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… Expand
View PDF on arXiv
62 Citations
Highly Influential Citations
3
Background Citations
24
Methods Citations
9
Results Citations
3

62 Citations

Citation Type

Citation Type

FPTAS for optimizing polynomials over the mixed-integer points of polytopes in fixed dimension
TLDR
Using a weaker notion of approximation, the existence of a fully polynomial-time approximation scheme (FPTAS) for the problem of maximizing a non-negativePolynomial over mixed-integer sets in convex polytopes, when the number of variables is fixed is shown. Expand
Integer quadratic programming in the plane
We show that the problem of minimizing a quadratic polynomial with integer coefficients over the integer points in a general two-dimensional rational polyhedron is solvable in time bounded by a… Expand
FPTAS for mixed-integer polynomial optimization with a fixed number of variables
We show the existence of an FPTAS for the problem of maximizing a non-negative polynomial over mixed-integer sets in convex polytopes, when the number of variables is fixed.
Norm bounds and underestimators for unconstrained polynomial integer minimization
TLDR
The radius of a p-norm ball which contains all integer minimizers is proved to be smaller than the radius known from the literature, and a new class of underestimators of the polynomial function is derived. Expand
When is rounding allowed ? A new approach to integer nonlinear optimization
In this paper we present a new approach for solving unrestricted integer nonlinear programming problems. More precisely, we investigate in which cases one can solve the integer version of a nonlinear… Expand
Minimizing Cubic and Homogeneous Polynomials over Integers in the Plane
TLDR
It is shown that the problem of minimizing a homogeneous polynomial of any fixed degree over the integer points in a bounded polyhedron in a real vector space of dimension two is solvable inPolynomial time. Expand
When is rounding allowed in integer nonlinear optimization?
TLDR
The strong rounding property is investigated which allows rounding a solution to the continuous relaxation to the next integer solution and in turn yields that the integer version can be solved in the same time complexity as its continuous relaxation for arbitrary dimensions. Expand
The complexity of generating functions for integer points in polyhedra and beyond
Motivated by the formula for the sum of the geometric series, we consider various classes  of sets S �¼ Zd of integer points for which an a priori �glong�h Laurent series or polynomial m�¸S xm can… Expand
Integer Convex Minimization in Low Dimensions
In this dissertation we discuss several approaches to solve integer and mixedinteger convex minimization problems. That is, we try to minimize a convex function over a convex set with the additional… Expand
On the Complexity of Nonlinear Mixed-Integer Optimization
This is a survey on the computational complexity of nonlinear mixedinteger optimization. It highlights a selection of important topics, ranging from incomputability results that arise from number… Expand
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 24 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. Expand
Minimizing Polynomial Functions
  • P. Parrilo, B. Sturmfels
  • Mathematics, Computer Science
  • 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. Expand
Integer Programming with a Fixed Number of Variables
  • H. Lenstra
  • Mathematics, Computer Science
  • Math. Oper. Res.
  • 1983
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
  • A. Barvinok
  • Computer Science, Mathematics
  • Math. Comput.
  • 2006
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. Expand
Global Optimization with Polynomials and the Problem of Moments
  • J. Lasserre
  • Computer Science, Mathematics
  • SIAM J. Optim.
  • 2001
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. Expand
A PTAS for the minimization of polynomials of fixed degree over the simplex
TLDR
It is shown that the bounds pΔ(k) yield a polynomial time approximation scheme for the minimization of polynomials of fixed degree d on the simplex, extending an earlier result of Bomze and De Klerk for degree d = 2. Expand
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. Expand
A polynomial time algorithm for counting integral points in polyhedra when the dimension is fixed
  • A. Barvinok
  • Computer Science
  • Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science
  • 1993
We prove that for any dimension d there exists a polynomial time algorithm for counting integral points in polyhedra in the d-dimensional Euclidean space. Previously such algorithms were known for… Expand
Effective lattice point counting in rational convex polytopes
TLDR
LattE , a computer package for lattice point enumeration which contains the first implementation of A. Barvinok’s algorithm, is described and it is proved that these kinds of symbolic–algebraic ideas surpass the traditional branch-and-bound enumeration and in some instances LattE is the only software capable of counting. Expand
A polynomial time algorithm for counting integral points in polyhedra when the dimension is fixed
We prove that for any dimension d there exists a polynomial time algorithm for counting integral points in polyhedra in the d-dimensional Euclidean space. Previously such algorithms were known for… Expand
...
1
2
3
...