January 12 - 13, 2007
Barvinok's algorithm for counting
- Generating functions and non-linear integer optimization
- Other Rational Functions (Vergne, Lasserre,Nesterov)
Barvinok's algorithm for counting
- Original approach
- Homogenized algorithm
- Integer linear programming in fixed dimension
We consider the problem (P) of minimizing a polynomial function
over a semialgebraic set defined by polynomial inequalities and equations.
This is a hard problem, which includes well known NP-hard instances such as
0/1 linear programming, the partition problem for integer sequences, or
matrix copositivity.
While it is hard to test whether a polynomial is nonnegative, one can test
efficiently whether it can be written as a sum of squares of polynomials
using semidefinite programming. Based on this paradigm, one can formulate
tractable semidefinite programming relaxations for (P) by replacing the
hard `nonnegativity' condition by the tractable 'sum of squares'
condition. The corresponding dual semidefinite programs involve positive
semidefinite moment matrices, which reflects the classical duality theory
between positive polynomials and moment sequences.
The objective of this tutorial is to present in detail the main properties
of these semidefinite programming relaxations: asymptotic/finite
convergence, optimality certificate, and extraction of global optimum
solutions for (P), and to review the underlying mathematical tools:
representation theorems for positive polynomials from real algebraic
geometry, results about the truncated moment problem, and the algebraic
eigenvalue method for solving systems of polynomial equations. These
characteristic features are implemented in GloptiPoly, a solver for
polynomial optimization developped by Henrion and Lasserre, and will be
demonstrated on examples. Additional topics that may be covered if time
allows include: various algebraic approaches to unconstrained polynomial
minimization, link to combinatorial methods for 0/1 polynomial
optimization, techniques for exploiting symmetry, sparsity, etc.
Part II
- The Gröebner fan.
- Total dual integrality.
- Group relaxations.
Part I
- Linear programming and triangulations.
- The integer program and toric ideals.
- Test sets, Gröebner bases and initial ideals.