Polynomial-time Algorithm

We present a new polynomial-time algorithm for linear programming. In the worst case, the algorithm requires O(tf'SL) 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(n~'~). We prove that given a polytope P and a strictly interior point a E P, 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.