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The Mehrotra-type predictor-corrector variant of the
homogeneous path-following algorithm is as follows.
(Remarks (a) and (b) are based on computational experience rather
than our having any explanation at this time.)
- (a) The numerical instability of the Schur complement equation
arising from the homogeneous algorithms appears to be much
more severe than that of the infeasible path-following
algorithms as decreases to zero. We overcome
this difficulty by first setting and
then computing in (28) when
is smaller than . In essence, this
amounts to switching to the infeasible path-following
algorithms where the Schur complement equation (9)
is numerically more stable.
- (b) For the homogeneous algorithms, it seems not desirable
to correct for the primal infeasibility so as
keep the primal infeasibility below a certain
small level once that level has been reached.
The effect of such a correction can be quite erratic,
in contrast to the case of the infeasible path-following
- (c) Once again, there are other termination criteria: lack
of progress or short step-lengths. Here we do not test
possible infeasibility in the way we did for our
infeasible-interior-point algorithms, because we
have a specific termination criterion ((b) in the
descriptions above) to detect infeasibility.