Characterization of the smoothness and curvature of a marginal function for a trust-region problem

This paper studies the smoothness and curvature of a marginal function for a trust-region problem. In this problem, a quadratic function is minimized over an ellipsoid. The marginal function considered is obtained by perturbing the trust radius, i.e., by changing the size of the ellipsoidal constraint. The values of the marginal function and of its first and second derivatives are explicitly calculated in all possible scenarios. A complete study of the smoothness and curvature of this marginal function is given. The main motivation for this work arises from an application in Statistics.