A note on using alternative second-order models for the subproblems arising in barrier function methods for minimization

Summary.

Inequality constrained minimization problems are often solved by considering a sequence of parameterized barrier functions. Each barrier function is approximately minimized and the relevant parameters subsequently adjusted. It is common for the estimated solution to one barrier function problem to be used as a starting estimate for the next. However, this has unfortunate repercussions for the standard Newton-like methods applied to the barrier subproblem. In this note, we consider a class of alternative Newton methods which attempt to avoid such difficulties. Such schemes have already proved of use in the Harwell Subroutine Library quadratic programming codes {\tt VE14} and {\tt VE19}.