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Lieven Vandenberghe and Martin S. Andersen (2015), "Chordal Graphs and Semidefinite Optimization", Foundations and TrendsĀ® in Optimization: Vol. 1: No. 4, pp 241-433. http://dx.doi.org/10.1561/2400000006

© 2015 L. Vandenberghe and M. S. Andersen

Optimization, Graphical models, Operations research, Statistical Signal Processing: tree-structured methods

Semidefinite optimization, Semidefinite programming, Convex optimization, Sparse matrices, Matrix completion problems, Graph theory

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**In this article:**

1. Introduction

2. Graphs

3. Chordal Graphs

4. Perfect Elimination Ordering

5. Combinatorial Optimization

6. Graph Elimination

7. Discrete Applications of Graph Elimination

8. Sparse Matrices

9. Positive Semidefinite Matrices

10. Positive Semidefinite Matrix Completion

11. Correlation and Euclidean Distance Matrices

12. Partial Separability in Convex Optimization

13. Conic Optimization

14. Sparse Semidefinite Optimization

Acknowledgments

Notation

References

Chordal graphs play a central role in techniques for exploiting sparsity in large semidefinite optimization problems and in related convex optimization problems involving sparse positive semidefinite matrices. Chordal graph properties are also fundamental to several classical results in combinatorial optimization, linear algebra, statistics, signal processing, machine learning, and nonlinear optimization. This survey covers the theory and applications of chordal graphs, with an emphasis on algorithms developed in the literature on sparse Cholesky factorization. These algorithms are formulated as recursions on elimination trees, supernodal elimination trees, or clique trees associated with the graph. The best known example is the multifrontal Cholesky factorization algorithm, but similar algorithms can be formulated for a variety of related problems, including the computation of the partial inverse of a sparse positive definite matrix, positive semidefinite and Euclidean distance matrix completion problems, and the evaluation of gradients and Hessians of logarithmic barriers for cones of sparse positive semidefinite matrices and their dual cones. The purpose of the survey is to show how these techniques can be applied in algorithms for sparse semidefinite optimization, and to point out the connections with related topics outside semidefinite optimization, such as probabilistic networks, matrix completion problems, and partial separability in nonlinear optimization.

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1. Introduction

2. Graphs

3. Chordal Graphs

4. Perfect Elimination Orderings

5. Combinatorial Optimization

6. Graph Elimination

7. Discrete Applications of Graph Elimination

8. Sparse Matrices

9. Positive Semidefinite Matrices

10. Positive Semidefinite Matrix Completion

11. Correlation and Euclidean Distance Matrices

12. Partial Separability in Convex Optimization

13. Conic Optimization

14. Sparse Semidefinite Optimization

Acknowledgments

Notation

References

Chordal graphs play a central role in techniques for exploiting sparsity in large semidefinite optimization problems, and in related convex optimization problems involving sparse positive semidefinite matrices. Chordal graph properties are also fundamental to several classical results in combinatorial optimization, linear algebra, statistics, signal processing, machine learning, and nonlinear optimization.

*Chordal Graphs and Semidefinite Optimization* covers the theory and applications of chordal graphs, with an emphasis on
algorithms developed in the literature on sparse Cholesky factorization. These algorithms are formulated as recursions on elimination trees,
supernodal elimination trees, or clique trees associated with the graph. The best known example is the multifrontal Cholesky factorization
algorithm but similar algorithms can be formulated for a variety of related problems, such as the computation of the partial inverse of a
sparse positive definite matrix, positive semidefinite and Euclidean distance matrix completion problems, and the evaluation of gradients and
Hessians of logarithmic barriers for cones of sparse positive semidefinite matrices and their dual cones. This monograph shows how these
techniques can be applied in algorithms for sparse semidefinite optimization. It also points out the connections with related topics outside
semidefinite optimization, such as probabilistic networks, matrix completion problems, and partial separability in nonlinear optimization.