Rounding of convex sets and efficient gradient methods for linear programming problems

In this paper we propose new efficient gradient schemes for two non-trivial classes of linear programming problems. These schemes are designed to compute approximate solutions withrelative accuracy . We prove that the upper complexity bound for both ln schemes is O( n m ln n) iterations of a gradient-type method, where n and m, (n < m), are the sizes of the corresponding linear programming problems. The proposed schemes are based on preliminary computation of an ellipsoidal rounding for some polytopes in Rn . In both cases this computation can be performed very efficiently, in O(n2 m ln m) operations at most.