 Mb
Release : 2024-01-28
Category : Mathematics
ISBN : 9783031386596

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Polynomial Optimization, Moments, and Applications by Michal Kočvara,Bernard Mourrain,Cordian Riener Pdf

Polynomial optimization is a fascinating field of study that has revolutionized the way we approach nonlinear problems described by polynomial constraints. The applications of this field range from production planning processes to transportation, energy consumption, and resource control. This introductory book explores the latest research developments in polynomial optimization, presenting the results of cutting-edge interdisciplinary work conducted by the European network POEMA. For the past four years, experts from various fields, including algebraists, geometers, computer scientists, and industrial actors, have collaborated in this network to create new methods that go beyond traditional paradigms of mathematical optimization. By exploiting new advances in algebra and convex geometry, these innovative approaches have resulted in significant scientific and technological advancements. This book aims to make these exciting developments accessible to a wider audience by gathering high-quality chapters on these hot topics. Aimed at both aspiring and established researchers, as well as industry professionals, this book will be an invaluable resource for anyone interested in polynomial optimization and its potential for real-world applications.

Author : Alexander Prestel,Charles Delzell
Publisher : Unknown
Page : 280 pages
File Size : 55,7 Mb
Release : 2014-01-15
Category : Electronic
ISBN : 3662046490

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Positive Polynomials by Alexander Prestel,Charles Delzell Pdf

Author : Jean-Bernard Lasserre
Publisher : World Scientific
Page : 384 pages
File Size : 52,5 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

Author : Miguel F. Anjos,Jean B. Lasserre
Publisher : Springer Science & Business Media
Page : 955 pages
File Size : 51,5 Mb
Release : 2011-11-19
Category : Business & Economics
ISBN : 9781461407690

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Handbook on Semidefinite, Conic and Polynomial Optimization by Miguel F. Anjos,Jean B. Lasserre Pdf

Semidefinite and conic optimization is a major and thriving research area within the optimization community. Although semidefinite optimization has been studied (under different names) since at least the 1940s, its importance grew immensely during the 1990s after polynomial-time interior-point methods for linear optimization were extended to solve semidefinite optimization problems. Since the beginning of the 21st century, not only has research into semidefinite and conic optimization continued unabated, but also a fruitful interaction has developed with algebraic geometry through the close connections between semidefinite matrices and polynomial optimization. This has brought about important new results and led to an even higher level of research activity. This Handbook on Semidefinite, Conic and Polynomial Optimization provides the reader with a snapshot of the state-of-the-art in the growing and mutually enriching areas of semidefinite optimization, conic optimization, and polynomial optimization. It contains a compendium of the recent research activity that has taken place in these thrilling areas, and will appeal to doctoral students, young graduates, and experienced researchers alike. The Handbook’s thirty-one chapters are organized into four parts: Theory, covering significant theoretical developments as well as the interactions between conic optimization and polynomial optimization; Algorithms, documenting the directions of current algorithmic development; Software, providing an overview of the state-of-the-art; Applications, dealing with the application areas where semidefinite and conic optimization has made a significant impact in recent years.

Author : Mihai Putinar,Seth Sullivant
Publisher : Springer Science & Business Media
Page : 382 pages
File Size : 50,8 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.

Author : Branislav Bošanský,Cleotilde Gonzalez,Stefan Rass,Arunesh Sinha
Publisher : Springer Nature
Page : 385 pages
File Size : 51,5 Mb
Release : 2021-10-30
Category : Computers
ISBN : 9783030903701

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Decision and Game Theory for Security by Branislav Bošanský,Cleotilde Gonzalez,Stefan Rass,Arunesh Sinha Pdf

This book constitutes the refereed proceedings of the 12th International Conference on Decision and Game Theory for Security, GameSec 2021,held in October 2021. Due to COVID-19 pandemic the conference was held virtually. The 20 full papers presented were carefully reviewed and selected from 37 submissions. The papers focus on Theoretical Foundations in Equilibrium Computation; Machine Learning and Game Theory; Ransomware; Cyber-Physical Systems Security; Innovations in Attacks and Defenses.

Author : Jean Bernard Lasserre
Publisher : Cambridge University Press
Page : 355 pages
File Size : 43,8 Mb
Release : 2015-02-19
Category : Mathematics
ISBN : 9781107060579

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An Introduction to Polynomial and Semi-Algebraic Optimization by Jean Bernard Lasserre Pdf

The first comprehensive introduction to the powerful moment approach for solving global optimization problems.

Author : Jean Bernard Lasserre
Publisher : Cambridge University Press
Page : 0 pages
File Size : 51,9 Mb
Release : 2015-02-19
Category : Mathematics
ISBN : 110763069X

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An Introduction to Polynomial and Semi-Algebraic Optimization by Jean Bernard Lasserre Pdf

This is the first comprehensive introduction to the powerful moment approach for solving global optimization problems (and some related problems) described by polynomials (and even semi-algebraic functions). In particular, the author explains how to use relatively recent results from real algebraic geometry to provide a systematic numerical scheme for computing the optimal value and global minimizers. Indeed, among other things, powerful positivity certificates from real algebraic geometry allow one to define an appropriate hierarchy of semidefinite (SOS) relaxations or LP relaxations whose optimal values converge to the global minimum. Several extensions to related optimization problems are also described. Graduate students, engineers and researchers entering the field can use this book to understand, experiment with and master this new approach through the simple worked examples provided.

Author : David Handelman
Publisher : American Mathematical Soc.
Page : 79 pages
File Size : 43,8 Mb
Release : 1985
Category : Mathematics
ISBN : 9780821823224

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Positive Polynomials and Product Type Actions of Compact Groups by David Handelman Pdf

Author : Hoai An Le Thi,Tao Pham Dinh,Ngoc Thanh Nguyen
Publisher : Springer
Page : 528 pages
File Size : 54,7 Mb
Release : 2015-05-04
Category : Technology & Engineering
ISBN : 9783319181615

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Modelling, Computation and Optimization in Information Systems and Management Sciences by Hoai An Le Thi,Tao Pham Dinh,Ngoc Thanh Nguyen Pdf

This proceedings set contains 85 selected full papers presented at the 3rd International Conference on Modelling, Computation and Optimization in Information Systems and Management Sciences - MCO 2015, held on May 11–13, 2015 at Lorraine University, France. The present part I of the 2 volume set includes articles devoted to Combinatorial optimization and applications, DC programming and DCA: thirty years of Developments, Dynamic Optimization, Modelling and Optimization in financial engineering, Multiobjective programming, Numerical Optimization, Spline Approximation and Optimization, as well as Variational Principles and Applications.

Author : Grigoriy Blekherman,Pablo A. Parrilo,Rekha R. Thomas
Publisher : SIAM
Page : 487 pages
File Size : 44,7 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.