Table of Contents

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. Ackowledgments. Notation. References