Convex Optimization & Euclidean Distance Geometry

Convex Optimization & Euclidean Distance Geometry PDF Author: Jon Dattorro
Publisher: Meboo Publishing USA
ISBN: 0976401304
Category : Mathematics
Languages : en
Pages : 776

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Book Description
The study of Euclidean distance matrices (EDMs) fundamentally asks what can be known geometrically given onlydistance information between points in Euclidean space. Each point may represent simply locationor, abstractly, any entity expressible as a vector in finite-dimensional Euclidean space.The answer to the question posed is that very much can be known about the points;the mathematics of this combined study of geometry and optimization is rich and deep.Throughout we cite beacons of historical accomplishment.The application of EDMs has already proven invaluable in discerning biological molecular conformation.The emerging practice of localization in wireless sensor networks, the global positioning system (GPS), and distance-based pattern recognitionwill certainly simplify and benefit from this theory.We study the pervasive convex Euclidean bodies and their various representations.In particular, we make convex polyhedra, cones, and dual cones more visceral through illustration, andwe study the geometric relation of polyhedral cones to nonorthogonal bases biorthogonal expansion.We explain conversion between halfspace- and vertex-descriptions of convex cones,we provide formulae for determining dual cones,and we show how classic alternative systems of linear inequalities or linear matrix inequalities and optimality conditions can be explained by generalized inequalities in terms of convex cones and their duals.The conic analogue to linear independence, called conic independence, is introducedas a new tool in the study of classical cone theory; the logical next step in the progression:linear, affine, conic.Any convex optimization problem has geometric interpretation.This is a powerful attraction: the ability to visualize geometry of an optimization problem.We provide tools to make visualization easier.The concept of faces, extreme points, and extreme directions of convex Euclidean bodiesis explained here, crucial to understanding convex optimization.The convex cone of positive semidefinite matrices, in particular, is studied in depth.We mathematically interpret, for example,its inverse image under affine transformation, and we explainhow higher-rank subsets of its boundary united with its interior are convex.The Chapter on "Geometry of convex functions",observes analogies between convex sets and functions:The set of all vector-valued convex functions is a closed convex cone.Included among the examples in this chapter, we show how the real affinefunction relates to convex functions as the hyperplane relates to convex sets.Here, also, pertinent results formultidimensional convex functions are presented that are largely ignored in the literature;tricks and tips for determining their convexityand discerning their geometry, particularly with regard to matrix calculus which remains largely unsystematizedwhen compared with the traditional practice of ordinary calculus.Consequently, we collect some results of matrix differentiation in the appendices.The Euclidean distance matrix (EDM) is studied,its properties and relationship to both positive semidefinite and Gram matrices.We relate the EDM to the four classical axioms of the Euclidean metric;thereby, observing the existence of an infinity of axioms of the Euclidean metric beyondthe triangle inequality. We proceed byderiving the fifth Euclidean axiom and then explain why furthering this endeavoris inefficient because the ensuing criteria (while describing polyhedra)grow linearly in complexity and number.Some geometrical problems solvable via EDMs,EDM problems posed as convex optimization, and methods of solution arepresented;\eg, we generate a recognizable isotonic map of the United States usingonly comparative distance information (no distance information, only distance inequalities).We offer a new proof of the classic Schoenberg criterion, that determines whether a candidate matrix is an EDM. Our proofrelies on fundamental geometry; assuming, any EDM must correspond to a list of points contained in some polyhedron(possibly at its vertices) and vice versa.It is not widely known that the Schoenberg criterion implies nonnegativity of the EDM entries; proved here.We characterize the eigenvalues of an EDM matrix and then devisea polyhedral cone required for determining membership of a candidate matrix(in Cayley-Menger form) to the convex cone of Euclidean distance matrices (EDM cone); \ie,a candidate is an EDM if and only if its eigenspectrum belongs to a spectral cone for EDM^N.We will see spectral cones are not unique.In the chapter "EDM cone", we explain the geometric relationship betweenthe EDM cone, two positive semidefinite cones, and the elliptope.We illustrate geometric requirements, in particular, for projection of a candidate matrixon a positive semidefinite cone that establish its membership to the EDM cone. The faces of the EDM cone are described,but still open is the question whether all its faces are exposed as they are for the positive semidefinite cone.The classic Schoenberg criterion, relating EDM and positive semidefinite cones, isrevealed to be a discretized membership relation (a generalized inequality, a new Farkas''''''''-like lemma)between the EDM cone and its ordinary dual. A matrix criterion for membership to the dual EDM cone is derived thatis simpler than the Schoenberg criterion.We derive a new concise expression for the EDM cone and its dual involvingtwo subspaces and a positive semidefinite cone."Semidefinite programming" is reviewedwith particular attention to optimality conditionsof prototypical primal and dual conic programs,their interplay, and the perturbation method of rank reduction of optimal solutions(extant but not well-known).We show how to solve a ubiquitous platonic combinatorial optimization problem from linear algebra(the optimal Boolean solution x to Ax=b)via semidefinite program relaxation.A three-dimensional polyhedral analogue for the positive semidefinite cone of 3X3 symmetricmatrices is introduced; a tool for visualizing in 6 dimensions.In "EDM proximity"we explore methods of solution to a few fundamental and prevalentEuclidean distance matrix proximity problems; the problem of finding that Euclidean distance matrix closestto a given matrix in the Euclidean sense.We pay particular attention to the problem when compounded with rank minimization.We offer a new geometrical proof of a famous result discovered by Eckart \& Young in 1936 regarding Euclideanprojection of a point on a subset of the positive semidefinite cone comprising all positive semidefinite matriceshaving rank not exceeding a prescribed limit rho.We explain how this problem is transformed to a convex optimization for any rank rho.

Convex Optimization & Euclidean Distance Geometry

Convex Optimization & Euclidean Distance Geometry PDF Author: Jon Dattorro
Publisher: Meboo Publishing USA
ISBN: 0976401304
Category : Mathematics
Languages : en
Pages : 776

Get Book Here

Book Description
The study of Euclidean distance matrices (EDMs) fundamentally asks what can be known geometrically given onlydistance information between points in Euclidean space. Each point may represent simply locationor, abstractly, any entity expressible as a vector in finite-dimensional Euclidean space.The answer to the question posed is that very much can be known about the points;the mathematics of this combined study of geometry and optimization is rich and deep.Throughout we cite beacons of historical accomplishment.The application of EDMs has already proven invaluable in discerning biological molecular conformation.The emerging practice of localization in wireless sensor networks, the global positioning system (GPS), and distance-based pattern recognitionwill certainly simplify and benefit from this theory.We study the pervasive convex Euclidean bodies and their various representations.In particular, we make convex polyhedra, cones, and dual cones more visceral through illustration, andwe study the geometric relation of polyhedral cones to nonorthogonal bases biorthogonal expansion.We explain conversion between halfspace- and vertex-descriptions of convex cones,we provide formulae for determining dual cones,and we show how classic alternative systems of linear inequalities or linear matrix inequalities and optimality conditions can be explained by generalized inequalities in terms of convex cones and their duals.The conic analogue to linear independence, called conic independence, is introducedas a new tool in the study of classical cone theory; the logical next step in the progression:linear, affine, conic.Any convex optimization problem has geometric interpretation.This is a powerful attraction: the ability to visualize geometry of an optimization problem.We provide tools to make visualization easier.The concept of faces, extreme points, and extreme directions of convex Euclidean bodiesis explained here, crucial to understanding convex optimization.The convex cone of positive semidefinite matrices, in particular, is studied in depth.We mathematically interpret, for example,its inverse image under affine transformation, and we explainhow higher-rank subsets of its boundary united with its interior are convex.The Chapter on "Geometry of convex functions",observes analogies between convex sets and functions:The set of all vector-valued convex functions is a closed convex cone.Included among the examples in this chapter, we show how the real affinefunction relates to convex functions as the hyperplane relates to convex sets.Here, also, pertinent results formultidimensional convex functions are presented that are largely ignored in the literature;tricks and tips for determining their convexityand discerning their geometry, particularly with regard to matrix calculus which remains largely unsystematizedwhen compared with the traditional practice of ordinary calculus.Consequently, we collect some results of matrix differentiation in the appendices.The Euclidean distance matrix (EDM) is studied,its properties and relationship to both positive semidefinite and Gram matrices.We relate the EDM to the four classical axioms of the Euclidean metric;thereby, observing the existence of an infinity of axioms of the Euclidean metric beyondthe triangle inequality. We proceed byderiving the fifth Euclidean axiom and then explain why furthering this endeavoris inefficient because the ensuing criteria (while describing polyhedra)grow linearly in complexity and number.Some geometrical problems solvable via EDMs,EDM problems posed as convex optimization, and methods of solution arepresented;\eg, we generate a recognizable isotonic map of the United States usingonly comparative distance information (no distance information, only distance inequalities).We offer a new proof of the classic Schoenberg criterion, that determines whether a candidate matrix is an EDM. Our proofrelies on fundamental geometry; assuming, any EDM must correspond to a list of points contained in some polyhedron(possibly at its vertices) and vice versa.It is not widely known that the Schoenberg criterion implies nonnegativity of the EDM entries; proved here.We characterize the eigenvalues of an EDM matrix and then devisea polyhedral cone required for determining membership of a candidate matrix(in Cayley-Menger form) to the convex cone of Euclidean distance matrices (EDM cone); \ie,a candidate is an EDM if and only if its eigenspectrum belongs to a spectral cone for EDM^N.We will see spectral cones are not unique.In the chapter "EDM cone", we explain the geometric relationship betweenthe EDM cone, two positive semidefinite cones, and the elliptope.We illustrate geometric requirements, in particular, for projection of a candidate matrixon a positive semidefinite cone that establish its membership to the EDM cone. The faces of the EDM cone are described,but still open is the question whether all its faces are exposed as they are for the positive semidefinite cone.The classic Schoenberg criterion, relating EDM and positive semidefinite cones, isrevealed to be a discretized membership relation (a generalized inequality, a new Farkas''''''''-like lemma)between the EDM cone and its ordinary dual. A matrix criterion for membership to the dual EDM cone is derived thatis simpler than the Schoenberg criterion.We derive a new concise expression for the EDM cone and its dual involvingtwo subspaces and a positive semidefinite cone."Semidefinite programming" is reviewedwith particular attention to optimality conditionsof prototypical primal and dual conic programs,their interplay, and the perturbation method of rank reduction of optimal solutions(extant but not well-known).We show how to solve a ubiquitous platonic combinatorial optimization problem from linear algebra(the optimal Boolean solution x to Ax=b)via semidefinite program relaxation.A three-dimensional polyhedral analogue for the positive semidefinite cone of 3X3 symmetricmatrices is introduced; a tool for visualizing in 6 dimensions.In "EDM proximity"we explore methods of solution to a few fundamental and prevalentEuclidean distance matrix proximity problems; the problem of finding that Euclidean distance matrix closestto a given matrix in the Euclidean sense.We pay particular attention to the problem when compounded with rank minimization.We offer a new geometrical proof of a famous result discovered by Eckart \& Young in 1936 regarding Euclideanprojection of a point on a subset of the positive semidefinite cone comprising all positive semidefinite matriceshaving rank not exceeding a prescribed limit rho.We explain how this problem is transformed to a convex optimization for any rank rho.

Euclidean Distance Geometry

Euclidean Distance Geometry PDF Author: Leo Liberti
Publisher: Springer
ISBN: 3319607928
Category : Mathematics
Languages : en
Pages : 141

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Book Description
This textbook, the first of its kind, presents the fundamentals of distance geometry: theory, useful methodologies for obtaining solutions, and real world applications. Concise proofs are given and step-by-step algorithms for solving fundamental problems efficiently and precisely are presented in Mathematica®, enabling the reader to experiment with concepts and methods as they are introduced. Descriptive graphics, examples, and problems, accompany the real gems of the text, namely the applications in visualization of graphs, localization of sensor networks, protein conformation from distance data, clock synchronization protocols, robotics, and control of unmanned underwater vehicles, to name several. Aimed at intermediate undergraduates, beginning graduate students, researchers, and practitioners, the reader with a basic knowledge of linear algebra will gain an understanding of the basic theories of distance geometry and why they work in real life.

Convex Optimization

Convex Optimization PDF Author: Stephen P. Boyd
Publisher: Cambridge University Press
ISBN: 9780521833783
Category : Business & Economics
Languages : en
Pages : 744

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Book Description
Convex optimization problems arise frequently in many different fields. This book provides a comprehensive introduction to the subject, and shows in detail how such problems can be solved numerically with great efficiency. The book begins with the basic elements of convex sets and functions, and then describes various classes of convex optimization problems. Duality and approximation techniques are then covered, as are statistical estimation techniques. Various geometrical problems are then presented, and there is detailed discussion of unconstrained and constrained minimization problems, and interior-point methods. The focus of the book is on recognizing convex optimization problems and then finding the most appropriate technique for solving them. It contains many worked examples and homework exercises and will appeal to students, researchers and practitioners in fields such as engineering, computer science, mathematics, statistics, finance and economics.

Convex Optimization Et Euclidean Distance Geometry

Convex Optimization Et Euclidean Distance Geometry PDF Author:
Publisher:
ISBN:
Category :
Languages : en
Pages : 710

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Book Description


Inference and Learning from Data

Inference and Learning from Data PDF Author: Ali H. Sayed
Publisher: Cambridge University Press
ISBN: 1009218123
Category : Computers
Languages : en
Pages : 1105

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Book Description
Discover core topics in inference and learning with the first volume of this extraordinary three-volume set.

Cooperative Localization and Navigation

Cooperative Localization and Navigation PDF Author: Chao Gao
Publisher: CRC Press
ISBN: 0429016670
Category : Technology & Engineering
Languages : en
Pages : 807

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Book Description
This book captures the latest results and techniques for cooperative localization and navigation drawn from a broad array of disciplines. It provides the reader with a generic and comprehensive view of modeling, strategies, and state estimation methodologies in that fields. It discusses the most recent research and novel advances in that direction, exploring the design of algorithms and architectures, benefits, and challenging aspects, as well as a potential broad array of disciplines, including wireless communication, indoor localization, robotics, emergency rescue, motion analysis, etc.

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

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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.

Adaptive Wireless Communications

Adaptive Wireless Communications PDF Author: Daniel W. Bliss
Publisher: Cambridge University Press
ISBN: 1107067731
Category : Technology & Engineering
Languages : en
Pages : 619

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Book Description
Adopting a balanced mix of theory, algorithms and practical design issues, this comprehensive volume explores cutting-edge applications in adaptive wireless communications and the implications these techniques have for future wireless network performance. Presenting practical concerns in the context of different strands from information theory, parameter estimation theory, array processing and wireless communication, the authors present a complete picture of the field. Topics covered include advanced multiple-antenna adaptive processing, ad hoc networking, MIMO, MAC protocols, space-time coding, cellular networks and cognitive radio, with the significance and effects of both internal and external interference a recurrent theme throughout. A broad, self-contained technical introduction to all the necessary mathematics, statistics, estimation theory and information theory is included, and topics are accompanied by a range of engaging end-of-chapter problems. With solutions available online, this is the perfect self-study resource for students of advanced wireless systems and wireless industry professionals.

Proceedings of Sixth International Congress on Information and Communication Technology

Proceedings of Sixth International Congress on Information and Communication Technology PDF Author: Xin-She Yang
Publisher: Springer Nature
ISBN: 9811623805
Category : Technology & Engineering
Languages : en
Pages : 1002

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Book Description
This book gathers selected high-quality research papers presented at the Sixth International Congress on Information and Communication Technology, held at Brunel University, London, on February 25–26, 2021. It discusses emerging topics pertaining to information and communication technology (ICT) for managerial applications, e-governance, e-agriculture, e-education and computing technologies, the Internet of things (IoT) and e-mining. Written by respected experts and researchers working on ICT, the book offers a valuable asset for young researchers involved in advanced studies. The book is presented in four volumes.

Elementary Cluster Analysis

Elementary Cluster Analysis PDF Author: James C. Bezdek
Publisher: CRC Press
ISBN: 1000794784
Category : Science
Languages : en
Pages : 516

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Book Description
The availability of packaged clustering programs means that anyone with data can easily do cluster analysis on it. But many users of this technology don't fully appreciate its many hidden dangers. In today's world of "grab and go algorithms," part of my motivation for writing this book is to provide users with a set of cautionary tales about cluster analysis, for it is very much an art as well as a science, and it is easy to stumble if you don't understand its pitfalls. Indeed, it is easy to trip over them even if you do! The parenthetical word usually in the title is very important, because all clustering algorithms can and do fail from time to time. Modern cluster analysis has become so technically intricate that it is often hard for the beginner or the non-specialist to appreciate and understand its many hidden dangers. Here's how Yogi Berra put it, and he was right: In theory there's no difference between theory and practice. In practice, there is ~Yogi Berra This book is a step backwards, to four classical methods for clustering in small, static data sets that have all withstood the tests of time. The youngest of the four methods is now almost 50 years old: Gaussian Mixture Decomposition (GMD, 1898) SAHN Clustering (principally single linkage (SL, 1909)) Hard c-means (HCM, 1956, also widely known as (aka) "k-means") Fuzzy c-means (FCM, 1973, reduces to HCM in a certain limit) The dates are the first known writing (to me, anyway) about these four models. I am (with apologies to Marvel Comics) very comfortable in calling HCM, FCM, GMD and SL the Fantastic Four. Cluster analysis is a vast topic. The overall picture in clustering is quite overwhelming, so any attempt to swim at the deep end of the pool in even a very specialized subfield requires a lot of training. But we all start out at the shallow end (or at least that's where we should start!), and this book is aimed squarely at teaching toddlers not to be afraid of the water. There is no section of this book that, if explored in real depth, cannot be expanded into its own volume. So, if your needs are for an in-depth treatment of all the latest developments in any topic in this volume, the best I can do - what I will try to do anyway - is lead you to the pool, and show you where to jump in.