  In this paper, we develop new continuous and discrete relaxations for
nonlinear expressions in an MINLP. In contrast to factorable programming, our
techniques utilize the inner-function structure by encapsulating it in a
polyhedral set, using a technique first proposed in [12]. We tighten the
relaxations derived in [33,13] and obtain new relaxations for functions that
could not be treated using prior techniques. We develop new
discretization-based mixed-integer programming relaxations that yield tighter
relaxations than similar relaxations in the literature. These relaxations
utilize the simplotope that captures inner-function structure to generalize the
incremental formulation of [8] to multivariate functions. In particular, when
the outer-function is supermodular, our formulations require exponentially
fewer continuous variables than any previously known formulation.
