Integer polynomial optimization in fixed dimension

Mathematics of Operations Research

Volumn 31, Issue 1, 2006, Pages 147-153

  De Loera, Jesús A ^a   Hemmecke, Raymond ^b   Köppe, Matthias ^b   Weismantel, Robert ^b  

^a UNIVERSITY OF CALIFORNIA   (United States)

^b OTTO VON GUERICKE UNIVERSITY   (Germany)

Author keywords

Approximation algorithms; Computational complexity; FPTAS; Integer nonlinear programming; Integer programming in fixed dimension; Rational functions

Indexed keywords

COMPUTATIONAL COMPLEXITY; INTEGER PROGRAMMING; NONLINEAR PROGRAMMING; OPERATIONS RESEARCH; OPTIMIZATION; PROBLEM SOLVING;

APPROXIMATION ALGORITHMS; FPTAS; INTEGER NONLINEAR PROGRAMMING; INTEGER PROGRAMMING IN FIXED DIMENSION; RATIONAL FUNCTIONS;

POLYNOMIAL APPROXIMATION;

EID: 33646746073     PISSN: 0364765X     EISSN: 15265471     Source Type: Journal    
DOI: 10.1287/moor.1050.0169     Document Type: Article

References (16)

1. Barvinok, A. I. 1994. Polynomial time algorithm for counting integral points in polyhedra when the dimension is fixed. Math. Oper. Res. 19 769-779.
2. Barvinok, A. I., J. Pommersheim. 1999. An algorithmic theory of lattice points in polyhedra. New Perspectives in Algebraic Combinatorics (Berkeley, CA, 1996-1997), Math. Sci. Res. Inst. Publ., Vol. 38. Cambridge University Press, Cambridge, UK, 91-147.
3. Barvinok, A. I. 2006. Computing the Ehrhart quasi-polynomial of a rational simplex. Math. Comput. Forthcoming.
4. de Klerk, E., M. Laurent, P. Parrilo. 2006. A PTAS for the minimization of polynomials of fixed degree over the simplex. Theoret. Comput. Sci. Forthcoming.
5. De Loera, J. A., R. Hemmecke, J. Tauzer, R. Yoshida. 2004. Effective lattice point counting in rational convex polytopes. J. Symbolic Comput. 38(4) 1273-1302.
6. De Loera, J. A., D. Haws, R. Hemmecke, P. Huggins, B. Sturmfels, R. Yoshida, 2004. Short rational functions for toric algebra and applications. J Symbolic Comput. 38(2) 959-973.
7. Garey, M. R., D. S. Johnson. 1979. Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, San Francisco, CA.
8. Huggins, P. M. 2004. Lattice point enumeration via rational functions and applications to optimization and statistics. Senior undergraduate thesis, Department of Mathematics, University of California, Davis, CA.
9. Jeroslow, R. G. 1973. There cannot be any algorithm for integer programming with quadratic constraints. Oper. Res. 21(1) 221-224.
10. Jones, J. P. 1982. Universal Diophantine equation. J. Symbolic Logic 47(3) 403-410.
11. Khachiyan, L., L. Porkolab. 2000. Integer optimization in convex semialgebraic sets. Discrete Comput. Geom. 23 207-224.
12. Lasserre, J. B. 2001. Global optimization with polynomials and the problem of moments. SIAM J. Optim. 11 796-817.
13. Laurent, M. 2003. A comparison of the Sherali-Adams, Lovász- Schrijver and Lasserre relaxations for 0-1 programming. Math. Oper. Res. 28(3) 470-496.
14. Lenstra, H. W., Jr. 1983. Integer programming with a fixed number of variables. Math. Oper. Res. 8 538-548.
15. Matiyasevich, Y. 1993. Hilben's Tenth Problem. The MIT Press, Cambridge, London, UK.
16. Parrilo, P. A., B. Sturmfels. 2003. Minimizing polynomial functions. S. Basu, L. Gonzalez-Vega, eds. Proc. DIMACS Workshop on Algorithmic and Quant. Aspects of Real Algebraic Geometry in Math. Comput. Sci., March 2001. American Mathematical Society. Providence, RI, 83-100.