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A cutting plane algorithm for convex programming that uses analytic centers

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Abstract

Anoracle for a convex setS ⊂ ℝn accepts as input any pointz in ℝn, and ifz ∈S, then it returns ‘yes’, while ifz ∉S, then it returns ‘no’ along with a separating hyperplane. We give a new algorithm that finds a feasible point inS in cases where an oracle is available. Our algorithm uses the analytic center of a polytope as test point, and successively modifies the polytope with the separating hyperplanes returned by the oracle. The key to establishing convergence is that hyperplanes judged to be ‘unimportant’ are pruned from the polytope. If a ball of radius 2−L is contained inS, andS is contained in a cube of side 2L+1, then we can show our algorithm converges after O(nL 2) iterations and performs a total of O(n 4 L 3+TnL 2) arithmetic operations, whereT is the number of arithmetic operations required for a call to the oracle. The bound is independent of the number of hyperplanes generated in the algorithm. An important application in which an oracle is available is minimizing a convex function overS.

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Author notes

  1. David S. Atkinson

    Present address: Department of Mathematics, Western Kentucky University, 42101, Bowling Green, KY, USA

Authors and Affiliations

  1. Department of Mathematics, University of Illinois at Urbana-Champaign, 61801, Urbana, IL, USA

    David S. Atkinson

  2. Department of Computer Science, University of Illinois at Urbana-Champaign, 61801, Urbana, IL, USA

    Pravin M. Vaidya

Authors
  1. David S. Atkinson
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  2. Pravin M. Vaidya
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Additional information

Supported by the National Science Foundation under Grant CCR-9057481PYI.

Supported by the National Science Foundation under Grants CCR-9057481 and CCR-9007195.

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Atkinson, D.S., Vaidya, P.M. A cutting plane algorithm for convex programming that uses analytic centers. Mathematical Programming 69, 1–43 (1995). https://doi.org/10.1007/BF01585551

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  • DOI: https://doi.org/10.1007/BF01585551

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