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.
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