A Variant of the Vavasis--Ye Layered-Step Interior-Point Algorithm for Linear Programming

In this paper we present a variant of Vavasis and Ye’s layered-step path following primaldual interior-point algorithm for linear programming. Our algorithm is a predictor-corrector type algorithm which uses from time to time the least layered squares (LLS) direction in place of the affine scaling direction. It has the same iteration-complexity bound of Vavasis and Ye’s algorithm, namely O(n3.5 log(χ̄A + n)) where n is the number of nonnegative variables and χ̄A is a certain condition number associated with the constraint matrix A. Vavasis and Ye’s algorithm requires explicit knowledge of χ̄A (which is very hard to compute or even estimate) in order to compute the layers for the LLS direction. In contrast, our algorithm uses the affine scaling direction at the current iterate to determine the layers for the LLS direction, and hence does not require the knowledge of χ̄A. A variant with similar properties and with the same complexity has been developed by Megiddo, Mizuno and Tsuchiya. However, their algorithm needs to compute n LLS directions on every iteration while ours computes at most one LLS direction on any given iteration.