A Primal-Dual Accelerated Interior Point Method Whose Running Time Depends Only on $A$ (*)

Stephen A. Vavasis (Cornell) and Yinyu Ye (Iowa)

We propose a primal-dual ``layered-step'' interior point (LIP) algorithm for linear programming with data given by real numbers. This algorithm follows the central path, either with short steps or with a new type of step called a ``layered least squares'' (LLS) step. The algorithm returns an exact optimum after a finite number of steps---in particular, after $O(n^{3.5}c(A))$ iterations, where $c(A)$ is a function of the coefficient matrix. The LLS steps can be thought of as accelerating a classical path-following interior point method. One consequence of the new method is a new characterization of the central path: we show that it composed of at most $n^2$ alternating straight and curved segments. If the LIP algorithm is applied to integer data, we get as another corollary a new proof of a well-known theorem by Tardos that linear programming can be solved in strongly polynomial time provided that $A$ contains small-integer entries.

(*) This paper represents a simplification and primal-dual version of an earlier manuscript ``An accelerated interior point method whose running depends only on $A$'' by the same authors.