A New Iteration-Complexity Bound for the MTY Predictor-Corrector Algorithm

In this paper we present a new iteration-complexity bound for the Mizuno-Todd-Ye predictorcorrector (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{cTx : Ax = b, x ≥ 0} with decision variable x ∈ , 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(n2 log(log(μ0/η)) + n log(χ̄A + n)) iterations, where χ̄A is a scaling invariant condition number associated with the matrix A. More specifically, χ̄A 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 + n logL)) 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.