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 ).
Suppose is updated to by
Then it can be shown that the scaled primal and dual infeasibilities, defined respectively by
satisfy the relation
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 .