Low Rank Semidefinite Programming
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Low Rank Semidefinite Programming
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Author : Alex Lemon
language : en
Publisher:
Release Date : 2016
Low Rank Semidefinite Programming written by Alex Lemon and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016 with Ranking and selection (Statistics) categories.
Finding low-rank solutions of semidefinite programs is important in many applications. For example, semidefinite programs that arise as relaxations of polynomial optimization problems are exact relaxations when the semidefinite program has a rank-1 solution. Unfortunately, computing a minimum-rank solution of a semidefinite program is an NP-hard problem. In this paper we review the theory of low-rank semidefinite programming, presenting theorems that guarantee the existence of a low-rank solution, heuristics for computing low-rank solutions, and algorithms for finding low-rank approximate solutions. Then we present applications of the theory to trust-region problems and signal processing.
Computational Enhancements And Applications In Low Rank Semidefinite Programming
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Author : Changhui Choi
language : en
Publisher:
Release Date : 2007
Computational Enhancements And Applications In Low Rank Semidefinite Programming written by Changhui Choi and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2007 with Combinatorial optimization categories.
The low-rank semidefinite programming solver SDPLR, originally developed by Burer and Monteiro, is an algorithm for solving large-scale semidefinite programming problems. In this thesis, we present computational enhancements and some applications in SDPLR, which include the hypergraph minimum-bisection problem and a special case of the Euclidean distance problem that arises in protein 3-dimensional conformation. The hypergraph minimum bisection is an important combinatorial optimization problem that arises in digital circuit design. In this thesis we present the first mathematical formulation of the hypergraph minimum bisection problem, its semidefinite programming relaxation, and computational results on the digital circuit benchmark problems. The protein 3-dimensional conformation problem is to predict the 3-dimensional structure that a protein takes in a certain fluid. We consider solving one of the sub-problems of the protein 3-dimensional conformation problem, which can be considered as a special case of the Euclidean distance problem. This problem can be formulated as a quadratically constrained quadratic program, and its semidefinite programming relaxation is considered. We report successful computational results on this problem using SDPLR and local search techniques.
Low Rank Structure In Semidefinite Programming And Sum Of Squares Optimization In Signal Processing
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Author : Tae Jung Roh
language : en
Publisher:
Release Date : 2007
Low Rank Structure In Semidefinite Programming And Sum Of Squares Optimization In Signal Processing written by Tae Jung Roh and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2007 with categories.
Much of the recent work in this field has centered around optimization problems involving nonnegative polynomial constraints. The basic observation is that sum-of-squares formulations (or relaxations) of such problems can be solved by semidefinite programming. In practice, however, the semidefinite programs that result from this approach are often challenging for general-purpose solvers due to the presence of large auxiliary matrix variables. It is therefore of interest to develop specialized algorithms for semidefinite programs derived from sum-of-squares formulations.
Combinatorial Conditions For Low Rank Solutions In Semidefinite Programming
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Author :
language : en
Publisher:
Release Date : 2013
Combinatorial Conditions For Low Rank Solutions In Semidefinite Programming written by and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013 with categories.
A Semidefinite Programming Method For Graph Realization And Low Rank Matrix Completion Problem
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Author : Zhisu Zhu
language : en
Publisher:
Release Date : 2011
A Semidefinite Programming Method For Graph Realization And Low Rank Matrix Completion Problem written by Zhisu Zhu and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011 with categories.
Owing to their high accuracy and ease of formulation, there has been great interest in applying convex optimization techniques, particularly semidefinite programming (SDP) relaxation, to the graph realization and sensor network localization problems in recent years. A drawback of such techniques is that the resulting convex program is often expensive to solve. In order to speed up computation, various edge sparsification heuristics have been proposed, whose aim is to reduce the number of edges in the input graph. Although these heuristics do reduce the size of the convex program and hence make it faster to solve, they are often ad hoc in nature and do not preserve the realization (or localization) properties of the input. As such, one often has to face a tradeoff between solution accuracy and computational effort. In this thesis, we propose a novel edge sparsification heuristic that can provably preserve the realization (or localization) properties of the original input. At the heart of our heuristic is a graph decomposition procedure that allows us to identify certain sparse generically universally rigid subgraphs of the input graph. Our computational results show that the proposed approach can significantly reduce the computational and memory complexities of SDP-based algorithms for solving the graph realization and sensor network localization problems. Moreover, it compares favorably with existing speedup approaches in terms of both accuracy and solution time. The graph realization problem indeed aims to reconstruct a matrix from a sampling of its entries, which can be viewed as a special case of the well-studied matrix completion problem. The main objective of the matrix completion problem is to design an efficient algorithm that can reconstruct a matrix by inspecting only a small number of its entries. Although, generally speaking, this is an impossible task, Candes and co-authors have recently shown that under a so-called incoherence assumption, a rank r n x n matrix can be reconstructed using SDP after one inspects O(nr log6 n) of its entries. We first provide an equivalent SDP formulation based on chordal decomposition, which has smaller SDP cones. Then we propose an alternative approach that can reconstruct a larger class of matrices by inspecting a significantly smaller number of the entries. Specifically, we first introduce a class of matrices, which we call stable matrices, and show that it includes all those that satisfy the incoherence assumption. Then, we propose a randomized basis pursuit (RBP) algorithm and show that it can reconstruct a stable rank r n x n matrix after inspecting O(nr log n) of its entries. Our sampling bound is only a logarithmic factor away from the information-theoretic limit and is essentially optimal.
Low Rank Semidefinite Programming
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Author : Alex Lemon
language : en
Publisher: Now Publishers
Release Date : 2016-05-04
Low Rank Semidefinite Programming written by Alex Lemon and has been published by Now Publishers this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016-05-04 with Mathematics categories.
Finding low-rank solutions of semidefinite programs is important in many applications. For example, semidefinite programs that arise as relaxations of polynomial optimization problems are exact relaxations when the semidefinite program has a rank-1 solution. Unfortunately, computing a minimum-rank solution of a semidefinite program is an NP-hard problem. This monograph reviews the theory of low-rank semidefinite programming, presenting theorems that guarantee the existence of a low-rank solution, heuristics for computing low-rank solutions, and algorithms for finding low-rank approximate solutions. It then presents applications of the theory to trust-region problems and signal processing.
Scalable Convex Optimization Methods For Semidefinite Programming
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Author : Alp Yurtsever
language : en
Publisher:
Release Date : 2019
Scalable Convex Optimization Methods For Semidefinite Programming written by Alp Yurtsever and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019 with categories.
Mots-clés de l'auteur: Convex optimization ; semidefinite programming ; low-rank matrix optimization ; primal-dual methods ; conditional gradient methods ; low-rank matrix sketching.
Minimum Rank Positive Semidefinite Matrix Completion With Chordal Sparsity Pattern
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Author : Xin Jiang
language : en
Publisher:
Release Date : 2017
Minimum Rank Positive Semidefinite Matrix Completion With Chordal Sparsity Pattern written by Xin Jiang and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017 with categories.
In recent years, semidefinite programming has been an important topic in the area of convex optimization, and several methods for exploiting the sparse structure in semidefinite programming problems have been developed. Some methods have been proposed to transform the standard semidefinite program into a conic optimization problem with respect to the cone of positive semidefinite completable matrices, and to take advantage of the sparsity pattern of the completable matrices. However, the problem arises of how to recover an optimal solution for the original semidefinite program, \ie, how to find a positive semidefinite completion for the positive semidefinite completable solution. In particular, a low-rank completion is of great interest in many applications. In general, it is difficult to determine the minimum rank among all positive semidefinite completions. However, if the sparsity pattern is chordal, efficient algorithms are known for constructing a positive semidefinite matrix completion with minimum rank. In the thesis, we investigate this completion approach as an inexpensive post-processing technique for semidefinite relaxations of nonconvex quadratic problems. We test the method on semidefinite relaxations of the optimal power flow problem. By numerical experiments, we show that the completion results substantially reduce the rank of the solution for the semidefinite relaxation.
Handbook Of Semidefinite Programming
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Author : Henry Wolkowicz
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
Handbook Of Semidefinite Programming written by Henry Wolkowicz and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Business & Economics categories.
Semidefinite programming (SDP) is one of the most exciting and active research areas in optimization. It has and continues to attract researchers with very diverse backgrounds, including experts in convex programming, linear algebra, numerical optimization, combinatorial optimization, control theory, and statistics. This tremendous research activity has been prompted by the discovery of important applications in combinatorial optimization and control theory, the development of efficient interior-point algorithms for solving SDP problems, and the depth and elegance of the underlying optimization theory. The Handbook of Semidefinite Programming offers an advanced and broad overview of the current state of the field. It contains nineteen chapters written by the leading experts on the subject. The chapters are organized in three parts: Theory, Algorithms, and Applications and Extensions.
Recent Scalability Improvements For Semidefinite Programming With Applications In Machine Learning Control And Robotics
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Author : Anirudha Majumdar
language : en
Publisher:
Release Date : 2020
Recent Scalability Improvements For Semidefinite Programming With Applications In Machine Learning Control And Robotics written by Anirudha Majumdar and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020 with categories.
Historically, scalability has been a major challenge for the successful application of semidefinite programming in fields such as machine learning, control, and robotics. In this article, we survey recent approaches to this challenge, including those that exploit structure (e.g., sparsity and symmetry) in a problem, those that produce low-rank approximate solutions to semidefinite programs, those that use more scalable algorithms that rely on augmented Lagrangian techniques and the alternating-direction method of multipliers, and those that trade off scalability with conservatism (e.g., by approximating semidefinite programs with linear and second-order cone programs). For each class of approaches, we provide a high-level exposition, an entry point to the corresponding literature, and examples drawn from machine learning, control, or robotics. We also present a list of software packages that implement many of the techniques discussed in the review. Our hope is that this article will serve as a gateway to the rich and exciting literature on scalable semidefinite programming for both theorists and practitioners.