A new polynomial-time algorithm for linear programming

Abstract

We present a new polynomial-time algorithm for linear programming. In the worst case, the algorithm requires O(n3.5 L) arithmetic operations on O(L) bit numbers, where n is the number of variables and L is the number of bits in the input. The running-time of this algorithm is better than the ellipsoid algorithm by a factor of O(n2.5). We prove that given a polytope P and a strictly interior point a εP, there is a projective transformation of the space that maps P, a to P′, a′ having the following property. The ratio of the radius of the smallest sphere with center a′, containing P′ to the radius of the largest sphere with center a′ contained in P′ is O(n). The algorithm consists of repeated application of such projective transformations each followed by optimization over an inscribed sphere to create a sequence of points which converges to the optimal solution in polynomial time. © 1984 Akadémiai Kiadó.