  In the context of bounding probability of small deviation, there are limited
general tools. However, such bounds have been widely applied in graph theory
and inventory management. We introduce a common approach to substantially
sharpen such inequality bounds by combining the semidefinite optimization
approach of moments problem and the Berry-Esseen theorem. As an application, we
improve the lower bound of Feige's conjecture from 0.14 to 0.1798.
