51,608 research outputs found
A new look at nonnegativity on closed sets and polynomial optimization
We first show that a continuous function f is nonnegative on a closed set
if and only if (countably many) moment matrices of some signed
measure with support equal to K, are all positive semidefinite
(if is compact is an arbitrary finite Borel measure with support
equal to K. In particular, we obtain a convergent explicit hierarchy of
semidefinite (outer) approximations with {\it no} lifting, of the cone of
nonnegative polynomials of degree at most . Wen used in polynomial
optimization on certain simple closed sets \K (like e.g., the whole space
, the positive orthant, a box, a simplex, or the vertices of the
hypercube), it provides a nonincreasing sequence of upper bounds which
converges to the global minimum by solving a hierarchy of semidefinite programs
with only one variable. This convergent sequence of upper bounds complements
the convergent sequence of lower bounds obtained by solving a hierarchy of
semidefinite relaxations
A discrete Farkas lemma
Given and , we consider the issue of
existence of a nonnegative integral solution to the system of
linear equations . We provide a discrete and explicit analogue of the
celebrated Farkas lemma for linear systems in and prove that checking
existence of integral solutions reduces to solving an explicit linear
programming problem of fixed dimension, known in advance.Comment: 9 pages; ICCSA 2003 conference, Montreal, May 200
Correction Bounds on measures satisfying moment conditions
The Annals of Applied Probability (2002) 12 1114-113
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