J. William Helton
The matricial relaxation of a linear matrix inequality.
Fri, 03/05/2010 - 16:01 — SomNambulAuthors: J. William Helton, Scott McCullough, Igor Klep
Subjects: Operator Algebras
AbstractGiven linear matrix inequalities (LMIs) L_1 and L_2, it is natural to ask:
(Q1) when does one dominate the other, that is, does L_1(X) PsD imply L_2(X)
PsD? (Q2) when do they have the same solution set? Such questions can be
NP-hard. This paper describes a natural relaxation of an LMI, based on
substituting matrices for the variables x_j. With this relaxation, the
domination questions (Q1) and (Q2) have elegant answers, indeed reduce to
constructible semidefinite programs. Assume there is an X such that L_1(X) and
L_2(X) are both PD, and suppose the positivity domain of L_1 is bounded.Non-Commutative Harmonic and Subharmonic Polynomials.
Tue, 09/29/2009 - 15:21 — SomNambulAuthors: J. William Helton, Daniel P. McAllaster, Joshua A. Hernandez
Subjects: Functional Analysis
AbstractThe paper introduces a notion of the Laplace operator of a polynomial p in
noncommutative variables x=(x_1,...,x_g). The Laplacian Lap[p,h] of p is a
polynomial in x and in a noncommuting variable h. When all variables commute we
have Lap[p,h]=h^2\Delta_x p where \Delta_x p is the usual Laplacian. A
symmetric polynomial in symmetric variables will be called harmonic if
Lap[p,h]=0 and subharmonic if the polynomial q(x,h):=Lap[p,h] takes positive
semidefinite matrix values whenever matrices X_1,..., X_g, H are substituted
for the variables x_1,...,x_g, h.Non-Commutative Harmonic and Subharmonic Polynomials.
Tue, 09/29/2009 - 15:21 — SomNambulAuthors: J. William Helton, Daniel P. McAllaster, Joshua A. Hernandez
Subjects: Functional Analysis
AbstractThe paper introduces a notion of the Laplace operator of a polynomial p in
noncommutative variables x=(x_1,...,x_g). The Laplacian Lap[p,h] of p is a
polynomial in x and in a noncommuting variable h. When all variables commute we
have Lap[p,h]=h^2\Delta_x p where \Delta_x p is the usual Laplacian. A
symmetric polynomial in symmetric variables will be called harmonic if
Lap[p,h]=0 and subharmonic if the polynomial q(x,h):=Lap[p,h] takes positive
semidefinite matrix values whenever matrices X_1,..., X_g, H are substituted
for the variables x_1,...,x_g, h.Every free basic semi-algebraic set has an LMI representation.
Tue, 09/01/2009 - 12:47 — SomNambulAuthors: J. William Helton, Scott McCullough
Subjects: Functional Analysis
AbstractThe (matrical) solution set of a Linear Matrix Inequality (LMI) is a convex
basic non-commutative semi-algebraic set. The main theorem of this paper is a
converse, a result which has implications for both semidefinite programming and
systems engineering. For p(x) a non-commutative polynomial in free variables x=
(x1, ... xg) we can substitute a tuple of symmetric matrices X= (X1, ... Xg)
for x and obtain a matrix p(X). Assume p is symmetric with p(0) invertible, let
Ip denote the set {X: p(X) is an invertible matrix}, and let Dp denote the
component of Ip containing 0.
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