Bounds on the expectation of a convex function of a multivariate random variable

The idea of bounding E[g(X)], where g is a convex function of a multivariate random variable X, has received considerable attention in the literature. The need for such a bound arises from the complexities associated with computing E[g(X)], especially in a stochastic programming setting, when the dimension of X is large. The first contribution in this area was Jensen's inequality (1906). The purpose here is to discuss the Ist order bounds derived by Madansky (1959), Ben-Tal and Hochman's (1972) refinement of the Madansky bound using additional information, and Gassman and Ziemba's (1986) extension of the Madansky bound. An example of bounding E[f(X)], where f is not necessarily convex, will also be discussed.