A new iteration-complexity bound for the MTY predictorcorrector algorithm

Abstract. In this paper we present a new iteration-complexity bound for the Mizuno–Todd–Ye predictor-corrector (MTY P-C) primal-dual interior-point algorithm for linear programming. The analysis of the paper is based on the important notion of crossover events introduced by Vavasis and Ye. For a standard form linear program min{cT x: Ax = b, x ≥ 0} with decision variable x ∈ n, we show that the MTY P-C algorithm, started from a well-centered interior-feasible solution with duality gap nµ0, finds an interior-feasible solution with duality gap less than nη in O(T (µ0/η)+n3.5 log(χ̄∗A)) iterations, where T (t) ≡ min{n2 log(log t), log t} for all t> 0 and χ̄∗A is a scaling invariant condition number associated with the matrix A. More specifically, χ̄∗A is the infimum of all the conditions numbers χ̄AD, where D varies over the set of positive diagonal matrices. Under the setting of the Turing machine model, our analysis yields an O(n3.5LA + min{n2 logL,L}) iteration-complexity bound for the MTY P-C algorithm to find a primal-dual optimal solution, where LA and L are the input sizes of the matrix A and the data (A, b, c), respectively. This contrasts well with the classical iteration-complexity bound for the MTY P-C algorithm, which depends linearly on L instead of logL