  In this paper, we define the upper (resp. lower) covariance under multiple
probabilities via a corresponding max-min-max (resp. min-max-min) optimization
problem and the related properties of covariances are obtained. In particular,
we propose a fast algorithm of calculation for upper and lower covariances
under the finite number of probabilities. As an application, our algorithm can
be used to solve a class of quadratic programming problem exactly, and we
obtain a probabilistic representation of such quadratic programming problem.
