The Graph Realization Problem
|
|
|
- Alexandrina Wilkerson
- 6 years ago
- Views:
Transcription
1 The Graph Realization Problem via Semi-Definite Programming A. Y. Alfakih Mathematics and Statistics University of Windsor The Graph Realization Problem p.1/21
2 The Graph Realization Problem Given an edge-weighted graph G = (V, E, ω), G is said to be realizable in R r iff p 1, p 2,..., p n R r such that p i p j 2 = ω ij (i, j) E. The Graph Realization Problem p.2/21
3 The Graph Realization Problem Given an edge-weighted graph G = (V, E, ω), G is said to be realizable in R r iff p 1, p 2,..., p n R r such that C 4 p i p j 2 = ω ij (i, j) E. The Graph Realization Problem p.2/21
4 The Graph Realization Problem Given an edge-weighted graph G = (V, E, ω), G is said to be realizable in R r iff p 1, p 2,..., p n R r such that p i p j 2 = ω ij (i, j) E C 4 p 1 p 2 p 4 p 3 The Graph Realization Problem p.2/21
5 The Graph Realization Problem Given an edge-weighted graph G = (V, E, ω), G is said to be realizable in R r iff p 1, p 2,..., p n R r such that p i p j 2 = ω ij (i, j) E C 4 p 1 p 4 p 2 p 3 The Graph Realization Problem p.2/21
6 GRP Complexity Given G = (V, E, ω) and integer k, the problem: Does G have a realization in R k, is NP-hard. The Graph Realization Problem p.3/21
7 GRP Complexity Given G = (V, E, ω) and integer k, the problem: Does G have a realization in R k, is NP-hard. Given G = (V, E, ω), the complexity of the problem: Does G have a realization in some Euclidean space, is still an open problem. The Graph Realization Problem p.3/21
8 GRP Complexity Given G = (V, E, ω) and integer k, the problem: Does G have a realization in R k, is NP-hard. Given G = (V, E, ω), the complexity of the problem: Does G have a realization in some Euclidean space, is still an open problem. Given G = (V, E, ω) and ǫ > 0, the problem: Does G have a realization, with tolerance ǫ, in some Euclidean space is polynomial. The Graph Realization Problem p.3/21
9 Among the many applications of GRP are: molecular conformation problems, multidimensional scaling, wireless sensor networks. The Graph Realization Problem p.4/21
10 Among the many applications of GRP are: molecular conformation problems, multidimensional scaling, wireless sensor networks. In this talk we will present: The Graph Realization Problem p.4/21
11 Among the many applications of GRP are: molecular conformation problems, multidimensional scaling, wireless sensor networks. In this talk we will present: A Semi-definite Programming model for the GRP. The Graph Realization Problem p.4/21
12 Among the many applications of GRP are: molecular conformation problems, multidimensional scaling, wireless sensor networks. In this talk we will present: A Semi-definite Programming model for the GRP. A characterization of the set of all realizations of graph G. The Graph Realization Problem p.4/21
13 Among the many applications of GRP are: molecular conformation problems, multidimensional scaling, wireless sensor networks. In this talk we will present: A Semi-definite Programming model for the GRP. A characterization of the set of all realizations of graph G. A randomized algorithm for obtaining low-dimensional realizations of graph G. The Graph Realization Problem p.4/21
14 Euclidean distance matrices (EDMs) An n n matrix D = (d ij ) is said to be an EDM iff p 1, p 2,..., p n R r such that d ij = p i p j 2 i, j = 1,..., n. The Graph Realization Problem p.5/21
15 Euclidean distance matrices (EDMs) An n n matrix D = (d ij ) is said to be an EDM iff p 1, p 2,..., p n R r such that d ij = p i p j 2 i, j = 1,..., n. The dimension of the affine span of p 1,..., p n is called the embedding dim of D. The Graph Realization Problem p.5/21
16 Example D = is an EDM. The Graph Realization Problem p.6/21
17 D = Example is an EDM. The points that generate D are: p 1 p 2 p 4 p 3 The Graph Realization Problem p.6/21
18 D = Example is an EDM. The points that generate D are: p 1 p 2 p 4 p 3 Embedding dim of D is 2. The Graph Realization Problem p.6/21
19 GRP = EDM Completion Problem Given a partial matrix A where only some of its elements are specified, The EDMCP is the problem of determining whether or not A can be completed into an EDM. The Graph Realization Problem p.7/21
20 GRP = EDM Completion Problem Given a partial matrix A where only some of its elements are specified, The EDMCP is the problem of determining whether or not A can be completed into an EDM. Let G = C 4 then A G = The Graph Realization Problem p.7/21
21 GRP = EDM Completion Problem Given a partial matrix A where only some of its elements are specified, The EDMCP is the problem of determining whether or not A can be completed into an EDM. A = Embedding dim = 2. The Graph Realization Problem p.7/21
22 GRP = EDM Completion Problem Given a partial matrix A where only some of its elements are specified, The EDMCP is the problem of determining whether or not A can be completed into an EDM. A = Embedding dim = 1. The Graph Realization Problem p.7/21
23 EDM Characterization (Schoenberg 35, Young and Householder 38) An n n symmetric D with diag(d) = 0 is EDM iff B := 1 2 JDJ 0 where J = I eet /n, e = (1,..., 1), and embedding dim of D = rank B. The Graph Realization Problem p.8/21
24 EDM Characterization (Schoenberg 35, Young and Householder 38) An n n symmetric D with diag(d) = 0 is EDM iff B := 1 2 JDJ 0 where J = I eet /n, e = (1,..., 1), and embedding dim of D = rank B. The points p 1,..., p n that generate D are given by the rows of P where B = PP T. The Graph Realization Problem p.8/21
25 SDP formulation of EDMCP (Alfakih,Khandani,Wolkowicz) min H (A K V (X)) 2 F subject to X 0. The Graph Realization Problem p.9/21
26 SDP formulation of EDMCP (Alfakih,Khandani,Wolkowicz) min H (A K V (X)) 2 F subject to X 0. H is the adjacency matrix of G and denotes Hadamard product. The Graph Realization Problem p.9/21
27 SDP formulation of EDMCP (Alfakih,Khandani,Wolkowicz) min H (A K V (X)) 2 F subject to X 0. rank X = r H is the adjacency matrix of G and denotes Hadamard product. The Graph Realization Problem p.9/21
28 SDP formulation of EDMCP (Alfakih,Khandani,Wolkowicz) min H (A K V (X)) 2 F subject to X 0. H is the adjacency matrix of G and denotes Hadamard product. This SDP can be solved efficiently using a primal-dual interior point algorithm. The Graph Realization Problem p.9/21
29 SDP formulation of EDMCP (Alfakih,Khandani,Wolkowicz) min H (A K V (X)) 2 F subject to X 0. H is the adjacency matrix of G and denotes Hadamard product. This SDP can be solved efficiently using a primal-dual interior point algorithm. The realization obtained will have the largest possible embedding dimension. The Graph Realization Problem p.9/21
30 Ω, Set of all Realizations of G Let X 1 be a given realization of G = (V, E, ω). The Graph Realization Problem p.10/21
31 Ω, Set of all Realizations of G Let X 1 be a given realization of G = (V, E, ω). Let Ω := {y : X(y) := X 1 + (i,j) E y ijm ij 0}. The Graph Realization Problem p.10/21
32 Ω, Set of all Realizations of G Let X 1 be a given realization of G = (V, E, ω). Let Ω := {y : X(y) := X 1 + (i,j) E y ijm ij 0}. The set of all realizations of G in R r is : {X(y) : y Ω, and rank X(y) = r}. The Graph Realization Problem p.10/21
33 Ω, Set of all Realizations of G Let X 1 be a given realization of G = (V, E, ω). Let Ω := {y : X(y) := X 1 + (i,j) E y ijm ij 0}. The set of all realizations of G in R r is : {X(y) : y Ω, and rank X(y) = r}. The set of all realizations of G in all spaces is : {X(y) : y Ω}. The Graph Realization Problem p.10/21
34 Properties of Ω Ω is a convex, closed and in general a non-polyhedral set. The Graph Realization Problem p.11/21
35 Properties of Ω Ω is a convex, closed and in general a non-polyhedral set. Ω always contains the origin. The Graph Realization Problem p.11/21
36 Properties of Ω Ω is a convex, closed and in general a non-polyhedral set. Ω always contains the origin. Ω is bounded whenever graph G is connected. The Graph Realization Problem p.11/21
37 Ω, An Example Given the following realization of C 4 : p 1 p 2 p 4 p X(0) 3 The Graph Realization Problem p.12/21
38 Ω, An Example Given the following realization of C 4 : p 1 p 2 p 4 p X(0) 3 y 2 y 24 y 13 y 1 Set Ω for C 4 The Graph Realization Problem p.12/21
39 Ω, An Example Given the following realization of C 4 : p 1 p 2 p 4 p X(0) 3 y 2 y 24 3 y 13 y 1 Set Ω for C 4 The Graph Realization Problem p.12/21
40 Ω, An Example Given the following realization of C 4 : p 1 p 2 y 2 y 24 y 3 13 y 1 p 4 p X(0) 3 p 4 p 1 p 3 p 2 X(y 2 ) Set Ω for C 4 The Graph Realization Problem p.12/21
41 Ω, An Example Given the following realization of C 4 : p 1 p 2 p 4 p X(0) 3 p 1 p 2 p 4 p 3 X(y 1 ) y 2 y 24 3 y 13 y 1 Set Ω for C 4 The Graph Realization Problem p.12/21
42 Faces of Ω The facial structure of Ω is closely related to that of the positive semidefinite cone. The Graph Realization Problem p.13/21
43 Faces of Ω The facial structure of Ω is closely related to that of the positive semidefinite cone. The relative interior of faces of Ω are characterized by Gale matrices. The Graph Realization Problem p.13/21
44 Gale Matrices [ ] Given p 1,..., p n R r Λ 1, let Λ = be the Λ 2 matrix whose[ columns form a] basis for p 1 p 2 p n nullspace of The Graph Realization Problem p.14/21
45 Gale Matrices [ ] Given p 1,..., p n R r Λ 1, let Λ = be the Λ 2 matrix whose[ columns form a] basis for p 1 p 2 p n nullspace of [ Z := ΛΛ 1 I n 1 r 1 = Λ 2 Λ 1 is called the Gale 1 matrix corresponding to the p i s. ] The Graph Realization Problem p.14/21
46 For the following configuration: p 1 p 2 p 4 p 3 The Graph Realization Problem p.15/21
47 For the following configuration: p 1 p 2 p 4 p 3 Z = [1, 1, 1, 1] T The Graph Realization Problem p.15/21
48 For the following configuration: p 1 p 2 p 4 p 3 Z = [1, 1, 1, 1] T Theorem (Alfakih): Let y 1, y 2 Ω and let F(y 1 ) denote the smallest face of Ω containing y 1. Then F(y 1 ) = F(y 2 ) iff the realizations X(y 1 ) and X(y 2 ) have the same Gale matrix. The Graph Realization Problem p.15/21
49 Faces of Ω, An Example y 2 y 1 Set Ω for C 4 The Graph Realization Problem p.16/21
50 Faces of Ω, An Example y 2 y 1 Set Ω for C 4 p 1 p 2 p 4 p 3 The Graph Realization Problem p.16/21
51 Faces of Ω, An Example y 2 y 1 Set Ω for C 4 p 1 p 2 p 4 p 3 The Graph Realization Problem p.16/21
52 Faces of Ω, An Example y 2 y 1 Set Ω for C 4 p 1 p 2 p 4 p 3 The Graph Realization Problem p.16/21
53 Faces of Ω, An Example y 2 y 1 Set Ω for C 4 p 1 p 2 p 4 p 3 The Graph Realization Problem p.16/21
54 Faces of Ω, An Example y 2 y 1 Set Ω for C 4 p 1 p 2 p 4 p 3 The Graph Realization Problem p.16/21
55 Faces of Ω, An Example y 2 y 1 Set Ω for C 4 p 1 p 2 p 4 p 3 The Graph Realization Problem p.16/21
56 Extreme Points of Ω Theorem(Barvinok, Pataki, Deza and Laurent): If a G = (V, E, ω) has a realization then it has a realization in R r where r 8 E The Graph Realization Problem p.17/21
57 Extreme Points of Ω Theorem(Barvinok, Pataki, Deza and Laurent): If a G = (V, E, ω) has a realization then it has a realization in R r where r 8 E Theorem: if ŷ is an extreme point of Ω then 8 E +1 1 rank X(ŷ) 2. The Graph Realization Problem p.17/21
58 Extreme Points of Ω Theorem(Barvinok, Pataki, Deza and Laurent): If a G = (V, E, ω) has a realization then it has a realization in R r where r 8 E Theorem: if ŷ is an extreme point of Ω then 8 E +1 1 rank X(ŷ) 2. Theorem(Barvinok): If G is not a union of a complete graph with r + 2 nodes and zero or more isolated nodes, then it has a realization in R r where r 8( E 1) The Graph Realization Problem p.17/21
59 Low-Dimensional Realizations Lemma (Alfakih): Assume Ω is full dimensional and let y Ω. If dim N Ω (y) s(s+1) 2 for some positive integer s, then rank X(y) n 1 s. The Graph Realization Problem p.18/21
60 Low-Dimensional Realizations Lemma (Alfakih): Assume Ω is full dimensional and let y Ω. If dim N Ω (y) s(s+1) 2 for some positive integer s, then rank X(y) n 1 s. The Graph Realization Problem p.18/21
61 Low-Dimensional Realizations Lemma (Alfakih): Assume Ω is full dimensional and let y Ω. If dim N Ω (y) s(s+1) 2 for some positive integer s, then rank X(y) n 1 s. The Graph Realization Problem p.18/21
62 Randomized Algorithm Input: Realization X 1 of G, positive integer t. The Graph Realization Problem p.19/21
63 Randomized Algorithm Input: Realization X 1 of G, positive integer t. for k = 1,..., t do The Graph Realization Problem p.19/21
64 Randomized Algorithm Input: Realization X 1 of G, positive integer t. for k = 1,..., t do 1- Let c k be random vector selected randomly from the unit sphere in R m. The Graph Realization Problem p.19/21
65 Randomized Algorithm Input: Realization X 1 of G, positive integer t. for k = 1,..., t do 1- Let c k be random vector selected randomly from the unit sphere in R m. 2- Solve the SDP problem: max {c kt y : y Ω} and let y k be the corresponding optimal solution. The Graph Realization Problem p.19/21
66 Randomized Algorithm Input: Realization X 1 of G, positive integer t. for k = 1,..., t do 1- Let c k be random vector selected randomly from the unit sphere in R m. 2- Solve the SDP problem: max {c kt y : y Ω} and let y k be the corresponding optimal solution. Pick y min = arg min {X(y k ) : k = 1,..., t}. The Graph Realization Problem p.19/21
67 c ij be m numbers randomly drawn from N(0, 1). Then c = 1 c c is uniform over the unit sphere in R m. The Graph Realization Problem p.20/21
68 c ij be m numbers randomly drawn from N(0, 1). Then c = 1 c c is uniform over the unit sphere in R m. The SDP is solved using SeDuMi 1.05R5 of Sturm. The Graph Realization Problem p.20/21
69 c ij be m numbers randomly drawn from N(0, 1). Then c = 1 c c is uniform over the unit sphere in R m. The SDP is solved using SeDuMi 1.05R5 of Sturm. Numerical tests on 3.06 GHz Pentium-4 PC with 496 MB of RAM. The Graph Realization Problem p.20/21
70 Numerical Tests n = 20 m = Barvinok bound = rank X(y ) The CPU time for optimizing c T y over Ω is in order of 1 sec. The Graph Realization Problem p.21/21
Semidefinite Facial Reduction for Euclidean Distance Matrix Completion
Semidefinite Facial Reduction for Euclidean Distance Matrix Completion Nathan Krislock, Henry Wolkowicz Department of Combinatorics & Optimization University of Waterloo First Alpen-Adria Workshop on Optimization
Spherical Euclidean Distance Embedding of a Graph
Spherical Euclidean Distance Embedding of a Graph Hou-Duo Qi University of Southampton Presented at Isaac Newton Institute Polynomial Optimization August 9, 2013 Spherical Embedding Problem The Problem:
On the eigenvalues of Euclidean distance matrices
Volume 27, N. 3, pp. 237 250, 2008 Copyright 2008 SBMAC ISSN 00-8205 www.scielo.br/cam On the eigenvalues of Euclidean distance matrices A.Y. ALFAKIH Department of Mathematics and Statistics University
Explicit Sensor Network Localization using Semidefinite Programming and Clique Reductions
Explicit Sensor Network Localization using Semidefinite Programming and Clique Reductions Nathan Krislock and Henry Wolkowicz Dept. of Combinatorics and Optimization University of Waterloo Haifa Matrix
A Unified Theorem on SDP Rank Reduction. yyye
SDP Rank Reduction Yinyu Ye, EURO XXII 1 A Unified Theorem on SDP Rank Reduction Yinyu Ye Department of Management Science and Engineering and Institute of Computational and Mathematical Engineering Stanford
Facial Reduction for Cone Optimization
Facial Reduction for Cone Optimization Henry Wolkowicz Dept. Combinatorics and Optimization University of Waterloo Tuesday Apr. 21, 2015 (with: Drusvyatskiy, Krislock, (Cheung) Voronin) Motivation: Loss
Facial reduction for Euclidean distance matrix problems
Facial reduction for Euclidean distance matrix problems Nathan Krislock Department of Mathematical Sciences Northern Illinois University, USA Distance Geometry: Theory and Applications DIMACS Center, Rutgers
SEMIDEFINITE PROGRAM BASICS. Contents
SEMIDEFINITE PROGRAM BASICS BRIAN AXELROD Abstract. A introduction to the basics of Semidefinite programs. Contents 1. Definitions and Preliminaries 1 1.1. Linear Algebra 1 1.2. Convex Analysis (on R n
SOCP Relaxation of Sensor Network Localization
SOCP Relaxation of Sensor Network Localization Paul Tseng Mathematics, University of Washington Seattle University of Vienna/Wien June 19, 2006 Abstract This is a talk given at Univ. Vienna, 2006. SOCP
Euclidean Distance Matrices, Semidefinite Programming, and Sensor Network Localization
Euclidean Distance Matrices, Semidefinite Programming, and Sensor Network Localization Abdo Y. Alfakih Miguel F. Anjos Veronica Piccialli Henry Wolkowicz November 2, 2009 University of Waterloo Department
Further Relaxations of the SDP Approach to Sensor Network Localization
Further Relaxations of the SDP Approach to Sensor Network Localization Zizhuo Wang, Song Zheng, Stephen Boyd, and inyu e September 27, 2006 Abstract Recently, a semidefinite programming (SDP) relaxation
Solving Euclidean Distance Matrix Completion Problems Via Semidefinite Programming*
Computational Optimization and Applications 12, 13 30 1999) c 1999 Kluwer Academic Publishers. Manufactured in The Netherlands. Solving Euclidean Distance Matrix Completion Problems Via Semidefinite Programming*
Theory of semidefinite programming for Sensor Network Localization
Math. Program. Ser. B DOI 10.1007/s10107-006-0040-1 FULL LENGTH PAPER Theory of semidefinite programming for Sensor Network Localization Anthony Man-Cho So Yinyu Ye Received: 22 November 2004 / Accepted:
Positive semidefinite rank
1/15 Positive semidefinite rank Hamza Fawzi (MIT, LIDS) Joint work with João Gouveia (Coimbra), Pablo Parrilo (MIT), Richard Robinson (Microsoft), James Saunderson (Monash), Rekha Thomas (UW) DIMACS Workshop
Fast ADMM for Sum of Squares Programs Using Partial Orthogonality
Fast ADMM for Sum of Squares Programs Using Partial Orthogonality Antonis Papachristodoulou Department of Engineering Science University of Oxford www.eng.ox.ac.uk/control/sysos antonis@eng.ox.ac.uk with
A new graph parameter related to bounded rank positive semidefinite matrix completions
Mathematical Programming manuscript No. (will be inserted by the editor) Monique Laurent Antonios Varvitsiotis A new graph parameter related to bounded rank positive semidefinite matrix completions the
Chordal structure in computer algebra: Permanents
Chordal structure in computer algebra: Permanents Diego Cifuentes Laboratory for Information and Decision Systems Electrical Engineering and Computer Science Massachusetts Institute of Technology Joint
Chordally Sparse Semidefinite Programs
Chordally Sparse Semidefinite Programs Raphael Louca Abstract We consider semidefinite programs where the collective sparsity pattern of the data matrices is characterized by a given chordal graph. Acyclic,
Further Relaxations of the SDP Approach to Sensor Network Localization
Further Relaxations of the SDP Approach to Sensor Network Localization Zizhuo Wang, Song Zheng, Stephen Boyd, and Yinyu Ye September 27, 2006; Revised May 2, 2007 Abstract Recently, a semi-definite programming
A Continuation Approach Using NCP Function for Solving Max-Cut Problem
A Continuation Approach Using NCP Function for Solving Max-Cut Problem Xu Fengmin Xu Chengxian Ren Jiuquan Abstract A continuous approach using NCP function for approximating the solution of the max-cut
Facial Reduction and Geometry on Conic Programming
Facial Reduction and Geometry on Conic Programming Masakazu Muramatsu UEC Bruno F. Lourenço, and Takashi Tsuchiya Seikei University GRIPS Based on the papers of LMT: 1. A structural geometrical analysis
Real solving polynomial equations with semidefinite programming
.5cm. Real solving polynomial equations with semidefinite programming Jean Bernard Lasserre - Monique Laurent - Philipp Rostalski LAAS, Toulouse - CWI, Amsterdam - ETH, Zürich LAW 2008 Real solving polynomial
On Representations of Graphs as Two-Distance Sets
arxiv:808.0595v [math.mg] 7 Aug 208 On Representations of Graphs as Two-Distance Sets A. Y. Alfakih Department of Mathematics and Statistics University of Windsor Windsor, Ontario N9B 3P4 Canada August
Research Reports on Mathematical and Computing Sciences
ISSN 1342-284 Research Reports on Mathematical and Computing Sciences Exploiting Sparsity in Linear and Nonlinear Matrix Inequalities via Positive Semidefinite Matrix Completion Sunyoung Kim, Masakazu
SOCP Relaxation of Sensor Network Localization
SOCP Relaxation of Sensor Network Localization Paul Tseng Mathematics, University of Washington Seattle IWCSN, Simon Fraser University August 19, 2006 Abstract This is a talk given at Int. Workshop on
Uniqueness of the Solutions of Some Completion Problems
Uniqueness of the Solutions of Some Completion Problems Chi-Kwong Li and Tom Milligan Abstract We determine the conditions for uniqueness of the solutions of several completion problems including the positive
Geometric Constraints II
Geometric Constraints II Realizability, Rigidity and Related theorems. Embeddability of Metric Spaces Section 1 Given the matrix D d i,j 1 i,j n corresponding to a metric space, give conditions under which
Singularity Degree of the Positive Semidefinite Matrix Completion Problem
Singularity Degree of the Positive Semidefinite Matrix Completion Problem Shin-ichi Tanigawa November 8, 2016 arxiv:1603.09586v2 [math.oc] 4 Nov 2016 Abstract The singularity degree of a semidefinite programming
Explicit Sensor Network Localization using Semidefinite Programming and Facial Reduction
Explicit Sensor Network Localization using Semidefinite Programming and Facial Reduction Nathan Krislock and Henry Wolkowicz Dept. of Combinatorics and Optimization University of Waterloo INFORMS, San
c 2017 Society for Industrial and Applied Mathematics
SIAM J. OPTIM. Vol. 27, No. 4, pp. 2301 2331 c 2017 Society for Industrial and Applied Mathematics NOISY EUCLIDEAN DISTANCE REALIZATION: ROBUST FACIAL REDUCTION AND THE PARETO FRONTIER D. DRUSVYATSKIY,
An Approximation Algorithm for Approximation Rank
An Approximation Algorithm for Approximation Rank Troy Lee Columbia University Adi Shraibman Weizmann Institute Conventions Identify a communication function f : X Y { 1, +1} with the associated X-by-Y
Approximate Farkas Lemmas in Convex Optimization
Approximate Farkas Lemmas in Convex Optimization Imre McMaster University Advanced Optimization Lab AdvOL Graduate Student Seminar October 25, 2004 1 Exact Farkas Lemma Motivation 2 3 Future plans The
Nonlinear Programming Models
Nonlinear Programming Models Fabio Schoen 2008 http://gol.dsi.unifi.it/users/schoen Nonlinear Programming Models p. Introduction Nonlinear Programming Models p. NLP problems minf(x) x S R n Standard form:
Lecture: Expanders, in theory and in practice (2 of 2)
Stat260/CS294: Spectral Graph Methods Lecture 9-02/19/2015 Lecture: Expanders, in theory and in practice (2 of 2) Lecturer: Michael Mahoney Scribe: Michael Mahoney Warning: these notes are still very rough.
Distance Geometry-Matrix Completion
Christos Konaxis Algs in Struct BioInfo 2010 Outline Outline Tertiary structure Incomplete data Tertiary structure Measure diffrence of matched sets Def. Root Mean Square Deviation RMSD = 1 n x i y i 2,
Positive Semidefinite Matrix Completion, Universal Rigidity and the Strong Arnold Property
Positive Semidefinite Matrix Completion, Universal Rigidity and the Strong Arnold Property M. Laurent a,b, A. Varvitsiotis a, a Centrum Wiskunde & Informatica (CWI), Science Park 123, 1098 XG Amsterdam,
Easy and not so easy multifacility location problems... (In 20 minutes.)
Easy and not so easy multifacility location problems... (In 20 minutes.) MINLP 2014 Pittsburgh, June 2014 Justo Puerto Institute of Mathematics (IMUS) Universidad de Sevilla Outline 1 Introduction (In
Dimension reduction for semidefinite programming
1 / 22 Dimension reduction for semidefinite programming Pablo A. Parrilo Laboratory for Information and Decision Systems Electrical Engineering and Computer Science Massachusetts Institute of Technology
A Simple Derivation of a Facial Reduction Algorithm and Extended Dual Systems
A Simple Derivation of a Facial Reduction Algorithm and Extended Dual Systems Gábor Pataki gabor@unc.edu Dept. of Statistics and OR University of North Carolina at Chapel Hill Abstract The Facial Reduction
LMI MODELLING 4. CONVEX LMI MODELLING. Didier HENRION. LAAS-CNRS Toulouse, FR Czech Tech Univ Prague, CZ. Universidad de Valladolid, SP March 2009
LMI MODELLING 4. CONVEX LMI MODELLING Didier HENRION LAAS-CNRS Toulouse, FR Czech Tech Univ Prague, CZ Universidad de Valladolid, SP March 2009 Minors A minor of a matrix F is the determinant of a submatrix
Lecture 1: April 24, 2013
TTIC/CMSC 31150 Mathematical Toolkit Spring 2013 Madhur Tulsiani Lecture 1: April 24, 2013 Scribe: Nikita Mishra 1 Applications of dimensionality reduction In this lecture we talk about various useful
A semidefinite relaxation scheme for quadratically constrained quadratic problems with an additional linear constraint
Iranian Journal of Operations Research Vol. 2, No. 2, 20, pp. 29-34 A semidefinite relaxation scheme for quadratically constrained quadratic problems with an additional linear constraint M. Salahi Semidefinite
Computing the Nearest Euclidean Distance Matrix with Low Embedding Dimensions
Computing the Nearest Euclidean Distance Matrix with Low Embedding Dimensions Hou-Duo Qi and Xiaoming Yuan September 12, 2012 and Revised May 12, 2013 Abstract Euclidean distance embedding appears in many
Inderjit Dhillon The University of Texas at Austin
Inderjit Dhillon The University of Texas at Austin ( Universidad Carlos III de Madrid; 15 th June, 2012) (Based on joint work with J. Brickell, S. Sra, J. Tropp) Introduction 2 / 29 Notion of distance
How to generate weakly infeasible semidefinite programs via Lasserre s relaxations for polynomial optimization
CS-11-01 How to generate weakly infeasible semidefinite programs via Lasserre s relaxations for polynomial optimization Hayato Waki Department of Computer Science, The University of Electro-Communications
Fast Algorithms for SDPs derived from the Kalman-Yakubovich-Popov Lemma
Fast Algorithms for SDPs derived from the Kalman-Yakubovich-Popov Lemma Venkataramanan (Ragu) Balakrishnan School of ECE, Purdue University 8 September 2003 European Union RTN Summer School on Multi-Agent
Matrix Rank Minimization with Applications
Matrix Rank Minimization with Applications Maryam Fazel Haitham Hindi Stephen Boyd Information Systems Lab Electrical Engineering Department Stanford University 8/2001 ACC 01 Outline Rank Minimization
Summer School: Semidefinite Optimization
Summer School: Semidefinite Optimization Christine Bachoc Université Bordeaux I, IMB Research Training Group Experimental and Constructive Algebra Haus Karrenberg, Sept. 3 - Sept. 7, 2012 Duality Theory
A notion of Total Dual Integrality for Convex, Semidefinite and Extended Formulations
A notion of for Convex, Semidefinite and Extended Formulations Marcel de Carli Silva Levent Tunçel April 26, 2018 A vector in R n is integral if each of its components is an integer, A vector in R n is
Lecture 1. 1 Conic programming. MA 796S: Convex Optimization and Interior Point Methods October 8, Consider the conic program. min.
MA 796S: Convex Optimization and Interior Point Methods October 8, 2007 Lecture 1 Lecturer: Kartik Sivaramakrishnan Scribe: Kartik Sivaramakrishnan 1 Conic programming Consider the conic program min s.t.
A Semidefinite Programming Approach to Tensegrity Theory and Realizability of Graphs
A Semidefinite Programming Approach to Tensegrity Theory and Realizability of Graphs Anthony Man Cho So, Yinyu Ye July 6, 2005 Abstract Recently, Connelly and Sloughter [12] have introduced the notion
A Quantum Interior Point Method for LPs and SDPs
A Quantum Interior Point Method for LPs and SDPs Iordanis Kerenidis 1 Anupam Prakash 1 1 CNRS, IRIF, Université Paris Diderot, Paris, France. September 26, 2018 Semi Definite Programs A Semidefinite Program
Conic optimization under combinatorial sparsity constraints
Conic optimization under combinatorial sparsity constraints Christoph Buchheim and Emiliano Traversi Abstract We present a heuristic approach for conic optimization problems containing sparsity constraints.
6.854J / J Advanced Algorithms Fall 2008
MIT OpenCourseWare http://ocw.mit.edu 6.85J / 8.5J Advanced Algorithms Fall 008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 8.5/6.85 Advanced Algorithms
The Shortest Vector Problem (Lattice Reduction Algorithms)
The Shortest Vector Problem (Lattice Reduction Algorithms) Approximation Algorithms by V. Vazirani, Chapter 27 - Problem statement, general discussion - Lattices: brief introduction - The Gauss algorithm
Introduction to Association Schemes
Introduction to Association Schemes Akihiro Munemasa Tohoku University June 5 6, 24 Algebraic Combinatorics Summer School, Sendai Assumed results (i) Vandermonde determinant: a a m =. a m a m m i
INTERIOR-POINT METHODS ROBERT J. VANDERBEI JOINT WORK WITH H. YURTTAN BENSON REAL-WORLD EXAMPLES BY J.O. COLEMAN, NAVAL RESEARCH LAB
1 INTERIOR-POINT METHODS FOR SECOND-ORDER-CONE AND SEMIDEFINITE PROGRAMMING ROBERT J. VANDERBEI JOINT WORK WITH H. YURTTAN BENSON REAL-WORLD EXAMPLES BY J.O. COLEMAN, NAVAL RESEARCH LAB Outline 2 Introduction
Semidefinite Programming
Semidefinite Programming Basics and SOS Fernando Mário de Oliveira Filho Campos do Jordão, 2 November 23 Available at: www.ime.usp.br/~fmario under talks Conic programming V is a real vector space h, i
SDP Relaxations for MAXCUT
SDP Relaxations for MAXCUT from Random Hyperplanes to Sum-of-Squares Certificates CATS @ UMD March 3, 2017 Ahmed Abdelkader MAXCUT SDP SOS March 3, 2017 1 / 27 Overview 1 MAXCUT, Hardness and UGC 2 LP
On positive duality gaps in semidefinite programming
On positive duality gaps in semidefinite programming Gábor Pataki arxiv:82.796v [math.oc] 3 Dec 28 January, 29 Abstract We present a novel analysis of semidefinite programs (SDPs) with positive duality
Applications of Convex Optimization on the Graph Isomorphism problem
Applications of Convex Optimization on the Graph Isomorphism problem 1 1 Department of Information and Computer Sciences UC Irvine March 8, 2011 Outline Motivation 1 Motivation The Graph Isomorphism problem
The Geometry of Matrix Rigidity
The Geometry of Matrix Rigidity Joseph M. Landsberg Jacob Taylor Nisheeth K. Vishnoi November 26, 2003 Abstract Consider the following problem: Given an n n matrix A and an input x, compute Ax. This problem
Positive semidefinite matrix approximation with a trace constraint
Positive semidefinite matrix approximation with a trace constraint Kouhei Harada August 8, 208 We propose an efficient algorithm to solve positive a semidefinite matrix approximation problem with a trace
Universal Rigidity and Edge Sparsification for Sensor Network Localization
Universal Rigidity and Edge Sparsification for Sensor Network Localization Zhisu Zhu Anthony Man Cho So Yinyu Ye September 23, 2009 Abstract Owing to their high accuracy and ease of formulation, there
Semidefinite programs and combinatorial optimization
Semidefinite programs and combinatorial optimization Lecture notes by L. Lovász Microsoft Research Redmond, WA 98052 lovasz@microsoft.com http://www.research.microsoft.com/ lovasz Contents 1 Introduction
Semidefinite and Second Order Cone Programming Seminar Fall 2001 Lecture 5
Semidefinite and Second Order Cone Programming Seminar Fall 2001 Lecture 5 Instructor: Farid Alizadeh Scribe: Anton Riabov 10/08/2001 1 Overview We continue studying the maximum eigenvalue SDP, and generalize
Characterizations of Strongly Regular Graphs: Part II: Bose-Mesner algebras of graphs. Sung Y. Song Iowa State University
Characterizations of Strongly Regular Graphs: Part II: Bose-Mesner algebras of graphs Sung Y. Song Iowa State University sysong@iastate.edu Notation K: one of the fields R or C X: a nonempty finite set
Continuous Optimisation, Chpt 9: Semidefinite Optimisation
Continuous Optimisation, Chpt 9: Semidefinite Optimisation Peter J.C. Dickinson DMMP, University of Twente p.j.c.dickinson@utwente.nl http://dickinson.website/teaching/2017co.html version: 28/11/17 Monday
LECTURE NOTE #11 PROF. ALAN YUILLE
LECTURE NOTE #11 PROF. ALAN YUILLE 1. NonLinear Dimension Reduction Spectral Methods. The basic idea is to assume that the data lies on a manifold/surface in D-dimensional space, see figure (1) Perform
4. Algebra and Duality
4-1 Algebra and Duality P. Parrilo and S. Lall, CDC 2003 2003.12.07.01 4. Algebra and Duality Example: non-convex polynomial optimization Weak duality and duality gap The dual is not intrinsic The cone
Lecture 5. 1 Goermans-Williamson Algorithm for the maxcut problem
Math 280 Geometric and Algebraic Ideas in Optimization April 26, 2010 Lecture 5 Lecturer: Jesús A De Loera Scribe: Huy-Dung Han, Fabio Lapiccirella 1 Goermans-Williamson Algorithm for the maxcut problem
Sensor Network Localization, Euclidean Distance Matrix Completions, and Graph Realization
Sensor Network Localization, Euclidean Distance Matrix Completions, and Graph Realization Yichuan Ding Nathan Krislock Jiawei Qian Henry Wolkowicz October 27, 2008 University of Waterloo Department of
arxiv: v1 [math.oc] 14 Oct 2014
![arxiv: v1 [math.oc] 14 Oct 2014](/thumbs/86/93331412.jpg "arxiv: v1 [math.oc] 14 Oct 2014")
arxiv:110.3571v1 [math.oc] 1 Oct 01 An Improved Analysis of Semidefinite Approximation Bound for Nonconvex Nonhomogeneous Quadratic Optimization with Ellipsoid Constraints Yong Hsia a, Shu Wang a, Zi Xu
Lecture Semidefinite Programming and Graph Partitioning
Approximation Algorithms and Hardness of Approximation April 16, 013 Lecture 14 Lecturer: Alantha Newman Scribes: Marwa El Halabi 1 Semidefinite Programming and Graph Partitioning In previous lectures,
AMS526: Numerical Analysis I (Numerical Linear Algebra)
AMS526: Numerical Analysis I (Numerical Linear Algebra) Lecture 3: Positive-Definite Systems; Cholesky Factorization Xiangmin Jiao Stony Brook University Xiangmin Jiao Numerical Analysis I 1 / 11 Symmetric
The maximal stable set problem : Copositive programming and Semidefinite Relaxations
The maximal stable set problem : Copositive programming and Semidefinite Relaxations Kartik Krishnan Department of Mathematical Sciences Rensselaer Polytechnic Institute Troy, NY 12180 USA kartis@rpi.edu
Linear Regression and Its Applications
Linear Regression and Its Applications Predrag Radivojac October 13, 2014 Given a data set D = {(x i, y i )} n the objective is to learn the relationship between features and the target. We usually start
Solving the MWT. Recall the ILP for the MWT. We can obtain a solution to the MWT problem by solving the following ILP:
Solving the MWT Recall the ILP for the MWT. We can obtain a solution to the MWT problem by solving the following ILP: max subject to e i E ω i x i e i C E x i {0, 1} x i C E 1 for all critical mixed cycles
TRINITY COLLEGE DUBLIN THE UNIVERSITY OF DUBLIN. School of Mathematics
JS and SS Mathematics JS and SS TSM Mathematics TRINITY COLLEGE DUBLIN THE UNIVERSITY OF DUBLIN School of Mathematics MA3484 Methods of Mathematical Economics Trinity Term 2015 Saturday GOLDHALL 09.30
Copositive Programming and Combinatorial Optimization
Copositive Programming and Combinatorial Optimization Franz Rendl http://www.math.uni-klu.ac.at Alpen-Adria-Universität Klagenfurt Austria joint work with I.M. Bomze (Wien) and F. Jarre (Düsseldorf) IMA
arxiv: v4 [math.oc] 12 Apr 2017
![arxiv: v4 [math.oc] 12 Apr 2017](/thumbs/94/118455111.jpg "arxiv: v4 [math.oc] 12 Apr 2017")
Exact duals and short certificates of infeasibility and weak infeasibility in conic linear programming arxiv:1507.00290v4 [math.oc] 12 Apr 2017 Minghui Liu Gábor Pataki April 14, 2017 Abstract In conic
Gram Spectrahedra. Lynn Chua UC Berkeley. joint work with Daniel Plaumann, Rainer Sinn, and Cynthia Vinzant
Gram Spectrahedra Lynn Chua UC Berkeley joint work with Daniel Plaumann, Rainer Sinn, and Cynthia Vinzant SIAM Conference on Applied Algebraic Geometry 017 1/17 Gram Matrices and Sums of Squares Let P
Chapter 6 - Orthogonality
Chapter 6 - Orthogonality Maggie Myers Robert A. van de Geijn The University of Texas at Austin Orthogonality Fall 2009 http://z.cs.utexas.edu/wiki/pla.wiki/ 1 Orthogonal Vectors and Subspaces http://z.cs.utexas.edu/wiki/pla.wiki/
Approximation Algorithms
Approximation Algorithms Chapter 26 Semidefinite Programming Zacharias Pitouras 1 Introduction LP place a good lower bound on OPT for NP-hard problems Are there other ways of doing this? Vector programs
Absolute Value Programming
O. L. Mangasarian Absolute Value Programming Abstract. We investigate equations, inequalities and mathematical programs involving absolute values of variables such as the equation Ax + B x = b, where A
Convex Optimization on Large-Scale Domains Given by Linear Minimization Oracles
Convex Optimization on Large-Scale Domains Given by Linear Minimization Oracles Arkadi Nemirovski H. Milton Stewart School of Industrial and Systems Engineering Georgia Institute of Technology Joint research
CSC Linear Programming and Combinatorial Optimization Lecture 10: Semidefinite Programming
CSC2411 - Linear Programming and Combinatorial Optimization Lecture 10: Semidefinite Programming Notes taken by Mike Jamieson March 28, 2005 Summary: In this lecture, we introduce semidefinite programming
SOLVING EUCLIDEAN DISTANCE MATRIX. University of Waterloo, Department of Combinatorics and Optimization, Waterloo, Ontario
Small Journal Name,?, 1{18 (199?) c 199? Kluwer Academic Publishers, Boston. Manufactured in The Netherlands. SOLVING EUCLIDEAN DISTANCE MATRIX COMPLETION PROBLEMS VIA SEMIDEFINITE PROGRAMMING ABDO Y.
Handout 6: Some Applications of Conic Linear Programming
ENGG 550: Foundations of Optimization 08 9 First Term Handout 6: Some Applications of Conic Linear Programming Instructor: Anthony Man Cho So November, 08 Introduction Conic linear programming CLP, and
CS675: Convex and Combinatorial Optimization Fall 2016 Combinatorial Problems as Linear and Convex Programs. Instructor: Shaddin Dughmi
CS675: Convex and Combinatorial Optimization Fall 2016 Combinatorial Problems as Linear and Convex Programs Instructor: Shaddin Dughmi Outline 1 Introduction 2 Shortest Path 3 Algorithms for Single-Source
On the matrix-cut rank of polyhedra
On the matrix-cut rank of polyhedra William Cook and Sanjeeb Dash Computational and Applied Mathematics Rice University August 5, 00 Abstract Lovász and Schrijver (99) described a semi-definite operator
Semidefinite and Second Order Cone Programming Seminar Fall 2001 Lecture 2
Semidefinite and Second Order Cone Programming Seminar Fall 2001 Lecture 2 Instructor: Farid Alizadeh Scribe: Xuan Li 9/17/2001 1 Overview We survey the basic notions of cones and cone-lp and give several
5 Flows and cuts in digraphs
5 Flows and cuts in digraphs Recall that a digraph or network is a pair G = (V, E) where V is a set and E is a multiset of ordered pairs of elements of V, which we refer to as arcs. Note that two vertices
minimize x x2 2 x 1x 2 x 1 subject to x 1 +2x 2 u 1 x 1 4x 2 u 2, 5x 1 +76x 2 1,
4 Duality 4.1 Numerical perturbation analysis example. Consider the quadratic program with variables x 1, x 2, and parameters u 1, u 2. minimize x 2 1 +2x2 2 x 1x 2 x 1 subject to x 1 +2x 2 u 1 x 1 4x
Problem 1 (Exercise 2.2, Monograph)
MS&E 314/CME 336 Assignment 2 Conic Linear Programming January 3, 215 Prof. Yinyu Ye 6 Pages ASSIGNMENT 2 SOLUTIONS Problem 1 (Exercise 2.2, Monograph) We prove the part ii) of Theorem 2.1 (Farkas Lemma
Mathematical Optimisation, Chpt 2: Linear Equations and inequalities
Mathematical Optimisation, Chpt 2: Linear Equations and inequalities Peter J.C. Dickinson p.j.c.dickinson@utwente.nl http://dickinson.website version: 12/02/18 Monday 5th February 2018 Peter J.C. Dickinson
U.C. Berkeley CS294: Spectral Methods and Expanders Handout 11 Luca Trevisan February 29, 2016
U.C. Berkeley CS294: Spectral Methods and Expanders Handout Luca Trevisan February 29, 206 Lecture : ARV In which we introduce semi-definite programming and a semi-definite programming relaxation of sparsest
A Hierarchy of Polyhedral Approximations of Robust Semidefinite Programs
A Hierarchy of Polyhedral Approximations of Robust Semidefinite Programs Raphael Louca Eilyan Bitar Abstract Robust semidefinite programs are NP-hard in general In contrast, robust linear programs admit
Directed Strongly Regular Graphs. Leif K. Jørgensen Aalborg University Denmark
Directed Strongly Regular Graphs Leif K. Jørgensen Aalborg University Denmark A. Duval 1988: A directed strongly regular graph with parameters v, k, t, λ, µ is a directed graph with the following properties:
Appendix PRELIMINARIES 1. THEOREMS OF ALTERNATIVES FOR SYSTEMS OF LINEAR CONSTRAINTS
Appendix PRELIMINARIES 1. THEOREMS OF ALTERNATIVES FOR SYSTEMS OF LINEAR CONSTRAINTS Here we consider systems of linear constraints, consisting of equations or inequalities or both. A feasible solution