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Author: Sabine Burgdorf
Publisher: Springer
ISBN: 3319333380
Category : Mathematics
Languages : en
Pages : 118
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Book Description
This book presents recent results on positivity and optimization of polynomials in non-commuting variables. Researchers in non-commutative algebraic geometry, control theory, system engineering, optimization, quantum physics and information science will find the unified notation and mixture of algebraic geometry and mathematical programming useful. Theoretical results are matched with algorithmic considerations; several examples and information on how to use NCSOStools open source package to obtain the results provided. Results are presented on detecting the eigenvalue and trace positivity of polynomials in non-commuting variables using Newton chip method and Newton cyclic chip method, relaxations for constrained and unconstrained optimization problems, semidefinite programming formulations of the relaxations and finite convergence of the hierarchies of these relaxations, and the practical efficiency of algorithms.
Author: Sabine Burgdorf
Publisher: Springer
ISBN: 3319333380
Category : Mathematics
Languages : en
Pages : 118
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Book Description
This book presents recent results on positivity and optimization of polynomials in non-commuting variables. Researchers in non-commutative algebraic geometry, control theory, system engineering, optimization, quantum physics and information science will find the unified notation and mixture of algebraic geometry and mathematical programming useful. Theoretical results are matched with algorithmic considerations; several examples and information on how to use NCSOStools open source package to obtain the results provided. Results are presented on detecting the eigenvalue and trace positivity of polynomials in non-commuting variables using Newton chip method and Newton cyclic chip method, relaxations for constrained and unconstrained optimization problems, semidefinite programming formulations of the relaxations and finite convergence of the hierarchies of these relaxations, and the practical efficiency of algorithms.
Author: Grigoriy Blekherman
Publisher: SIAM
ISBN: 1611972280
Category : Mathematics
Languages : en
Pages : 487
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Book Description
An accessible introduction to convex algebraic geometry and semidefinite optimization. For graduate students and researchers in mathematics and computer science.
Author: Mihai Putinar
Publisher: Springer Science & Business Media
ISBN: 0387096868
Category : Mathematics
Languages : en
Pages : 382
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Book Description
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: Konrad Schmüdgen
Publisher: Springer
ISBN: 3319645463
Category : Mathematics
Languages : en
Pages : 530
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Book Description
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.
Author: Didier Henrion
Publisher: Springer Science & Business Media
ISBN: 9783540239482
Category : Technology & Engineering
Languages : en
Pages : 332
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Book Description
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.
Author: Victor Magron
Publisher: World Scientific
ISBN: 1800612966
Category : Mathematics
Languages : en
Pages : 223
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Book Description
Many applications, including computer vision, computer arithmetic, deep learning, entanglement in quantum information, graph theory and energy networks, can be successfully tackled within the framework of polynomial optimization, an emerging field with growing research efforts in the last two decades. One key advantage of these techniques is their ability to model a wide range of problems using optimization formulations. Polynomial optimization heavily relies on the moment-sums of squares (moment-SOS) approach proposed by Lasserre, which provides certificates for positive polynomials. On the practical side, however, there is 'no free lunch' and such optimization methods usually encompass severe scalability issues. Fortunately, for many applications, including the ones formerly mentioned, we can look at the problem in the eyes and exploit the inherent data structure arising from the cost and constraints describing the problem.This book presents several research efforts to resolve this scientific challenge with important computational implications. It provides the development of alternative optimization schemes that scale well in terms of computational complexity, at least in some identified class of problems. It also features a unified modeling framework to handle a wide range of applications involving both commutative and noncommutative variables, and to solve concretely large-scale instances. Readers will find a practical section dedicated to the use of available open-source software libraries.This interdisciplinary monograph is essential reading for students, researchers and professionals interested in solving optimization problems with polynomial input data.
Author: Klaus Jansen
Publisher: Springer
ISBN: 3540278214
Category : Computers
Languages : en
Pages : 436
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Book Description
This book constitutes the joint refereed proceedings of the 7th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2004 and the 8th International Workshop on Randomization and Computation, RANDOM 2004, held in Cambridge, MA, USA in August 2004. The 37 revised full papers presented were carefully reviewed and selected from 87 submissions. Among the issues addressed are design and analysis of approximation algorithms, inapproximability results, approximation classes, online problems, graph algorithms, cuts, geometric computations, network design and routing, packing and covering, scheduling, game theory, design and analysis of randomised algorithms, randomized complexity theory, pseudorandomness, derandomization, probabilistic proof systems, error-correcting codes, and other applications of approximation and randomness.
Author: Miguel F. Anjos
Publisher: Springer Science & Business Media
ISBN: 1461407699
Category : Business & Economics
Languages : en
Pages : 955
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Book Description
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: Bernd Sturmfels
Publisher: American Mathematical Soc.
ISBN: 0821832514
Category : Mathematics
Languages : en
Pages : 162
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Book Description
Bridging a number of mathematical disciplines, and exposing many facets of systems of polynomial equations, Bernd Sturmfels's study covers a wide spectrum of mathematical techniques and algorithms, both symbolic and numerical.
Author: Themistocles M. Rassias
Publisher: World Scientific
ISBN: 9789810206147
Category : Mathematics
Languages : en
Pages : 658
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Book Description
This volume presents an account of some of the most important work that has been done on various research problems in the theory of polynomials of one and several variables and their applications. It is dedicated to P L Chebyshev, a leading Russian mathematician.
