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تعداد نتایج: 686

A Course in Convexity

By Alexander Barvinok
Publisher: American Mathematical Soc., 2002-11-19
Pages: 378
Language: en
ISBN_13:9780821829684
ISBN_10:0821829688
Subject: Mathematics

Convex Sets and Their Applications

جزئیات و درخواست

By Steven R. Lay
Publisher: Courier Corporation, 2007-01-01
Pages: 260
Language: en
ISBN_13:9780486458038
ISBN_10:0486458032
Subject: Mathematics

Geometry of Linear Matrix Inequalities

A Course in Convexity and Real Algebraic Geometry with a Vie...

جزئیات و درخواست

By Tim Netzer, Daniel Plaumann
Publisher: Springer Nature, 2023-06-07
Pages: 167
Language: en
ISBN_13:9783031264559
ISBN_10:303126455X
Subject: Mathematics

Undergraduate Convexity

From Fourier and Motzkin to Kuhn and Tucker

جزئیات و درخواست

By Niels Lauritzen
Publisher: World Scientific, 2013
Pages: 298
Language: en
ISBN_13:9789814412520
ISBN_10:981441252X
Subject: Mathematics

Convexity from the Geometric Point of View

جزئیات و درخواست

By Vitor Balestro
Publisher: Springer Nature,
Pages: 1195
Language: en
ISBN_13:9783031505072
ISBN_10:3031505077

Convex Optimization & Euclidean Distance Geometry

$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.$

جزئیات و درخواست

By Jon Dattorro
Publisher: Meboo Publishing USA, 2005
Pages: 776
Language: en
ISBN_13:9780976401308
ISBN_10:0976401304
Subject: Mathematics

Convex Bodies: The Brunn–Minkowski Theory

جزئیات و درخواست

By Rolf Schneider
Publisher: Cambridge University Press, 2014
Pages: 759
Language: en
ISBN_13:9781107601017
ISBN_10:1107601010
Subject: Mathematics

Lectures on Convex Geometry

جزئیات و درخواست

By Daniel Hug, Wolfgang Weil
Publisher: Springer Nature, 2020-08-27
Pages: 300
Language: en
ISBN_13:9783030501808
ISBN_10:3030501809
Subject: Mathematics

Lectures On Convex Sets (Second Edition)

جزئیات و درخواست

By Valeriu Soltan
Publisher: World Scientific, 2019-11-28
Pages: 611
Language: en
ISBN_13:9789811202131
ISBN_10:9811202133
Subject: Mathematics

Convex Optimization Theory

جزئیات و درخواست

By Dimitri Bertsekas
Publisher: Athena Scientific, 2009-06-01
Pages: 443
Language: en
ISBN_13:9781886529311
ISBN_10:1886529310
Subject: Mathematics

Convex Functions and Their Applications

A Contemporary Approach

جزئیات و درخواست

By Constantin P. Niculescu, Lars-Erik Persson
Publisher: Springer, 2018-06-08
Pages: 430
Language: en
ISBN_13:9783319783376
ISBN_10:3319783378
Subject: Mathematics

Proceedings ...: Convexity and applications; lectures by B. Grünbaum and V. Klee

جزئیات و درخواست

By Lincoln K. Durst
1967
Pages: 172
Language: en
OTHER:STANFORD:36105033275525
Subject: Geometry

Convex Optimization Techniques for Geometric Covering Problems

$The present thesis is a commencement of a generalization of covering results in specific settings, such as the Euclidean space or the sphere, to arbitrary compact metric spaces. In particular we consider coverings of compact metric spaces $(X,d)$ by balls of radius $r$. We are interested in the minimum number of such balls needed to cover $X$, denoted by $\Ncal(X,r)$. For finite $X$ this problem coincides with an instance of the combinatorial \textsc{set cover} problem, which is $\mathrm{NP}$-complete. We illustrate approximation techniques based on the moment method of Lasserre for finite graphs and generalize these techniques to compact metric spaces $X$ to obtain upper and lower bounds for $\Ncal(X,r)$. \\ The upper bounds in this thesis follow from the application of a greedy algorithm on the space $X$. Its approximation quality is obtained by a generalization of the analysis of Chv\'atal's algorithm for the weighted case of \textsc{set cover}. We apply this greedy algorithm to the spherical case $X=S^n$ and retrieve the best non-asymptotic bound of B\"or\"oczky and Wintsche. Additionally, the algorithm can be used to determine coverings of Euclidean space with arbitrary measurable objects having non-empty interior. The quality of these coverings slightly improves a bound of Nasz\'odi. \\ For the lower bounds we develop a sequence of bounds $\Ncal^t(X,r)$ that converge after finitely (say $\alpha\in\N$) many steps: $$\Ncal^1(X,r)\leq \ldots \leq \Ncal^\alpha(X,r)=\Ncal(X,r).$$ The drawback of this sequence is that the bounds $\Ncal^t(X,r)$ are increasingly difficult to compute, since they are the objective values of infinite-dimensional conic programs whose number of constraints and dimension of underlying cones grow accordingly to $t$. We show that these programs satisfy strong duality and derive a finite dimensional semidefinite program to approximate $\Ncal^2(S^2,r)$ to arbitrary precision. Our results rely in part on the moment methods developed by de Laat and Vallentin for the packing problem on topological packing graphs. However, in the covering problem we have to deal with two types of constraints instead of one type as in packing problems and consequently additional work is required.$

جزئیات و درخواست

By Jan Hendrik Rolfes
Publisher: BoD – Books on Demand, 2021-09-15
Pages: 128
Language: de
ISBN_13:9783754346754
ISBN_10:375434675X
Subject: Mathematics

Polytopes and Graphs

جزئیات و درخواست

By Guillermo Pineda Villavicencio
Publisher: Cambridge University Press, 2024-02-29
Pages: 482
Language: en
ISBN_13:9781009257787
ISBN_10:1009257781
Subject: Mathematics

Combinatorial Convexity and Algebraic Geometry

جزئیات و درخواست

By Guenter Ewald
Publisher: Springer Science & Business Media, 1996-10-03
Pages: 402
Language: en
ISBN_10:0387947558
ISBN_13:9780387947556
Subject: Mathematics

A Course on Optimization and Best Approximation

جزئیات و درخواست

By Richard B. Holmes
Publisher: Springer, 1972
Pages: 530
Language: en
OTHER:UCSC:32106013847568
Subject: Mathematics

Linear and Convex Optimization

A Mathematical Approach

جزئیات و درخواست

By Michael H. Veatch
Publisher: John Wiley & Sons, 2021-01-13
Pages: 384
Language: en
ISBN_13:9781119664048
ISBN_10:1119664047
Subject: Mathematics

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