  The focus in this work is on interior-point methods for
inequality-constrained quadratic programs, and particularly on the system of
nonlinear equations to be solved for each value of the barrier parameter.
Newton iterations give high quality solutions, but we are interested in
modified Newton systems that are computationally less expensive at the expense
of lower quality solutions. We propose a structured modified Newton approach
where each modified Jacobian is composed of a previous Jacobian, plus one
low-rank update matrix per succeeding iteration. Each update matrix is, for a
given rank, chosen such that the distance to the Jacobian at the current
iterate is minimized, in both 2-norm and Frobenius norm. The approach is
structured in the sense that it preserves the nonzero pattern of the Jacobian.
The choice of update matrix is supported by results in an ideal theoretical
setting. We also produce numerical results with a basic interior-point
implementation to investigate the practical performance within and beyond the
theoretical framework. In order to improve performance beyond the theoretical
framework, we also motivate and construct two heuristics to be added to the
method.
