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Genericity In Polynomial Optimization
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Book Synopsis Genericity In Polynomial Optimization by : Tien Son Pham
Download or read book Genericity In Polynomial Optimization written by Tien Son Pham and published by World Scientific. This book was released on 2016-12-22 with total page 261 pages. Available in PDF, EPUB and Kindle. Book excerpt: In full generality, minimizing a polynomial function over a closed semi-algebraic set requires complex mathematical equations. This book explains recent developments from singularity theory and semi-algebraic geometry for studying polynomial optimization problems. Classes of generic problems are defined in a simple and elegant manner by using only the two basic (and relatively simple) notions of Newton polyhedron and non-degeneracy conditions associated with a given polynomial optimization problem. These conditions are well known in singularity theory, however, they are rarely considered within the optimization community.Explanations focus on critical points and tangencies of polynomial optimization, Hölderian error bounds for polynomial systems, Frank-Wolfe-type theorem for polynomial programs and well-posedness in polynomial optimization. It then goes on to look at optimization for the different types of polynomials. Through this text graduate students, PhD students and researchers of mathematics will be provided with the knowledge necessary to use semi-algebraic geometry in optimization.
Book Synopsis Genericity in Polynomial Optimization by : Huy-Vui Hà
Download or read book Genericity in Polynomial Optimization written by Huy-Vui Hà and published by . This book was released on 2017 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Polynomial Optimization, Moments, and Applications by : Michal Kočvara
Download or read book Polynomial Optimization, Moments, and Applications written by Michal Kočvara and published by Springer Nature. This book was released on 2024-01-28 with total page 274 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Sparse Polynomial Optimization: Theory And Practice by : Victor Magron
Download or read book Sparse Polynomial Optimization: Theory And Practice written by Victor Magron and published by World Scientific. This book was released on 2023-04-25 with total page 223 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Approximation Methods for Polynomial Optimization by : Zhening Li
Download or read book Approximation Methods for Polynomial Optimization written by Zhening Li and published by Springer Science & Business Media. This book was released on 2012-07-25 with total page 129 pages. Available in PDF, EPUB and Kindle. Book excerpt: Polynomial optimization have been a hot research topic for the past few years and its applications range from Operations Research, biomedical engineering, investment science, to quantum mechanics, linear algebra, and signal processing, among many others. In this brief the authors discuss some important subclasses of polynomial optimization models arising from various applications, with a focus on approximations algorithms with guaranteed worst case performance analysis. The brief presents a clear view of the basic ideas underlying the design of such algorithms and the benefits are highlighted by illustrative examples showing the possible applications. This timely treatise will appeal to researchers and graduate students in the fields of optimization, computational mathematics, Operations Research, industrial engineering, and computer science.
Book Synopsis Semidefinite Relaxations Approach to Polynomial Optimization and Its Extensions by : Li Wang
Download or read book Semidefinite Relaxations Approach to Polynomial Optimization and Its Extensions written by Li Wang and published by . This book was released on 2014 with total page 119 pages. Available in PDF, EPUB and Kindle. Book excerpt: The goal of this thesis is to study a special nonlinear programming, namely, polynomial optimization in which both the objective and constraints are polynomials. This kind of problem is always NP-hard even if the objective is nonconvex quadratic and all constraints are linear. The semidefinite (SDP) relaxations approach, based on sum of squares representations, provides us with strong tools to solve polynomial optimization problems with finitely many constraints globally. We first review two SDP relaxation methods for solving polynomial optimization problems with finitely many constraints: the classic Lasserre's SDP relaxation and Jacobian SDP relaxation. In general, these methods relax the polynomial optimization problem as a sequence of SDPs whose optima are the lower bounds of the global minimum and converge to the global minimum under certain assumptions. We also prove that the assumption of nonsingularity in Jacobian SDP relaxation method can be weakened to have finite singularities. Then, we study the problem of minimizing a rational function. We reformulate the problem by the technique of homogenization, the original problem and the reformulated problem are shown to be equivalent under some generic conditions. The constraint set of the reformulated problem may not be compact, and Lasserre's SDP relaxation may not have finite convergence, so we apply Jacobian SDP relaxation to solve the reformulated polynomial optimization problem. Some numerical examples are presented to show the efficiency of this method. Next, we consider the problem of minimizing semi-infinite polynomial programming (SIPP). We propose an exchange algorithm with SDP relaxations to solve SIPP problems with a compact index set globally. And we extend the proposed method to SIPP problems with noncompact index set via homogenization. The reformulated problem is equivalent to original SIPP problem under some generic conditions. At last, we study the problem of finding best rank-1 approximations for both symmetric and nonsymmetric tensor. For symmetric tensors, this is equivalent to optimizing homogeneous polynomials over unit spheres; for nonsymmetric tensors, this is equivalent to optimizing multi-quadratic forms over multi-spheres. We use semidefinite relaxations approach to solve these polynomial optimization problems. Extensive numerical experiments are presented to show that this approach is practical in getting best rank-1 approximations.
Book Synopsis Optimization of Polynomials in Non-Commuting Variables by : Sabine Burgdorf
Download or read book Optimization of Polynomials in Non-Commuting Variables written by Sabine Burgdorf and published by Springer. This book was released on 2016-06-07 with total page 118 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis An Introduction to Polynomial and Semi-Algebraic Optimization by : Jean Bernard Lasserre
Download or read book An Introduction to Polynomial and Semi-Algebraic Optimization written by Jean Bernard Lasserre and published by Cambridge University Press. This book was released on 2015-02-19 with total page 355 pages. Available in PDF, EPUB and Kindle. Book excerpt: The first comprehensive introduction to the powerful moment approach for solving global optimization problems.
Book Synopsis Nondifferentiable Optimization and Polynomial Problems by : N.Z. Shor
Download or read book Nondifferentiable Optimization and Polynomial Problems written by N.Z. Shor and published by Springer Science & Business Media. This book was released on 2013-04-17 with total page 407 pages. Available in PDF, EPUB and Kindle. Book excerpt: Polynomial extremal problems (PEP) constitute one of the most important subclasses of nonlinear programming models. Their distinctive feature is that an objective function and constraints can be expressed by polynomial functions in one or several variables. Let :e = {:e 1, ... , :en} be the vector in n-dimensional real linear space Rn; n PO(:e), PI (:e), ... , Pm (:e) are polynomial functions in R with real coefficients. In general, a PEP can be formulated in the following form: (0.1) find r = inf Po(:e) subject to constraints (0.2) Pi (:e) =0, i=l, ... ,m (a constraint in the form of inequality can be written in the form of equality by introducing a new variable: for example, P( x) ~ 0 is equivalent to P(:e) + y2 = 0). Boolean and mixed polynomial problems can be written in usual form by adding for each boolean variable z the equality: Z2 - Z = O. Let a = {al, ... ,a } be integer vector with nonnegative entries {a;}f=l. n Denote by R[a](:e) monomial in n variables of the form: n R[a](:e) = IT :ef'; ;=1 d(a) = 2:7=1 ai is the total degree of monomial R[a]. Each polynomial in n variables can be written as sum of monomials with nonzero coefficients: P(:e) = L caR[a](:e), aEA{P) IX x Nondifferentiable optimization and polynomial problems where A(P) is the set of monomials contained in polynomial P.
Book Synopsis Moment and Polynomial Optimization by : Jiawang Nie
Download or read book Moment and Polynomial Optimization written by Jiawang Nie and published by SIAM. This book was released on 2023-06-15 with total page 484 pages. Available in PDF, EPUB and Kindle. Book excerpt: Moment and polynomial optimization is an active research field used to solve difficult questions in many areas, including global optimization, tensor computation, saddle points, Nash equilibrium, and bilevel programs, and it has many applications. The author synthesizes current research and applications, providing a systematic introduction to theory and methods, a comprehensive approach for extracting optimizers and solving truncated moment problems, and a creative methodology for using optimality conditions to construct tight Moment-SOS relaxations. This book is intended for applied mathematicians, engineers, and researchers entering the field. It can be used as a textbook for graduate students in courses on convex optimization, polynomial optimization, and matrix and tensor optimization.
Book Synopsis Global Optimization of Polynomial Functions and Applications by : Jiawang Nie
Download or read book Global Optimization of Polynomial Functions and Applications written by Jiawang Nie and published by . This book was released on 2006 with total page 324 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Handbook on Semidefinite, Conic and Polynomial Optimization by : Miguel F. Anjos
Download or read book Handbook on Semidefinite, Conic and Polynomial Optimization written by Miguel F. Anjos and published by Springer Science & Business Media. This book was released on 2011-11-19 with total page 955 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Polynomial Optimization written by and published by . This book was released on 2015 with total page 306 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Semidefinite Optimization and Convex Algebraic Geometry by : Grigoriy Blekherman
Download or read book Semidefinite Optimization and Convex Algebraic Geometry written by Grigoriy Blekherman and published by SIAM. This book was released on 2013-03-21 with total page 487 pages. Available in PDF, EPUB and Kindle. Book excerpt: An accessible introduction to convex algebraic geometry and semidefinite optimization. For graduate students and researchers in mathematics and computer science.
Book Synopsis Foundations of Generic Optimization by : M. Iglesias
Download or read book Foundations of Generic Optimization written by M. Iglesias and published by Springer Science & Business Media. This book was released on 2006-03-30 with total page 303 pages. Available in PDF, EPUB and Kindle. Book excerpt: The success of a genetic algorithm when applied to an optimization problem depends upon several features present or absent in the problem to be solved, including the quality of the encoding of data, the geometric structure of the search space, deception or epistasis. This book deals essentially with the latter notion, presenting, for the first time, a complete state-of-the-art of research on this notion, in a structured, completely self-contained and methodical way. In particular, it contains a refresher on the linear algebra used in the text as well as an elementary introductory chapter on genetic algorithms aimed at readers unacquainted with this notion. In this way, the monograph aims to serve a broad audience consisting of graduate and advanced undergraduate students in mathematics and computer science, as well as researchers working in the domains of optimization, artificial intelligence, theoretical computer science, combinatorics and evolutionary algorithms.
Book Synopsis Moment-sos Hierarchy, The: Lectures In Probability, Statistics, Computational Geometry, Control And Nonlinear Pdes by : Didier Henrion
Download or read book Moment-sos Hierarchy, The: Lectures In Probability, Statistics, Computational Geometry, Control And Nonlinear Pdes written by Didier Henrion and published by World Scientific. This book was released on 2020-11-04 with total page 248 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Moment-SOS hierarchy is a powerful methodology that is used to solve the Generalized Moment Problem (GMP) where the list of applications in various areas of Science and Engineering is almost endless. Initially designed for solving polynomial optimization problems (the simplest example of the GMP), it applies to solving any instance of the GMP whose description only involves semi-algebraic functions and sets. It consists of solving a sequence (a hierarchy) of convex relaxations of the initial problem, and each convex relaxation is a semidefinite program whose size increases in the hierarchy.The goal of this book is to describe in a unified and detailed manner how this methodology applies to solving various problems in different areas ranging from Optimization, Probability, Statistics, Signal Processing, Computational Geometry, Control, Optimal Control and Analysis of a certain class of nonlinear PDEs. For each application, this unconventional methodology differs from traditional approaches and provides an unusual viewpoint. Each chapter is devoted to a particular application, where the methodology is thoroughly described and illustrated on some appropriate examples.The exposition is kept at an appropriate level of detail to aid the different levels of readers not necessarily familiar with these tools, to better know and understand this methodology.
Book Synopsis A Conic Programming Approach to Polynomial Optimization Problems by : Luis F. Zuluaga
Download or read book A Conic Programming Approach to Polynomial Optimization Problems written by Luis F. Zuluaga and published by . This book was released on 2004 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt:
