As in the case of infeasible path-following algorithms, taking different step-lengths in the primal and dual updates under appropriate conditions can enhance the performance of homogeneous algorithms. Our purpose now is to establish one such condition.
Suppose we are given the search direction 
.
Let 
and 
be 
times the maximum possible
primal and dual step-lengths that can be taken for
the primal and dual updates, respectively (where 
).
Let
Suppose 
is updated to
by
Then it can be shown that the scaled primal and dual infeasibilities, defined respectively by
satisfy the relation
where
Our condition is basically determined by considering
reductions in the norms of the scaled infeasibilities.
To determine this condition, let us note that
the function 
,
, is decreasing if 
and increasing otherwise. Using this fact, we see that
the norms of the scaled infeasibilities 
are decreasing functions of the step-lengths
if and they are
increasing functions of the step-lengths otherwise.
To keep the possible amplifications in the
norms of the scaled infeasibilities to a minimum,
we set 
and 
to be
when 
;
otherwise, it is beneficial to take the maximum possible
primal and dual step-lengths so as to get the maximum
possible reductions in the scaled infeasibilities.
For the latter case, we take different step-lengths,
and 
, provided that
the resulting scaled total complementarity is also reduced, that is, if
when we substitute and
into (32) and (33).
To summarize, we take different step-lengths,
and ,
for the primal and dual updates only when
and (37) holds;
otherwise, we take the same step-length
for
and .
