A Penalty Method for Correlation Matrix Problems with Prescribed Constraints

Abstract

In this thesis, we apply the penalty technique to solve the nearest correlation matrix problem, i.e, we consider the penalized version of the former problem. To deal with the penalized problem, we first apply the essential idea of the majorization method by solving a sequence of unconstrained inner problems iteratively. Actually, the inner problem is generated by the Lagrangian dual approach based on the metric projection and the Moreau-Yosida regularization. Since the objective function in the inner problem is not twice continuously differentiable, we take advantage of the strongly semismooth to overcome this difficulty. Then we propose a semismooth Newton-CG method to solve the inner problem. Finally, we analyze the convergence properties of the semismooth Newton-CG method by using the constraint nondegeneracy. The numerical results reported indicate that our algorithm is efficient and robust.