FPTAS for optimizing polynomials over the mixed-integer points of polytopes in fixed dimension

• Published in: Mathematical programming Vol. 115; no. 2; pp. 273 - 290
• Main Authors: De Loera, Jesús A., Hemmecke, Raymond, Köppe, Matthias, Weismantel, Robert
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer-Verlag 01.10.2008; Springer; Springer Nature B.V
• Subjects: Algorithms; Applied sciences; Approximation; Calculus of Variations and Optimal Control; Optimization; Combinatorics; Exact sciences and technology; Full Length Paper; Integer programming; Linear programming; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematical programming; Mathematics; Mathematics and Statistics; Mathematics of Computing; Numerical Analysis; Operational research and scientific management; Operational research. Management science; Optimization; Polynomials; Theoretical; Integer programming in fixed dimension; 90C30; Mixed-integer nonlinear programming; 90C57; 90C11; Approximation algorithms; FPTAS; 90C60; Computational complexity; Convex set; Polynomial approximation; Convex polytope; Mixed integer programming; Non linear programming; Approximation algorithm; Computational complexity, Approximation algorithms; Optimization; Polynomial time; Integer programming; FPTAS, 90C11; Algorithm complexity; Mixed-integer nonlinear programming, Integer programming in fixed dimension; Mathematical programming
• ISSN: 0025-5610; 1436-4646
• DOI: 10.1007/s10107-007-0175-8

We show the existence of a fully polynomial-time approximation scheme (FPTAS) for the problem of maximizing a non-negative polynomial over mixed-integer sets in convex polytopes, when the number of variables is fixed. Moreover, using a weaker notion of approximation, we show the existence of a fully polynomial-time approximation scheme for the problem of maximizing or minimizing an arbitrary polynomial over mixed-integer sets in convex polytopes, when the number of variables is fixed.