Conditional Quadratic Semidefinite Programming:Examples and Methods

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.