《《2016 A variant of the Vavasis-Ye layered-step interior-point algorithm for linear programming》.pdf

A variant of the Vavasis-Ye layered-step interior-point algorithm for linear programming R. D. C. Monteiro∗ T. Tsuchiya† April, 2001 Abstract In this paper we present a variant of Vavasis and Ye’s layered-step path following primal- dual 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(χ¯A + n where n is the number of nonnegative variables and χ¯A is a certain condition number associated with the constraint matrix A. Vavasis and Ye’s algorithm requires explicit knowledge of χ¯A (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 χ¯A . 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. Key words: Interior-point algorithms, primal-dual algorithms, path-following, central path, lay- ered steps, condition number, polynomial complexity, predictor-corrector, affine scaling, strongly polynomial, linear programming. 1 Introduc