174 research outputs found
Convex Hulls of Algebraic Sets
This article describes a method to compute successive convex approximations
of the convex hull of a set of points in R^n that are the solutions to a system
of polynomial equations over the reals. The method relies on sums of squares of
polynomials and the dual theory of moment matrices. The main feature of the
technique is that all computations are done modulo the ideal generated by the
polynomials defining the set to the convexified. This work was motivated by
questions raised by Lov\'asz concerning extensions of the theta body of a graph
to arbitrary real algebraic varieties, and hence the relaxations described here
are called theta bodies. The convexification process can be seen as an
incarnation of Lasserre's hierarchy of convex relaxations of a semialgebraic
set in R^n. When the defining ideal is real radical the results become
especially nice. We provide several examples of the method and discuss
convergence issues. Finite convergence, especially after the first step of the
method, can be described explicitly for finite point sets.Comment: This article was written for the "Handbook of Semidefinite, Cone and
Polynomial Optimization: Theory, Algorithms, Software and Applications
Computation with Polynomial Equations and Inequalities arising in Combinatorial Optimization
The purpose of this note is to survey a methodology to solve systems of
polynomial equations and inequalities. The techniques we discuss use the
algebra of multivariate polynomials with coefficients over a field to create
large-scale linear algebra or semidefinite programming relaxations of many
kinds of feasibility or optimization questions. We are particularly interested
in problems arising in combinatorial optimization.Comment: 28 pages, survey pape
Recognizing Graph Theoretic Properties with Polynomial Ideals
Many hard combinatorial problems can be modeled by a system of polynomial
equations. N. Alon coined the term polynomial method to describe the use of
nonlinear polynomials when solving combinatorial problems. We continue the
exploration of the polynomial method and show how the algorithmic theory of
polynomial ideals can be used to detect k-colorability, unique Hamiltonicity,
and automorphism rigidity of graphs. Our techniques are diverse and involve
Nullstellensatz certificates, linear algebra over finite fields, Groebner
bases, toric algebra, convex programming, and real algebraic geometry.Comment: 20 pages, 3 figure
Semidefinite Characterization and Computation of Real Radical Ideals
For an ideal given by a set of generators, a new
semidefinite characterization of its real radical is
presented, provided it is zero-dimensional (even if is not). Moreover we
propose an algorithm using numerical linear algebra and semidefinite
optimization techniques, to compute all (finitely many) points of the real
variety as well as a set of generators of the real radical
ideal. The latter is obtained in the form of a border or Gr\"obner basis. The
algorithm is based on moment relaxations and, in contrast to other existing
methods, it exploits the real algebraic nature of the problem right from the
beginning and avoids the computation of complex components.Comment: 41 page
A unified approach to computing real and complex zeros of zero-dimensional ideals
In this paper we propose a unified methodology for computing the set of complex () or real () roots of an ideal in , assuming is finite. We show how moment matrices, defined in terms of a given set of generators of the ideal I, can be used to (numerically) find not only the real variety , as shown in the authors’ previous work, but also the complex variety , thus leading to a unified treatment of the algebraic and real algebraic problem. In contrast to the real algebraic version of the algorithm, the complex analogue only uses basic numerical linear algebra because it does not require positive semidefiniteness of the moment matrix and so avoids semidefinite programming techniques. The links between these algorithms and other numerical algebraic methods are outlined and their stopping criteria are related
Border Basis relaxation for polynomial optimization
A relaxation method based on border basis reduction which improves the
efficiency of Lasserre's approach is proposed to compute the optimum of a
polynomial function on a basic closed semi algebraic set. A new stopping
criterion is given to detect when the relaxation sequence reaches the minimum,
using a sparse flat extension criterion. We also provide a new algorithm to
reconstruct a finite sum of weighted Dirac measures from a truncated sequence
of moments, which can be applied to other sparse reconstruction problems. As an
application, we obtain a new algorithm to compute zero-dimensional minimizer
ideals and the minimizer points or zero-dimensional G-radical ideals.
Experimentations show the impact of this new method on significant benchmarks.Comment: Accepted for publication in Journal of Symbolic Computatio
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