  We consider a general conic mixed-binary set where each homogeneous conic
constraint $j$ involves an affine function of independent continuous variables
and an epigraph variable associated with a nonnegative function, $f_j$, of
common binary variables. Sets of this form naturally arise as substructures in
a number of applications including mean-risk optimization, chance-constrained
problems, portfolio optimization, lot-sizing and scheduling, fractional
programming, variants of the best subset selection problem, a class of sparse
semidefinite programs, and distributionally robust chance-constrained programs.
We give a convex hull description of this set that relies on simultaneous
characterization of the epigraphs of $f_j$'s, which is easy to do when all
functions $f_j$'s are submodular. Our result unifies and generalizes an
existing result in two important directions. First, it considers \emph{multiple
general convex cone} constraints instead of a single second-order cone type
constraint. Second, it takes \emph{arbitrary nonnegative functions} instead of
a specific submodular function obtained from the square root of an affine
function. We close by demonstrating the applicability of our results in the
context of a number of problem classes.
