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Modified interior-point method for large-and-sparse low-rank semidefinite programs

Richard Y. Zhang, Javad Lavaei
2017 2017 IEEE 56th Annual Conference on Decision and Control (CDC)  
In this paper, we describe a modified interior-point method for the efficient solution of large-and-sparse low-rank SDPs, which finds applications in graph theory, approximation theory, control theory,  ...  Given that the problem data is large-and-sparse, conjugate gradients (CG) can be used to avoid forming, storing, and factoring the large and fully-dense interior-point Hessian matrix, but the resulting  ...  In this paper, we present a modification to the standard interior-point method that makes it substantially more efficient for large-and-sparse low-rank SDPs.  ... 
doi:10.1109/cdc.2017.8264510 dblp:conf/cdc/ZhangL17 fatcat:dzyjlsrmb5cnzj6drn23wqvawa

A Survey of Recent Scalability Improvements for Semidefinite Programming with Applications in Machine Learning, Control, and Robotics [article]

Anirudha Majumdar, Georgina Hall, Amir Ali Ahmadi
2019 arXiv   pre-print
In this paper, we survey recent approaches for addressing this challenge including (i) approaches for exploiting structure (e.g., sparsity and symmetry) in a problem, (ii) approaches that produce low-rank  ...  approximate solutions to semidefinite programs, (iii) more scalable algorithms that rely on augmented Lagrangian techniques and the alternating direction method of multipliers, and (iv) approaches that  ...  ACKNOWLEDGMENTS We thank Cemil Dibek for his constructive feedback on the first draft of this manuscript.  ... 
arXiv:1908.05209v3 fatcat:g2vqfhv27vgddbv7l4xciywf4u

Implementation of interior point methods for mixed semidefinite and second order cone optimization problems

Jos F. Sturm
2002 Optimization Methods and Software  
There is a large number of implementational choices to be made for the primal-dual interior point method in the context of mixed semidefinite and second order cone optimization.  ...  point method was especially suited for solving semidefinite programming problems.  ...  I am therefore also grateful to Tamás Terlaky for suggesting in 2002 to contribute the then still unfinished paper to this special issue of Optimization Methods and Software.  ... 
doi:10.1080/1055678021000045123 fatcat:5v4qblu5orddlovttmugdgrzri

Implementation of nonsymmetric interior-point methods for linear optimization over sparse matrix cones

Martin S. Andersen, Joachim Dahl, Lieven Vandenberghe
2010 Mathematical Programming Computation  
We describe an implementation of nonsymmetric interior-point methods for linear cone programs defined by two types of matrix cones: the cone of positive semidefinite matrices with a given chordal sparsity  ...  We present experimental results of two implementations, one of which is based on an augmented system approach, and a comparison with publicly available interior-point solvers for semidefinite programming  ...  the original author(s) and source are credited.  ... 
doi:10.1007/s12532-010-0016-2 fatcat:2nrkpvzsfrb47cu2m6bnl5soua

A direct formulation for sparse PCA using semidefinite programming [article]

Alexandre d'Aspremont, Laurent El Ghaoui, Michael I. Jordan, Gert R. G. Lanckriet
2006 arXiv   pre-print
We use a modification of the classical variational representation of the largest eigenvalue of a symmetric matrix, where cardinality is constrained, and derive a semidefinite programming based relaxation  ...  We also discuss Nesterov's smooth minimization technique applied to the SDP arising in the direct sparse PCA method.  ...  Thanks to Andrew Mullhaupt and Francis Bach for useful suggestions.  ... 
arXiv:cs/0406021v3 fatcat:extchl4fondejmzsp3efgx2b7e

Exploiting Low-Rank Structure in Semidefinite Programming by Approximate Operator Splitting [article]

Mario Souto, Joaquim D. Garcia, Alvaro Veiga
2018 arXiv   pre-print
The main contribution of this work is to achieve a substantial speedup by effectively adjusting the proposed algorithm in order to exploit the low-rank property inherent to several semidefinite programming  ...  This work aims to reduce this scalability gap by proposing a novel proximal algorithm for solving general semidefinite programming problems.  ...  We extend many thanks to all members of LAMPS (Laboratory of Applied Mathematical Programming and Statistics), in special Thuener Silva and Raphael Saavedra, for the daily support and fruitful discussions  ... 
arXiv:1810.05231v3 fatcat:3uk6cjpedfcofjdcc5dahzkxjm

Scalable Semidefinite Relaxation for Maximum A Posterior Estimation [article]

Qixing Huang, Yuxin Chen, Leonidas Guibas
2014 arXiv   pre-print
Encouragingly, the proposed procedure allows solving SDR for large-scale problems, e.g., problems on a grid graph comprising hundreds of thousands of variables with multiple states per node.  ...  We have evaluated the performance of SDR on various benchmark datasets including OPENGM2 and PIC in terms of both the quality of the solutions and computation time.  ...  Acknowledgments This work has been supported in part by NSF grants FO-DAVA 808515 and CCF 1011228, AFOSR grant FA9550-12-1-0372, ONR MURI N00014-13-1-0341, and a Google research award.  ... 
arXiv:1405.4807v1 fatcat:tfvdczkthfbknbxsiqf5hcdrdy

Chordal Decomposition in Rank Minimized Semidefinite Programs with Applications to Subspace Clustering [article]

Jared Miller, Yang Zheng, Biel Roig-Solvas, Mario Sznaier, Antonis Papachristodoulou
2019 arXiv   pre-print
Implementations of rank-minimized SDPs through interior-point and first-order algorithms are discussed.  ...  Semidefinite programs (SDPs) often arise in relaxations of some NP-hard problems, and if the solution of the SDP obeys certain rank constraints, the relaxation will be tight.  ...  In this section, we modify Problem (3) to exploit its structure for adaptation in both interior point methods and first order methods. A.  ... 
arXiv:1904.10041v2 fatcat:h2s44rqafvgnbgsob2u5efl6re

Parallel Implementation of a Semidefinite Programming Solver Based on CSDP on a Distributed Memory Cluster

Ivan D. Ivanov, Etienne de Klerk
2007 Social Science Research Network  
Moreover, we show that very good parallel efficiency is obtained for large-scale problems where the number of linear equality constraints is very large compared to the block sizes of the positive semidefinite  ...  In this paper we present the algorithmic framework and practical aspects of implementing a parallel version of a primal-dual semidefinite programming solver on a distributed memory computer cluster.  ...  Acknowledgement The authors would like to thank Professors Masakazu Kojima and Katsuki Fujisawa for their kind assistance with the installation of SDPARA on the DAS3 Beowulf cluster at the Delft University  ... 
doi:10.2139/ssrn.987781 fatcat:ricg6mwpfzcp7fknttgirlnexa

Convex Optimization with Abstract Linear Operators

Steven Diamond, Stephen Boyd
2015 2015 IEEE International Conference on Computer Vision (ICCV)  
By combining the matrix-free modeling framework and cone solver, we obtain a general method for efficiently solving convex optimization problems involving fast linear transforms.  ...  We introduce a convex optimization modeling framework that transforms a convex optimization problem expressed in a form natural and convenient for the user into an equivalent cone program in a way that  ...  First-order methods are an alternative to interior-point methods that scale more easily to large cone programs, at the cost of lower accuracy.  ... 
doi:10.1109/iccv.2015.84 dblp:conf/iccv/DiamondB15 fatcat:yxb64zokrvgmtl3orzuvfoctaq

Sparse Semidefinite Programs with Guaranteed Near-Linear Time Complexity via Dualized Clique Tree Conversion [article]

Richard Y. Zhang, Javad Lavaei
2020 arXiv   pre-print
Assuming that ω≪ n, we prove that the per-iteration cost of an interior-point method is linear O(n) time and memory, so an ϵ-accurate and ϵ-feasible iterate is obtained after O(√(n)log(1/ϵ)) iterations  ...  We confirm our theoretical insights with numerical results on semidefinite programs as large as n=13659. (Supporting code at https://github.com/ryz-codes/dual_ctc )  ...  Finally, a reviewer noted that if the original problem (SDP) has a low-rank solution, then the interior-point method iterates approach a low-dimensional face of the semidefinite cone, which could present  ... 
arXiv:1710.03475v4 fatcat:l7kwecoftnhj5ire5nbk6cktdm

An Inexact Projected Gradient Method with Rounding and Lifting by Nonlinear Programming for Solving Rank-One Semidefinite Relaxation of Polynomial Optimization [article]

Heng Yang, Ling Liang, Luca Carlone, Kim-Chuan Toh
2021 arXiv   pre-print
method (sGS-APG) to generate a good initial point, and phase two using a modified limited-memory BFGS (L-BFGS) method to obtain an accurate solution.  ...  We consider solving high-order semidefinite programming (SDP) relaxations of nonconvex polynomial optimization problems (POPs) that often admit degenerate rank-one optimal solutions.  ...  For STRIDE, since the nonlinear programing (STLS) does not admit any manifold structure, we use fmincon with an interior point method as the nlp method.  ... 
arXiv:2105.14033v2 fatcat:xeklsleunnbefpnoo5mkyyi3eq

Chordal Decomposition in Rank Minimized Semidefinite Programs with Applications to Subspace Clustering

Jared Miller, Yang Zheng, Biel Roig-Solvas, Mario Sznaier, Antonis Papachristodoulou
2019 2019 IEEE 58th Conference on Decision and Control (CDC)  
Implementations of rank-minimized SDPs through interior-point and first-order algorithms are discussed.  ...  Decomposition methods based on chordal sparsity have already been applied to speed up the solution of sparse SDPs, but methods for dealing with rank constraints are underdeveloped.  ...  In this section, we modify Problem (3) to exploit its structure for adaptation in both interior point methods and first order methods. A.  ... 
doi:10.1109/cdc40024.2019.9029620 dblp:conf/cdc/Miller0RSP19 fatcat:zbfqbwkxgvh4pgcsyvls4xj2nu

HDSDP: Software for Semidefinite Programming [article]

Wenzhi Gao, Dongdong Ge, Yinyu Ye
2022 arXiv   pre-print
HDSDP is a numerical software solving the semidefinite programming problems.  ...  The main framework of HDSDP resembles the dual-scaling interior point solver DSDP[2] and several new features, especially a dual method based on the simplified homogeneous self-dual embedding, have been  ...  Finally, we sincerely respect the developers of DSDP for their precious suggestions [2] and their invaluable efforts getting DSDP through all along the way.  ... 
arXiv:2207.13862v1 fatcat:7t7t2vt43ngivkrzqhlktjrp3m

Space-Efficient Approximation Algorithms for MAXCUT and COLORING Semidefinite Programs [chapter]

Philip N. Klein, Hsueh-I Lu
1998 Lecture Notes in Computer Science  
For a graph with n nodes and m edges, previous work on solving its semidefinite relaxation for MAXCUT requires spaceÕ(n 2 ).  ...  The essential part of the best known approximation algorithm for graph MAXCUT is approximately solving MAXCUT's semidefinite relaxation.  ...  Nesterov and Nemirovsky showed [27] how to use the interior-point methods to solve semidefinite programs.  ... 
doi:10.1007/3-540-49381-6_41 fatcat:izpqs3ww75hoxpdt3yo2dcl7ke
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