3,199 research outputs found
Finding largest small polygons with GloptiPoly
A small polygon is a convex polygon of unit diameter. We are interested in
small polygons which have the largest area for a given number of vertices .
Many instances are already solved in the literature, namely for all odd ,
and for and 8. Thus, for even , instances of this problem
remain open. Finding those largest small polygons can be formulated as
nonconvex quadratic programming problems which can challenge state-of-the-art
global optimization algorithms. We show that a recently developed technique for
global polynomial optimization, based on a semidefinite programming approach to
the generalized problem of moments and implemented in the public-domain Matlab
package GloptiPoly, can successfully find largest small polygons for and
. Therefore this significantly improves existing results in the domain.
When coupled with accurate convex conic solvers, GloptiPoly can provide
numerical guarantees of global optimality, as well as rigorous guarantees
relying on interval arithmetic
GloptiPoly 3: moments, optimization and semidefinite programming
We describe a major update of our Matlab freeware GloptiPoly for parsing
generalized problems of moments and solving them numerically with semidefinite
programming
Convergence analysis of a Lasserre hierarchy of upper bounds for polynomial minimization on the sphere
We study the convergence rate of a hierarchy of upper bounds for polynomial
minimization problems, proposed by Lasserre [SIAM J. Optim. 21(3) (2011), pp.
864-885], for the special case when the feasible set is the unit (hyper)sphere.
The upper bound at level r of the hierarchy is defined as the minimal expected
value of the polynomial over all probability distributions on the sphere, when
the probability density function is a sum-of-squares polynomial of degree at
most 2r with respect to the surface measure.
We show that the exact rate of convergence is Theta(1/r^2), and explore the
implications for the related rate of convergence for the generalized problem of
moments on the sphere.Comment: 14 pages, 2 figure
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