Mathematical Programming

, Volume 61, Issue 1–3, pp 233–249 | Cite as

A recursive algorithm for finding the minimum norm point in a polytope and a pair of closest points in two polytopes

  • Kazuyuki Sekitani
  • Yoshitsugu Yamamoto


For a given pair of finite point setsP andQ in some Euclidean space we consider two problems: Problem 1 of finding the minimum Euclidean norm point in the convex hull ofP and Problem 2 of finding a minimum Euclidean distance pair of points in the convex hulls ofP andQ. We propose a finite recursive algorithm for these problems. The algorithm is not based on the simplicial decomposition of convex sets and does not require to solve systems of linear equations.

Key words

Minimum norm point minimum distance pair of points recursive algorithm convex quadratic program 


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  1. [1]
    M.D. Canon and C.D. Cullum, “The determination of optimum separating hyperplanes I. A finite step procedure,” RC 2023, IBM Watson Research Center (Yorktown Heights, NY, 1968).Google Scholar
  2. [2]
    M.D. Canon and C.D. Cullum, “A tight upper bound on the rate of convergence of the Frank—Wolfe algorithm,”SIAM Journal on Control 6 (1968) 509–516.Google Scholar
  3. [3]
    M. Frank and P. Wolfe, “An algorithm for quadratic programming,”Naval Research Logistics Quarterly 3 (1956) 95–110.Google Scholar
  4. [4]
    R.M. Freund, “Dual gauge programs, with applications to quadratic programming and the minimumnorm problem,”Mathematical Programming 38 (1987) 47–67.Google Scholar
  5. [5]
    S. Fujishige and P. Zhan, “A dual algorithm for finding the minimum-norm point in a polytope,”Journal of the Operations Research Society of Japan 33 (1990) 188–195.Google Scholar
  6. [6]
    S. Fujishige and P. Zhan, “A dual algorithm for finding a nearest pair of points in two polytopes,” DP No. 470, Institute of Socio-Economic Planning, University of Tsukuba (Tsukuba, 1991).Google Scholar
  7. [7]
    O.L. Mangasarian and R. De Leone, “Error bounds for strongly convex program and (super)linearly convergent iterative schemes for the least 2-norm solution of linear programs,”Applied Mathematics and Optimization 17 (1988) 1–14.Google Scholar
  8. [8]
    O.L. Mangasarian and T.-H. Shiau, “Error bounds for monotone linear complementarity problems,”Mathematical Programming 36 (1986) 81–89.Google Scholar
  9. [9]
    P. Wolfe, “Finding the nearest point in a polytope,”Mathematical Programming 11 (1976) 128–149.Google Scholar

Copyright information

© The Mathematical Programming Society, Inc. 1993

Authors and Affiliations

  • Kazuyuki Sekitani
    • 1
  • Yoshitsugu Yamamoto
    • 1
  1. 1.Institute of Socio-Economic PlanningUniversity of TsukubaTsukuba, IbarakiJapan

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