Counterexample to a Conjecture on an Infeasible Interior-Point Method

Roos proved that the devised full-step infeasible algorithm has O(n) worst-case iteration complexity. This complexity bound depends linearly on a parameter ¯ κ(ζ), which is proved to be less than √ 2n. Based on extensive computational evidence (hundreds of thousands of randomly generated problems), Roos conjectured that ¯ κ(ζ) = 1 (Conjecture 5.1 in the above-mentioned paper), which would yield an O(√ n) iteration full-Newton step infeasible interior-point algorithm. In this paper we present an example showing that ¯ κ(ζ) is in the order of √ n, the same order as that proved in Roos's original paper. In other words, the conjecture is false.