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

Inequality constrained minimization problems are often solved byconsidering a sequence of parameterized barrier functions. Each barrierfunction is approximately minimized and the relevant parameterssubsequently adjusted.It is common for the estimated solution to one barrier function problemto be used as a starting estimate for the next. However, this hasunfortunate repercussions for the standard Newton-like methods applied tothe barrier subproblem.In this note, we consider a class of alternative Newton methods whichattempt to avoid such difficulties.Such schemes have already proved of use in the Harwell Subroutine Libraryquadratic programming codes {\tt VE14} and {\tt VE19}.