Abstract:

The conditional quadratic semidefinite programming (cQSDP) refers to
a class of matrix optimization problems whose matrix variables are required to be
positive semidefinite on a subspace, and the objectives are quadratic. The chief
purpose of this paper is to focus on two primal examples of cQSDP: the problem of
matrix completion/approximation on a subspace and the Euclidean distance matrix
problem. For the latter problem, we review some classical contributions and
establish certain links among them. Moreover, we develop a semismooth Newton
method for a special class of cQSDP and establish its quadratic convergence under
the condition of constraint nondegeneracy. We also include an application in calibrating
the correlation matrix in Libor market models. We hope this work will
stimulate new research in cQSDP.

Key words: Matrix optimization ,   Conditional semidefinite programming ,  Euclidean distance matrix , Semismooth Newton method