Smallest Boxes Research Articles

Numerical solution for bounding feasible point sets

Finding feasible points is important in optimization. There are currently two major classes of algorithms to deal with the problem of feasible points. The first class of algorithms (of local nature) is to find an approximate feasible point. Given a neighbourhood of an approximate feasible point, the second class of algorithms is to prove whether a feasible point exists inside this neighbourhood. To the best of our knowledge, no methods have been practically implemented to efficiently find the smallest boxes for bounding the feasible points defined by a system of nonlinear and nonconvex inequalities, unless the feasible set is convex. In this paper, we will present a numerical method to find the smallest boxes for bounding the feasible point sets defined by a nonlinear and nonconvex inequality and/or a system of nonlinear and nonconvex inequalities. Two examples have been synthetically constructed and used to show that the proposed numerical method can indeed correctly find all the smallest bounding boxes at any given accuracy efficiently. A brief comparison with relevant techniques will be discussed. Our method may also be thought of as the first solid theoretical basis for multisection and multisplitting in global optimization, when compared with those empirical ones in the literature.

Gamma‐Ray Burst Counterparts: Optical Data

The surest solution of the gamma-ray burst (GRB) mystery is to find an unambiguous low-energy quiescent counterpart. To this end, we have intensively searched within the smallest GRB error boxes for any counterpart candidates. This paper reports on 255 hr of exposure with ground-based telescopes since 1981. We report our results in the U, B, V, R. I, J, H, and K bands. We find the usual array of mildly unusual sources in the boxes, but none is sufficiently unusual to suggest a causal connection. We find that the smallest boxes are empty to fairly deep magnitudes. This fact can be of significance since virtually all cosmological models place bright bursters inside normal host galaxies at moderate distances. To allow for quantitative evaluation of the predictions of these models, we have compiled a list of limits on the brighest galaxy inside each of the 26 regions in various bands. This list was compiled from our own results as well as from the published literature. The limits on host galaxy luminosities from these data are substantially more restrictive than the limits from recent optical transients because the bursts we report on are much brighter than the bursts with optical transients.