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Sparse Polynomial Optimization: Theory And Practice

Sparse Polynomial Optimization: Theory And Practice PDF Author: Victor Magron
Publisher: World Scientific
ISBN: 1800612966
Category : Mathematics
Languages : en
Pages : 223

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.

Sparse Polynomial Optimization: Theory And Practice

Sparse Polynomial Optimization: Theory And Practice PDF Author: Victor Magron
Publisher: World Scientific
ISBN: 1800612966
Category : Mathematics
Languages : en
Pages : 223

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.

Sparse Polynomial Optimization

Sparse Polynomial Optimization PDF Author: Victor Magron
Publisher: Wspc (Europe)
ISBN: 9781800612945
Category : Mathematical optimization
Languages : en
Pages : 0

Book Description
"For the first time, a detailed account of the theory of sparse polynomial optimization along with numerous illustrations of the theory in explicit examples are shown A practical section is dedicated to the use of available open-source software libraries to help people tackle their own problems. Suited for graduate students, engineers and researchers in optimization, applied mathematics, quantum physics, deep learning, power systems, etc. This book is complementary to The Moment-SOS Hierarchy by Henrion, Korda and Lasserre, in the sense that it does focus on theoretical and practical tools to solve large-scale polynomial optimization problems in an efficient way"--

Polynomial Optimization, Moments, and Applications

Polynomial Optimization, Moments, and Applications PDF Author: Michal Kočvara
Publisher: Springer Nature
ISBN: 3031386590
Category : Mathematics
Languages : en
Pages : 274

Book Description
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.

Sparse Optimization Theory and Methods

Sparse Optimization Theory and Methods PDF Author: Yun-Bin Zhao
Publisher: CRC Press
ISBN: 1351624148
Category : Business & Economics
Languages : en
Pages : 222

Book Description
Seeking sparse solutions of underdetermined linear systems is required in many areas of engineering and science such as signal and image processing. The efficient sparse representation becomes central in various big or high-dimensional data processing, yielding fruitful theoretical and realistic results in these fields. The mathematical optimization plays a fundamentally important role in the development of these results and acts as the mainstream numerical algorithms for the sparsity-seeking problems arising from big-data processing, compressed sensing, statistical learning, computer vision, and so on. This has attracted the interest of many researchers at the interface of engineering, mathematics and computer science. Sparse Optimization Theory and Methods presents the state of the art in theory and algorithms for signal recovery under the sparsity assumption. The up-to-date uniqueness conditions for the sparsest solution of underdertemined linear systems are described. The results for sparse signal recovery under the matrix property called range space property (RSP) are introduced, which is a deep and mild condition for the sparse signal to be recovered by convex optimization methods. This framework is generalized to 1-bit compressed sensing, leading to a novel sign recovery theory in this area. Two efficient sparsity-seeking algorithms, reweighted l1-minimization in primal space and the algorithm based on complementary slackness property, are presented. The theoretical efficiency of these algorithms is rigorously analysed in this book. Under the RSP assumption, the author also provides a novel and unified stability analysis for several popular optimization methods for sparse signal recovery, including l1-mininization, Dantzig selector and LASSO. This book incorporates recent development and the author’s latest research in the field that have not appeared in other books.

Formal Methods

Formal Methods PDF Author: André Platzer
Publisher: Springer Nature
ISBN: 3031711629
Category :
Languages : en
Pages : 692

Book Description


Sparse Polynomial Approximation of High-Dimensional Functions

Sparse Polynomial Approximation of High-Dimensional Functions PDF Author: Ben Adcock
Publisher: SIAM
ISBN: 161197688X
Category : Mathematics
Languages : en
Pages : 310

Book Description
Over seventy years ago, Richard Bellman coined the term “the curse of dimensionality” to describe phenomena and computational challenges that arise in high dimensions. These challenges, in tandem with the ubiquity of high-dimensional functions in real-world applications, have led to a lengthy, focused research effort on high-dimensional approximation—that is, the development of methods for approximating functions of many variables accurately and efficiently from data. This book provides an in-depth treatment of one of the latest installments in this long and ongoing story: sparse polynomial approximation methods. These methods have emerged as useful tools for various high-dimensional approximation tasks arising in a range of applications in computational science and engineering. It begins with a comprehensive overview of best s-term polynomial approximation theory for holomorphic, high-dimensional functions, as well as a detailed survey of applications to parametric differential equations. It then describes methods for computing sparse polynomial approximations, focusing on least squares and compressed sensing techniques. Sparse Polynomial Approximation of High-Dimensional Functions presents the first comprehensive and unified treatment of polynomial approximation techniques that can mitigate the curse of dimensionality in high-dimensional approximation, including least squares and compressed sensing. It develops main concepts in a mathematically rigorous manner, with full proofs given wherever possible, and it contains many numerical examples, each accompanied by downloadable code. The authors provide an extensive bibliography of over 350 relevant references, with an additional annotated bibliography available on the book’s companion website (www.sparse-hd-book.com). This text is aimed at graduate students, postdoctoral fellows, and researchers in mathematics, computer science, and engineering who are interested in high-dimensional polynomial approximation techniques.

SOFSEM 2007: Theory and Practice of Computer Science

SOFSEM 2007: Theory and Practice of Computer Science PDF Author: Jan van Leeuwen
Publisher: Springer
ISBN: 3540695079
Category : Computers
Languages : en
Pages : 956

Book Description
This book constitutes the refereed proceedings of the 33rd Conference on Current Trends in Theory and Practice of Computer Science, SOFSEM 2007, held in Harrachov, Czech Republic in January 2007. The 69 revised full papers, presented together with 11 invited contributions were carefully reviewed and selected from 283 submissions. The papers were organized in four topical tracks.

Iterative Methods for Sparse Linear Systems

Iterative Methods for Sparse Linear Systems PDF Author: Yousef Saad
Publisher: SIAM
ISBN: 0898715342
Category : Mathematics
Languages : en
Pages : 537

Book Description
Mathematics of Computing -- General.

Semidefinite Optimization and Convex Algebraic Geometry

Semidefinite Optimization and Convex Algebraic Geometry PDF Author: Grigoriy Blekherman
Publisher: SIAM
ISBN: 1611972280
Category : Mathematics
Languages : en
Pages : 487

Book Description
An accessible introduction to convex algebraic geometry and semidefinite optimization. For graduate students and researchers in mathematics and computer science.

Handbook on Semidefinite, Conic and Polynomial Optimization

Handbook on Semidefinite, Conic and Polynomial Optimization PDF Author: Miguel F. Anjos
Publisher: Springer Science & Business Media
ISBN: 1461407699
Category : Business & Economics
Languages : en
Pages : 955

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.