  We study the mixed-integer epigraph of a special class of convex functions
with non-convex indicator constraints, which are often used to impose logical
constraints on the support of the solutions. The class of functions we consider
are defined as compositions of low-dimensional nonlinear functions with affine
functions Extended formulations describing the convex hull of such sets can
easily be constructed via disjunctive programming, although a direct
application of this method often yields prohibitively large formulations, whose
size is exponential in the number of variables. In this paper, we propose a new
disjunctive representation of the sets under study, which leads to compact
formulations with size exponential in the dimension of the nonlinear function,
but polynomial in the number of variables. Moreover, we show how to project out
the additional variables for the case of dimension one, recovering or
generalizing known results for the convex hulls of such sets (in the original
space of variables). Our computational results indicate that the proposed
approach can significantly improve the performance of solvers in structured
problems.
