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 with relative accuracy δ. We prove that the upper complexity bound for both schemes is O((√(n ln m)/δ)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 R n . In both cases, this computation can be performed very efficiently, in O(n 2 m ln m) operations at most.

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