6 research outputs found
A Tensor Analogy of Yuan's Theorem of the Alternative and Polynomial Optimization with Sign structure
Yuan's theorem of the alternative is an important theoretical tool in
optimization, which provides a checkable certificate for the infeasibility of a
strict inequality system involving two homogeneous quadratic functions. In this
paper, we provide a tractable extension of Yuan's theorem of the alternative to
the symmetric tensor setting. As an application, we establish that the optimal
value of a class of nonconvex polynomial optimization problems with suitable
sign structure (or more explicitly, with essentially non-positive coefficients)
can be computed by a related convex conic programming problem, and the optimal
solution of these nonconvex polynomial optimization problems can be recovered
from the corresponding solution of the convex conic programming problem.
Moreover, we obtain that this class of nonconvex polynomial optimization
problems enjoy exact sum-of-squares relaxation, and so, can be solved via a
single semidefinite programming problem.Comment: acceted by Journal of Optimization Theory and its application, UNSW
preprint, 22 page
Copositive certificates of non-negativity for polynomials on semialgebraic sets
A certificate of non-negativity is a way to write a given function so that
its non-negativity becomes evident. Certificates of non-negativity are
fundamental tools in optimization, and they underlie powerful algorithmic
techniques for various types of optimization problems. We propose certificates
of non-negativity of polynomials based on copositive polynomials. The
certificates we obtain are valid for generic semialgebraic sets and have a
fixed small degree, while commonly used sums-of-squares (SOS) certificates are
guaranteed to be valid only for compact semialgebraic sets and could have large
degree. Optimization over the cone of copositive polynomials is not tractable,
but this cone has been well studied. The main benefit of our copositive
certificates of non-negativity is their ability to translate results known
exclusively for copositive polynomials to more general semialgebraic sets. In
particular, we show how to use copositive polynomials to construct structured
(e.g., sparse) certificates of non-negativity, even for unstructured
semialgebraic sets. Last but not least, copositive certificates can be used to
obtain not only hierarchies of tractable lower bounds, but also hierarchies of
tractable upper bounds for polynomial optimization problems.Comment: 27 pages, 1 figur