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Improved complexity results on solving real-number linear feasibility problems

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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 A x = b and x ≥ 0, can be solved in in O(n 2.5 c(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(n 1.5 c(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.

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Authors and Affiliations

  1. Department of Management Science and Engineering, Stanford University, Stanford, CA, 94305, USA

    Yinyu Ye

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  1. Yinyu Ye
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Correspondence to Yinyu Ye.

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This author was supported in part by NSF Grants DMS-9703490 and DMS-0306611

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Ye, Y. Improved complexity results on solving real-number linear feasibility problems. Math. Program. 106, 339–363 (2006). https://doi.org/10.1007/s10107-005-0610-7

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