Complexity of integer quasiconvex polynomial optimization

  • Authors:

  • Sebastian Heinz

  • Affiliations:

  • Institut für Mathematik, Mathematisch-Naturwissenschaftliche Fakultät II, Humboldt-Universität zu Berlin, Berlin, Germany

  • Venue:
  • Journal of Complexity - Festschrift for the 70th birthday of Arnold Schönhage
  • Year:
  • 2005

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Abstract

We study a particular case of integer polynomial optimization: Minimize a polynomial F° on the set of integer points described by an inequality system F1 ≤ 0,...,Fs≤0, where F°, F1,...,Fs are quasiconvex polynomials in n variables with integer coefficients.We design an algorithm solving this problem that belongs to the time-complexity class O(s)ċlO(1)ċ dO(n)ċ2O(n3, where d≥2 is an upper bound for the total degree of the polynomials involved and l denotes the maximum binary length of all coefficients. The algorithm is polynomial for a fixed number n of variables and represents a direct generalization of Lenstra's algorithm [Math. Oper. Res. 8 (1983) 538-548] in integer linear optimization. In the considered case, our complexity-result improves the algorithm given by Khachiyan and Porkolab [Discrete Comput. Geom. 23 (2000) 207-224] for integer optimization on convex semialgebraic sets.