457 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
Lowering electricity prices through deregulation
A wave of regulatory reform is now transforming the U.S. electricity industry. As state and federal authorities allow independent power producers to compete with utilities in supplying electricity, consumers are paying close attention to the effects of this change on their energy bills. Although deregulation poses significant structural challenges, the introduction of competitive pressures should ultimately lead to efficiency gains for the industry and cost savings for households and businesses.Electric utilities ; Prices
The toric Hilbert scheme of a rank two lattice is smooth and irreducible
The toric Hilbert scheme of a lattice L in Z^n is the multigraded Hilbert
scheme parameterizing all ideals in k[x_1,...,x_n] with Hilbert function value
one for every degree in the grading monoid N^n/L. In this paper we show that if
L is two-dimensional, then the toric Hilbert scheme of L is smooth and
irreducible. This result is false for lattices of dimension three and higher as
the toric Hilbert scheme of a rank three lattice can be reducible.Comment: 19 pages, 4 figure
The Hilbert Zonotope and a Polynomial Time Algorithm for Universal Grobner Bases
We provide a polynomial time algorithm for computing the universal Gr\"obner
basis of any polynomial ideal having a finite set of common zeros in fixed
number of variables. One ingredient of our algorithm is an effective
construction of the state polyhedron of any member of the Hilbert scheme
Hilb^d_n of n-long d-variate ideals, enabled by introducing the Hilbert
zonotope H^d_n and showing that it simultaneously refines all state polyhedra
of ideals on Hilb^d_n
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