2,634 research outputs found
A Semidefinite Approach for Truncated K-Moment Problems
A truncated moment sequence (tms) of degree d is a vector indexed by
monomials whose degree is at most d. Let K be a semialgebraic set.The truncated
K-moment problem (TKMP) is: when does a tms y admit a positive Borel measure
supported? This paper proposes a semidefinite programming (SDP) approach for
solving TKMP. When K is compact, we get the following results: whether a tms y
of degree d admits a K-measure or notcan be checked via solving a sequence of
SDP problems; when y admits no K-measure, a certificate will be given; when y
admits a K-measure, a representing measure for y would be obtained from solving
the SDP under some necessary and some sufficient conditions. Moreover, we also
propose a practical SDP method for finding flat extensions, which in our
numerical experiments always finds a finitely atomic representing measure for a
tms when it admits one
Linear Optimization with Cones of Moments and Nonnegative Polynomials
Let A be a finite subset of N^n and R[x]_A be the space of real polynomials
whose monomial powers are from A. Let K be a compact basic semialgebraic set of
R^n such that R[x]_A contains a polynomial that is positive on K. Denote by
P_A(K) the cone of polynomials in R[x]_A that are nonnegative on K. The dual
cone of P_A(K) is R_A(K), the set of all A-truncated moment sequences in R^A
that admit representing measures supported in K. Our main results are: i) We
study the properties of P_A(K) and R_A(K) (like interiors, closeness, duality,
memberships), and construct a convergent hierarchy of semidefinite relaxations
for each of them. ii) We propose a semidefinite algorithm for solving linear
optimization problems with the cones P_A(K) and R_A(K), and prove its
asymptotic and finite convergence; a stopping criterion is also given. iii) We
show how to check whether P_A(K) and R_A(K) intersect affine subspaces; if they
do, we show to get get a point in the intersections; if they do not, we prove
certificates for the non-intersecting
The truncated tracial moment problem
We present tracial analogs of the classical results of Curto and Fialkow on
moment matrices. A sequence of real numbers indexed by words in non-commuting
variables with values invariant under cyclic permutations of the indexes, is
called a tracial sequence. We prove that such a sequence can be represented
with tracial moments of matrices if its corresponding moment matrix is positive
semidefinite and of finite rank. A truncated tracial sequence allows for such a
representation if and only if one of its extensions admits a flat extension.
Finally, we apply the theory via duality to investigate trace-positive
polynomials in non-commuting variables.Comment: 21 page
Positivstellensatz and flat functionals on path *-algebras
We consider the class of non-commutative *-algebras which are path algebras
of doubles of quivers with the natural involutions. We study the problem of
extending positive truncated functionals on such *-algebras. An analog of the
solution of the truncated Hamburger moment problem by Curto and Fialkow for
path *-algebras is presented and non-commutative positivstellensatz is proved.
We aslo present an analog of the flat extension theorem of Curto and Fialkow
for this class of algebras.Comment: corrected typo
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