Integer Polynomial Optimization in Fixed Dimension

@article{Loera2006IntegerPO,
  title={Integer Polynomial Optimization in Fixed Dimension},
  author={Jes{\'u}s A. De Loera and Raymond Hemmecke and Matthias K{\"o}ppe and Robert Weismantel},
  journal={Math. Oper. Res.},
  year={2006},
  volume={31},
  pages={147-153}
}
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… 
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References

SHOWING 1-10 OF 22 REFERENCES
Integer Optimization on Convex Semialgebraic Sets
TLDR
Lenstra's theorem on the polynomial-time solvability of linear integer programming in fixed dimension to semidefinite integer programming is extended.
Minimizing Polynomial Functions
  • P. Parrilo, B. Sturmfels
  • Computer Science, Mathematics
    Algorithmic and Quantitative Aspects of Real Algebraic Geometry in Mathematics and Computer Science
  • 2001
TLDR
It is demonstrated that existing algebraic methods are dramatically outperformed by a relaxation technique, due to N.Z. Shor and the first author, which involves sums of squares and semidefinite programming.
Integer Programming with a Fixed Number of Variables
It is shown that the integer linear programming problem with a fixed number of variables is polynomially solvable. The proof depends on methods from geometry of numbers.
Computing the Ehrhart quasi-polynomial of a rational simplex
TLDR
A polynomial time algorithm to compute any fixed number of the highest coefficients of the Ehrhart quasi-polynomial of a rational simplex for rational polytopes of a fixed dimension is presented.
Global Optimization with Polynomials and the Problem of Moments
TLDR
It is shown that the problem of finding the unconstrained global minimum of a real-valued polynomial p(x): R n to R, in a compact set K defined byPolynomial inequalities reduces to solving an (often finite) sequence of convex linear matrix inequality (LMI) problems.
There Cannot be any Algorithm for Integer Programming with Quadratic Constraints
TLDR
It is shown that no computing device can be programmed to compute the optimum criterion value for all problems in this class of integer programming problems in which squares of variables may occur in the constraints.
An Algorithmic Theory of Lattice Points in Polyhedra
We discuss topics related to lattice points in rational polyhedra, including efficient enumeration of lattice points, “short” generating functions for lattice points in rational polyhedra, relations
Convex Combinatorial Optimization
TLDR
The convex combinatorial optimization problem is introduced, and it is shown that it is strongly polynomial time solvable over any edge-guaranteed family.
Short rational generating functions for lattice point problems
Abstract. We prove that for any fixed d the generating function of the projectionof the set of integer points in a rational d-dimensional polytope can be computed inpolynomial time. As a corollary, we
...
...