A conversion of an SDP having free variables into the standard form SDP
Computational Optimization and Applications
ACM Transactions on Mathematical Software (TOMS)
On Listing, Sampling, and Counting the Chordal Graphs with Edge Constraints
COCOON '08 Proceedings of the 14th annual international conference on Computing and Combinatorics
Covariance selection for nonchordal graphs via chordal embedding
Optimization Methods & Software - Mathematical programming in data mining and machine learning
A parallel interior point decomposition algorithm for block angular semidefinite programs
Computational Optimization and Applications
On listing, sampling, and counting the chordal graphs with edge constraints
Theoretical Computer Science
Universal rigidity: towards accurate and efficient localization of wireless networks
INFOCOM'10 Proceedings of the 29th conference on Information communications
Exploiting structured sparsity in large scale semidefinite programming problems
ICMS'10 Proceedings of the Third international congress conference on Mathematical software
Universal Rigidity and Edge Sparsification for Sensor Network Localization
SIAM Journal on Optimization
Convergent SDP-relaxations for polynomial optimization with sparsity
ICMS'06 Proceedings of the Second international conference on Mathematical Software
ACM Transactions on Mathematical Software (TOMS)
Welfare-maximizing correlated equilibria using Kantorovich polynomials with sparsity
Journal of Global Optimization
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A critical disadvantage of primal-dual interior-point methods compared to dual interior-point methods for large scale semidefinite programs (SDPs) has been that the primal positive semidefinite matrix variable becomes fully dense in general even when all data matrices are sparse. Based on some fundamental results about positive semidefinite matrix completion, this article proposes a general method of exploiting the aggregate sparsity pattern over all data matrices to overcome this disadvantage. Our method is used in two ways. One is a conversion of a sparse SDP having a large scale positive semidefinite matrix variable into an SDP having multiple but smaller positive semidefinite matrix variables to which we can effectively apply any interior-point method for SDPs employing a standard block-diagonal matrix data structure. The other way is an incorporation of our method into primal-dual interior-point methods which we can apply directly to a given SDP. In Part II of this article, we will investigate an implementation of such a primal-dual interior-point method based on positive definite matrix completion, and report some numerical results.