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A predictor-corrector method for
for semidefinite linear programming

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Chih-Jen Lin and Romesh Saigal

In this paper we present a generalization of the predictor corrector
method of linear programming problem to semidefinite linear programming problem.
We consider a direction which, we show, belongs to a family of directions
presented by Kojima, Shindoh and Hara. This direction has also been analyzed by
Zhang, who claims a similar result for a generalization of Mizuno, Todd and Ye
method for the linear programming problem. We show that starting with the
initial complementary slackness violation of $t_0$, in
$(\mbox{log}(\frac{\epsilon}{t_0})\sqrt{n})$ of the predictor corrector method,
the complementary slackness violation can be reduced to less than or equal
to $\epsilon>0$. We also analyse a modified corrector direction in which the
linear system to be solved differs from that of the predictor in only the right
hand side, and obtain a similar bound. We then use this modified corrector step
in an implementable method which is shown to take a total of
$(\mbox{log}(\frac{\epsilon}{t_0})\sqrt{n}\mbox{log}(n))$ predictor and
corrector steps. Since it takes $O(n^3)$ multiplications during each predictor
or corrector step, after $O(\mbox{log}(\frac{\epsilon}{t_0})n^{3.5})$
multiplications the complementary slackness violation is reduced to less than or
equal to $\epsilon$.
Technical Report, Department of Industrial Engineering, University
of Michigan, October, 1995.