Integer Polynomial Optimization in Fixed Dimension

Authors:
Jess A. De Loera;Raymond Hemmecke;Matthias Kppe;Robert Weismantel
Affiliations:
Department of Mathematics, University of California, Davis, California 95616, USA;Otto-von-Guericke-Universitt Magdeburg, FMA/IMO, Universittsplatz 2, 39106 Magdeburg, Germany;Otto-von-Guericke-Universitt Magdeburg, FMA/IMO, Universittsplatz 2, 39106 Magdeburg, Germany;Otto-von-Guericke-Universitt Magdeburg, FMA/IMO, Universittsplatz 2, 39106 Magdeburg, Germany
Venue:
Mathematics of Operations Research
Year:
2006

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Abstract

We classify, according to their computational complexity, integer optimization problems whose constraints and objective functions are polynomials with integer coefficients, and the number of variables is fixed. For the optimization of an integer polynomial over the lattice points of a convex polytope, we show an algorithm to compute lower and upper bounds for the optimal value. For polynomials that are nonnegative over the polytope, these sequences of bounds lead to a fully polynomial-time approximation scheme for the optimization problem.