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Computing an Integer Point of a Class of Convex Sets

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  • C. Dang

Abstract

A simplicial algorithm is proposed for computing an integer point of a convex set C⊂Rn satisfying $$x = \max \{ x^1 ,x^2 \} = (\max \{ _1^1 ,x_1^2 \} ,...,\max \{ x_n^1 ,x_n^2 \} ^T \in C,$$ with $$x^1 = (x_1^1 ,x_2^1 ,...,x_n^1 )^T \in C,{\text{ }}x^2 = (x_1^2 ,x_2^2 ,...,x_n^2 )^T \in C.$$ The algorithm subdivides R n into integer simplices and assigns an integer labelto each integer point of R n. Starting at an arbitraryinteger point, the algorithm follows a finite simplicial path that leads either to an integer point of C or to the conclusion that C has no integer point.

Suggested Citation

  • C. Dang, 2001. "Computing an Integer Point of a Class of Convex Sets," Journal of Optimization Theory and Applications, Springer, vol. 108(2), pages 333-348, February.
    • Handle: RePEc:spr:joptap:v:108:y:2001:i:2:d:10.1023_a:1026438301292
    DOI: 10.1023/A:1026438301292
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    References listed on IDEAS

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    1. Talman, A.J.J. & van der Laan, G., 1979. "A restart algorithm for computing fixed points without an extra dimension," Other publications TiSEM 1f2102f8-e6da-4e9c-a2ed-9, Tilburg University, School of Economics and Management.
    2. Herbert E. Scarf, 2008. "Production Sets with Indivisibilities Part I: Generalities," Palgrave Macmillan Books, in: Zaifu Yang (ed.), Herbert Scarf’s Contributions to Economics, Game Theory and Operations Research, chapter 2, pages 7-38, Palgrave Macmillan.
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    5. Herbert E. Scarf, 1967. "The Approximation of Fixed Points of a Continuous Mapping," Cowles Foundation Discussion Papers 216R, Cowles Foundation for Research in Economics, Yale University.
    6. Chuangyin Dang & Hans van Maaren, 1998. "A Simplicial Approach to the Determination of an Integer Point of a Simplex," Mathematics of Operations Research, INFORMS, vol. 23(2), pages 403-415, May.
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