Improved complexity results on solving real-number linear feasibility problems

Authors:
Yinyu Ye
Affiliations:
Department of Management Science and Engineering, Stanford University, 94305, Stanford, CA, USA
Venue:
Mathematical Programming: Series A and B
Year:
2006

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Abstract

We present complexity results on solving real-number standard linear programs LP(A,b,c), where the constraint matrix ** the right-hand-side vector ** and the objective coefficient vector ** are real. In particular, we present a two-layered interior-point method and show that LP(A,b,0), i.e., the linear feasibility problem Ax = b and x ≥ 0, can be solved in in O(n2.5c(A)) interior-point method iterations. Here 0 is the vector of all zeros and c(A) is the condition measure of matrix A defined in [25]. This complexity iteration bound is reduced by a factor n from that for general LP(A, b, c) in [25]. We also prove that the iteration bound will be further reduced to O(n1.5c(A)) for LP(A, 0, 0), i.e., for the homogeneous linear feasibility problem. These results are surprising since the classical view has been that linear feasibility would be as hard as linear programming.