Local and Linear Convergence of an Algorithm for Solving A Sparse Minimization Problem

For an unconstrained minimization problem with a sparse Hessian, a symmetric version of Schubert's update is given which preserves the sparseness structure defined by the Hessian. At each iteration of the algorithm there are two sparse linear systems to be solved. These have the same sparseness structure defined by the Hessian. The differences between succeeding approximations to the Hessian and the Hessian at the solution are related by a careful evaluation of the difference in the Frobenius norm. This relation is used in proving the local and linear convergence of the algorithm