The Projected Gradient Method for Least Squares Matrix Approximations with Spectral Constraints

The problems of computing least squares approximations for various types of real and symmetric matrices subject to spectral constraints share a common structure. This paper describes a general procedure in using the projected gradient method. It is shown that the projected gradient of the objective function on the manifold of constraints usually can be formulated explicitly. This gives rise to the construction of a descent flow that can be followed numerically. The explicit form also facilitates the computation of the second-order optimality conditions. Examples of applications are discussed. With slight modifications, the procedure can be extended to solve least squares problems for general matrices subject to singular-value constraints.