A simple duality proof in convex quadratic programming with a quadratic constraint, and some applications

In this paper a simple derivation of duality is presented for convex quadratic programs with a convex quadratic constraint. This problem arises in a number of applications including trust region subproblems of nonlinear programming, regularized solution of ill-posed least squares problems, and ridge regression problems in statistical analysis. In general, the dual problem is a concave maximization problem with a linear equality constraint. We apply the duality result to: (1) the trust region subproblem, (2) the smoothing of empirical functions, and (3) to piecewise quadratic trust region subproblems arising in nonlinear robust Huber M-estimation problems in statistics. The results are obtained from a straightforward application of Lagrange duality.