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تعداد نتایج: 1259

Conjugate Duality in Convex Optimization

By Radu Ioan-Bot
Publisher: Springer, 2011-03-03
Pages: 164
Language: en
ISBN_10:364204915X
ISBN_13:9783642049156
Subject: Business & Economics

Conjugate Duality in Convex Optimization

جزئیات و درخواست

By Radu Ioan Bot
Publisher: Springer Science & Business Media, 2009-12-24
Pages: 171
Language: en
ISBN_13:9783642049002
ISBN_10:3642049001
Subject: Business & Economics

Conjugate Duality and Optimization

جزئیات و درخواست

By R. Tyrrell Rockafellar
Publisher: SIAM, 1974-01-01
Pages: 80
Language: en
ISBN_10:1611970520
ISBN_13:9781611970524
Subject: Technology & Engineering

Convex Duality and Financial Mathematics

جزئیات و درخواست

By Peter Carr, Qiji Jim Zhu
Publisher: Springer, 2018-07-18
Pages: 162
Language: en
ISBN_13:9783319924922
ISBN_10:3319924923
Subject: Mathematics

Convex Optimization

جزئیات و درخواست

By Stephen P. Boyd, Lieven Vandenberghe
Publisher: Cambridge University Press, 2004-03-08
Pages: 744
Language: en
ISBN_10:0521833787
ISBN_13:9780521833783
Subject: Business & Economics

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

By Heinz H. Bauschke, Patrick L. Combettes
Publisher: Springer, 2017-02-28
Pages: 624
Language: en
ISBN_13:9783319483115
ISBN_10:3319483110
Subject: Mathematics

Convex Analysis and Nonlinear Optimization

Theory and Examples

جزئیات و درخواست

By Jonathan Borwein, Adrian S. Lewis
Publisher: Springer Science & Business Media, 2010-05-05
Pages: 316
Language: en
ISBN_13:9780387312569
ISBN_10:0387312560
Subject: Mathematics

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

Semi-Infinite Programming

Recent Advances

جزئیات و درخواست

By Miguel Ángel Goberna, Marco A. López
Publisher: Springer Science & Business Media, 2013-11-11
Pages: 392
Language: en
ISBN_13:9781475734034
ISBN_10:1475734034
Subject: Computers

Optimality Conditions in Convex Optimization

A Finite-Dimensional View

جزئیات و درخواست

By Anulekha Dhara, Joydeep Dutta
Publisher: CRC Press, 2011-10-17
Pages: 446
Language: en
ISBN_13:9781439868225
ISBN_10:1439868220
Subject: Business & Economics

Network Flows and Monotropic Optimization

جزئیات و درخواست

By R. Tyrell Rockafellar
Publisher: Athena Scientific, 1999-06-01
Pages: 632
Language: en
ISBN_13:9781886529069
ISBN_10:188652906X
Subject: Mathematics

Convexity and Duality in Optimization

Proceedings of the Symposium on Convexity and Duality in Opt...

جزئیات و درخواست

By Jacob Ponstein
Publisher: Springer Science & Business Media, 2012-12-06
Pages: 151
Language: en
ISBN_13:9783642456107
ISBN_10:3642456103
Subject: Business & Economics

Discrete Convex Analysis

جزئیات و درخواست

By Kazuo Murota
Publisher: SIAM, 2003-01-01
Pages: 411
Language: en
ISBN_10:0898718503
ISBN_13:9780898718508
Subject: Mathematics

First-order and Stochastic Optimization Methods for Machine Learning

جزئیات و درخواست

By Guanghui Lan
Publisher: Springer Nature, 2020-05-15
Pages: 591
Language: en
ISBN_13:9783030395681
ISBN_10:3030395685
Subject: Mathematics

A Mathematical View of Interior-point Methods in Convex Optimization

جزئیات و درخواست

By James Renegar
Publisher: SIAM, 2001-01-01
Pages: 124
Language: en
ISBN_10:0898718813
ISBN_13:9780898718812
Subject: Mathematics

Convex Analysis

(PMS-28)

جزئیات و درخواست

By Ralph Tyrell Rockafellar
Publisher: Princeton University Press, 2015-04-29
Pages: 470
Language: en
ISBN_13:9781400873173
ISBN_10:1400873177
Subject: Mathematics

Lectures on Modern Convex Optimization

Analysis, Algorithms, and Engineering Applications

جزئیات و درخواست

By Aharon Ben-Tal, Arkadi Nemirovski
Publisher: SIAM, 2001-01-01
Pages: 500
Language: en
ISBN_13:9780898714913
ISBN_10:0898714915
Subject: Technology & Engineering

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