23 research outputs found
A refined error analysis for fixed-degree polynomial optimization over the simplex
We consider the problem of minimizing a fixed-degree polynomial over the
standard simplex. This problem is well known to be NP-hard, since it contains
the maximum stable set problem in combinatorial optimization as a special case.
In this paper, we revisit a known upper bound obtained by taking the minimum
value on a regular grid, and a known lower bound based on P\'olya's
representation theorem. More precisely, we consider the difference between
these two bounds and we provide upper bounds for this difference in terms of
the range of function values. Our results refine the known upper bounds in the
quadratic and cubic cases, and they asymptotically refine the known upper bound
in the general case.Comment: 13 page
An Alternative Perspective on Copositive and Convex Relaxations of Nonconvex Quadratic Programs
We study convex relaxations of nonconvex quadratic programs. We identify a
family of so-called feasibility preserving convex relaxations, which includes
the well-known copositive and doubly nonnegative relaxations, with the property
that the convex relaxation is feasible if and only if the nonconvex quadratic
program is feasible. We observe that each convex relaxation in this family
implicitly induces a convex underestimator of the objective function on the
feasible region of the quadratic program. This alternative perspective on
convex relaxations enables us to establish several useful properties of the
corresponding convex underestimators. In particular, if the recession cone of
the feasible region of the quadratic program does not contain any directions of
negative curvature, we show that the convex underestimator arising from the
copositive relaxation is precisely the convex envelope of the objective
function of the quadratic program, providing another proof of Burer's
well-known result on the exactness of the copositive relaxation. We also
present an algorithmic recipe for constructing instances of quadratic programs
with a finite optimal value but an unbounded doubly nonnegative relaxation.Comment: 26 page
A new certificate for copositivity
In this article, we introduce a new method of certifying any copositive matrix to be copositive. This is done through the use of a theorem by Hadeler and the Farkas Lemma. For a given copositive matrix this certificate is constructed by solving finitely many linear systems, and can be subsequently checked by checking finitely many linear inequalities. In some cases, this certificate can be relatively small, even when the matrix generates an extreme ray of the copositive cone which is not positive semidefinite plus nonnegative. This certificate can also be used to generate the set of minimal zeros of a copositive matrix. In the final section of this paper we introduce a set of newly discovered extremal copositive matrices
Positive semidefinite approximations to the cone of copositive kernels
It has been shown that the maximum stable set problem in some infinite
graphs, and the kissing number problem in particular, reduces to a minimization
problem over the cone of copositive kernels. Optimizing over this infinite
dimensional cone is not tractable, and approximations of this cone have been
hardly considered in literature. We propose two convergent hierarchies of
subsets of copositive kernels, in terms of non-negative and positive definite
kernels. We use these hierarchies and representation theorems for invariant
positive definite kernels on the sphere to construct new SDP-based bounds on
the kissing number. This results in fast-to-compute upper bounds on the kissing
number that lie between the currently existing LP and SDP bounds.Comment: 29 pages, 2 tables, 1 figur