Full Read and Download Reference eBook
Real Algebraic Geometry And Optimization Book in PDF, ePub and Kindle version is available to download in english. Read online anytime anywhere directly from your device. Click on the download button below to get a free pdf file of Real Algebraic Geometry And Optimization book. This book definitely worth reading, it is an incredibly well-written.
Author : Grigoriy Blekherman
Publisher : SIAM
Page : 487 pages
File Size : 35,87 MB
Release : 2013-03-21
Category : Mathematics
ISBN : 1611972280
An accessible introduction to convex algebraic geometry and semidefinite optimization. For graduate students and researchers in mathematics and computer science.
Author : Thorsten Theobald
Publisher : American Mathematical Society
Page : 312 pages
File Size : 31,66 MB
Release : 2024-04-17
Category : Mathematics
ISBN : 147047431X
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 : Saugata Basu
Publisher : Springer Science & Business Media
Page : 602 pages
File Size : 45,52 MB
Release : 2013-03-09
Category : Mathematics
ISBN : 3662053551
In this first-ever graduate textbook on the algorithmic aspects of real algebraic geometry, the main ideas and techniques presented form a coherent and rich body of knowledge, linked to many areas of mathematics and computing. Mathematicians already aware of real algebraic geometry will find relevant information about the algorithmic aspects. Researchers in computer science and engineering will find the required mathematical background. This self-contained book is accessible to graduate and undergraduate students.
Author : Saugata Basu
Publisher : Springer Science & Business Media
Page : 665 pages
File Size : 28,53 MB
Release : 2007-04-21
Category : Mathematics
ISBN : 3540330992
This is the first graduate textbook on the algorithmic aspects of real algebraic geometry. The main ideas and techniques presented form a coherent and rich body of knowledge. Mathematicians will find relevant information about the algorithmic aspects. Researchers in computer science and engineering will find the required mathematical background. Being self-contained the book is accessible to graduate students and even, for invaluable parts of it, to undergraduate students. This second edition contains several recent results on discriminants of symmetric matrices and other relevant topics.
Author : Claus Scheiderer
Publisher : Springer
Page : 0 pages
File Size : 21,44 MB
Release : 2024-10-24
Category : Mathematics
ISBN : 9783031692123
This textbook is designed for a one-year graduate course in real algebraic geometry, with a particular focus on positivity and sums of squares of polynomials. The first half of the book features a thorough introduction to ordered fields and real closed fields, including the Tarski–Seidenberg projection theorem and transfer principle. Classical results such as Artin's solution to Hilbert's 17th problem and Hilbert's theorems on sums of squares of polynomials are presented in detail. Other features include careful introductions to the real spectrum and to the geometry of semialgebraic sets. The second part studies Archimedean positivstellensätze in great detail and in various settings, together with important applications. The techniques and results presented here are fundamental to contemporary approaches to polynomial optimization. Important results on sums of squares on projective varieties are covered as well. The last part highlights applications to semidefinite programming and polynomial optimization, including recent research on semidefinite representation of convex sets. Written by a leading expert and based on courses taught for several years, the book assumes familiarity with the basics of commutative algebra and algebraic varieties, as can be covered in a one-semester first course. Over 350 exercises, of all levels of difficulty, are included in the book.
Author : Mihai Putinar
Publisher : Springer Science & Business Media
Page : 382 pages
File Size : 43,32 MB
Release : 2008-12-10
Category : Mathematics
ISBN : 0387096868
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 : Jesus A. De Loera
Publisher : SIAM
Page : 341 pages
File Size : 18,41 MB
Release : 2012-01-01
Category : Mathematics
ISBN : 9781611972443
This book presents recent advances in the mathematical theory of discrete optimization, particularly those supported by methods from algebraic geometry, commutative algebra, convex and discrete geometry, generating functions, and other tools normally considered outside the standard curriculum in optimization.
Author : Cynthia Leslie Vinzant
Publisher :
Page : 254 pages
File Size : 38,94 MB
Release : 2011
Category :
ISBN :
In the past twenty years, a strong interplay has developed between convex optimization and algebraic geometry. Algebraic geometry provides necessary tools to analyze the behavior of solutions, the geometry of feasible sets, and to develop new relaxations for hard non-convex problems. On the other hand, numerical solvers for convex optimization have led to new fast algorithms in real algebraic geometry. In Chapter 1, we introduce some of the necessary background in convex optimization and real algebraic geometry and discuss some of the important results and questions in their intersection. One of the biggest of which is: when can a convex closed semialgebraic set be the feasible set of a semidefinite program and how can one construct such a representation? In Chapter 2, we explore the consequences of an ideal having a real radical initial ideal, both for the geometry its real variety and as an application to sums of squares representations of polynomials. We show that if the initial ideal of an ideal is real radical for a vector in the tropical variety, then this vectors belongs to logarithmic set of its real variety. We also give algebraic sufficient conditions for a ray to be in the logarithmic limit set of a more general semialgebraic set. If, in addition, the ray has positive coordinates, then the corresponding quadratic module is stable, which has consequences for problems in polynomial optimization. In particular, if an ideal has a is real radical initial ideal for some positive weight vector, then the preorder generated by the ideal is stable. This provides a method for checking the conditions for stability given by Powers and Scheiderer. In Chapter 3, we examine fundamental objects in convex algebraic geometry, such as definite determinantal representations and sums of squares, in the special case of plane quartics. A smooth quartic curve in the complex projective plane has 36 inequivalent representations as a symmetric determinant of linear forms and 63 representations as a sum of three squares. These correspond to Cayley octads and Steiner complexes respectively. We present exact algorithms for computing these objects from the 28 bitangents. This expresses Vinnikov quartics as spectrahedra and positive quartics as Gram matrices. We explore the geometry of Gram spectrahedra and discuss methods for computing determinantal representations. Interwoven are many examples and an exposition of much of the 19th century theory of plane quartics. In Chapter 4, we study real algebraic curves that control interior point methods in linear programming. The central curve of a linear program is an algebraic curve specified by linear and quadratic constraints arising from complementary slackness. It is the union of the various central paths for minimizing or maximizing the cost function over any region in the associated hyperplane arrangement. We determine the degree, arithmetic genus and defining prime ideal of the central curve, thereby answering a question of Bayer and Lagarias. These invariants, along with the degree of the Gauss image of the curve, are expressed in terms of the matroid of the input matrix. Extending work of Dedieu, Malajovich and Shub, this yields an instance-specific bound on the total curvature of the central path, a quantity relevant for interior point methods. The global geometry of central curves is studied in detail. Chapter 5 has two parts. In the first, we study the kt̂h symmetric trigonometric moment curve and its convex hull, the Barvinok-Novik orbitope. In 2008, Barvinok and Novik introduce these objects and show that there is some threshold so that for two points on S1̂ with arclength below this threshold the line segment between their lifts to the curve form an edge on the Barvinok-Novik orbitope and for points with arclength above this threshold, their lifts do not form an edge. They also give a lower bound for this threshold and conjecture that this bound is tight. Results of Smilansky prove tightness for k=2. Here we prove this conjecture for all k. In the second part, we discuss the convex hull of a general parametrized curve. These convex hulls can be written as spectrahedral shadows and, as we shall demonstrate, one can compute and effectively describe their faces.
Author : Saugata Basu
Publisher : Springer
Page : 662 pages
File Size : 35,94 MB
Release : 2009-09-02
Category : Mathematics
ISBN : 9783540821953
This is the first graduate textbook on the algorithmic aspects of real algebraic geometry. The main ideas and techniques presented form a coherent and rich body of knowledge. Mathematicians will find relevant information about the algorithmic aspects. Researchers in computer science and engineering will find the required mathematical background. Being self-contained the book is accessible to graduate students and even, for invaluable parts of it, to undergraduate students. This second edition contains several recent results on discriminants of symmetric matrices and other relevant topics.
Author : Jean Bernard Lasserre
Publisher : Cambridge University Press
Page : 355 pages
File Size : 16,41 MB
Release : 2015-02-19
Category : Mathematics
ISBN : 1316240398
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.