Computing sum of squares decompositions with rational coefficients

Authors:
Helfried Peyrl;Pablo A. Parrilo
Affiliations:
Automatic Control Laboratory, ETH Zürich, Physikstrasse 3, 8092 Zürich, Switzerland;Laboratory for Information and Decision Systems, Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, MA 02139-4307, USA
Venue:
Theoretical Computer Science
Year:
2008

Citing 7
Cited 12

Semidefinite programming

SIAM Review
Global Optimization with Polynomials and the Problem of Moments

SIAM Journal on Optimization
Convex Optimization

Convex Optimization
An Inequality for Circle Packings Proved by Semidefinite Programming

Discrete & Computational Geometry
Sums of Squares and Semidefinite Program Relaxations for Polynomial Optimization Problems with Structured Sparsity

SIAM Journal on Optimization
Sum of squares method for sensor network localization

Computational Optimization and Applications
The algebraic degree of semidefinite programming

Mathematical Programming: Series A and B

Computing the radius of positive semidefiniteness of a multivariate real polynomial via a dual of Seidenberg's method

Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation
Deciding reachability of the infimum of a multivariate polynomial

Proceedings of the 36th international symposium on Symbolic and algebraic computation
The minimum-rank gram matrix completion via modified fixed point continuation method

Proceedings of the 36th international symposium on Symbolic and algebraic computation
On the generation of positivstellensatz witnesses in degenerate cases

ITP'11 Proceedings of the Second international conference on Interactive theorem proving
Exact certification in global polynomial optimization via sums-of-squares of rational functions with rational coefficients

Journal of Symbolic Computation
Computing Rational Points in Convex Semialgebraic Sets and Sum of Squares Decompositions

SIAM Journal on Optimization
Certificates of impossibility of Hilbert-Artin representations of a given degree for definite polynomials and functions

Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation
Computing rational solutions of linear matrix inequalities

Proceedings of the 38th international symposium on International symposium on symbolic and algebraic computation
Convex algebraic geometry and semidefinite optimization

Proceedings of the 38th international symposium on International symposium on symbolic and algebraic computation
Algorithmic aspects of sums of Hermitian squares of noncommutative polynomials

Computational Optimization and Applications
Certification of bounds of non-linear functions: the templates method

CICM'13 Proceedings of the 2013 international conference on Intelligent Computer Mathematics
Equilibrium problems involving the Lorentz cone

Journal of Global Optimization

Quantified Score

Hi-index	5.23

Visualization

Abstract

Sum of squares (SOS) decompositions for nonnegative polynomials are usually computed numerically, using convex optimization solvers. Although the underlying floating point methods in principle allow for numerical approximations of arbitrary precision, the computed solutions will never be exact. In many applications such as geometric theorem proving, it is of interest to obtain solutions that can be exactly verified. In this paper, we present a numeric-symbolic method that exploits the efficiency of numerical techniques to obtain an approximate solution, which is then used as a starting point for the computation of an exact rational result. We show that under a strict feasibility assumption, an approximate solution of the semidefinite program is sufficient to obtain a rational decomposition, and quantify the relation between the numerical error versus the rounding tolerance needed. Furthermore, we present an implementation of our method for the computer algebra system Macaulay 2.