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 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{cTx:Ax = b, x ≥ 0} with decision variable x∈ ⁿ, e show that the MTY P-C algorithm, started from a well-centered interior-feasible solution with duality gap n μ₀, finds an interior-feasible solution with duality gap less than nη in O(T(μ₀/η)+n3.5log(χ^*A)) iterations, where T(t)≡min²log(log t) ,log t\ for all $t > 0$ and χ^*A is a scaling invariant condition number associated with the matrix A. More specifically, χ^*A is the infimum of all the conditions numbers χAD, 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+ min²log L,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 log L.