The lectures will give an introduction to sparse matrix completion problems and their applications in semidefinite optimization. Chordal graphs have an important place in the theory of several common types of symmetric matrix completion problems: maximum-determinant positive semidefinite completion, minimum-rank positive semidefinite completion, and Euclidean distance matrix completion. For sparsity patterns described by chordal graphs, these completion problems can be solved by explicit formulas or efficient recursive algorithms.

The lectures will cover the classical results from linear algebra and their convex geometry, and applications to algorithms for sparse semidefinite optimization (interior-point methods, decomposition and splitting algorithms, and first-order algorithms). Data structures and techniques from the sparse matrix literature, such as elimination trees, supernodal elimination trees, and multifrontal algorithms will play a unifying role in the derivation of key results and algorithms.