The Iteration-Complexity Upper Bound for the Mizuno-Todd-Ye Predictor-Corrector Algorithm is Tight

Abstract

It is an open question whether there is an interior-point algorithm for linear optimization problems with a lower iteration-complexity than the classical bound $$\mathcal {O}(\sqrt{n} \log (\frac{\mu _1}{\mu _0}))$$. This paper provides a negative answer to that question for a variant of the Mizuno-Todd-Ye predictor-corrector algorithm. In fact, we prove that for any $$\varepsilon >0$$, there is a redundant Klee-Minty cube for which the aforementioned algorithm requires $$n^{(\frac{1}{2}-\varepsilon )} $$ iterations to reduce the barrier parameter by at least a constant. This is provably the first case of an adaptive step interior-point algorithm where the classical iteration-complexity upper bound is shown to be tight.