Regularized Newton Methods for Minimization of Convex Quadratic Splines with Singular Hessians

Abstract

A quadratic spline function on ℝ n is a differentiable piecewise quadratic function. Many convex quadratic programming problems can be reformulated as the problem of unconstrained minimization of convex quadratic splines. Therefore, it is important to investigate efficient algorithms for the unconstrained minimization of a convex quadratic spline. In this paper, for a convex quadratic spline f(x) that has a matrix representation and is bounded below, we show that one can find a global minimizer of f(x) in finitely many iterations even though the global minimizers of f(x) might form an unbounded set. The new idea is to use a regularized Newton direction when a Hessian matrix of f(x) is singular. Applications to quadratic programming are also included.