Theory and algorithms for matrix problems with positive semidefinite constraints

This thesis presents new theoretical results and algorithms for two matrix problems with positive semidefinite constraints: it adds to the well-established nearest correlation matrix problem, and introduces a class of semidefinite Lagrangian subspaces. First, we propose shrinking, a method for restoring positive semidefiniteness of an indefinite matrix $M_0$ that computes the optimal parameter $\a_*$ in a convex combination of $M_0$ and a chosen positive semidefinite target matrix. We describe three algorithms for computing $\a_*$, and then focus on the case of keeping fixed a positive semidefinite leading principal submatrix of an indefinite approximation of a correlation matrix, showing how the structure can be exploited to reduce the cost of two algorithms. We describe how weights can be used to construct a natural choice of the target matrix and that they can be incorporated without any change to computational methods, which is in contrast to the nearest correlation matrix problem. Numerical experiments show that shrinking can be at least an order of magnitude faster than computing the nearest correlation matrix and so is preferable in time-critical applications. Second, we focus on estimating the distance in the Frobenius norm of a symmetric matrix $A$ to its nearest correlation matrix $\Ncm(A)$ without first computing the latter. The goal is to enable a user to identify an invalid correlation matrix relatively cheaply and to decide whether to revisit its construction or to compute a replacement. We present a few currently available lower and upper bounds for $\dcorr(A) = \normF{A - \Ncm(A)}$ and derive several new upper bounds, discuss the computational cost of all the bounds, and test their accuracy on a collection of invalid correlation matrices. The experiments show that several of our bounds are well suited to gauging the correct order of magnitude of $\dcorr(A)$, which is perfectly satisfactory for practical applications. Third, we show how Anderson acceleration can be used to speed up the convergence of the alternating projections method for computing the nearest correlation matrix, and that the acceleration remains effective when it is applied to the variants of the nearest correlation matrix problem in which specified elements are fixed or a lower bound is imposed on the smallest eigenvalue. This is particularly significant for the nearest correlation matrix problem with fixed elements because no Newton method with guaranteed convergence is available for it. Moreover, alternating projections is a general method for finding a point in the intersection of several sets and this appears to be the first demonstration that these methods can benefit from Anderson acceleration. Finally, we introduce semidefinite Lagrangian subspaces, describe their connection to the unique positive semidefinite solution of an algebraic Riccati equation, and show that these subspaces can be represented by a subset $\mathcal{I} \subseteq \{1,2,\dots, n\}$ and a Hermitian matrix $X\in\mathbb{C}^{n\times n}$ that is a generalization of a quasidefinite matrix. We further obtain a semidefiniteness-preserving version of an optimization algorithm introduced by Mehrmann and Poloni [\textit{SIAM J.\ Matrix Anal. \ Appl. }, 33(2012), pp.\ 780--805] to compute a pair $(\mathcal{I}_{\opt},X_{\opt})$ with $M = \max_{i,j} \abs{(X_{\opt})_{ij}}$ as small as possible, which improves numerical stability in several contexts.