  It has been well established that first order optimization methods can
converge to the maximal objective value of concave functions and provide
constant factor approximation guarantees for (non-convex/non-concave)
continuous submodular functions. In this work, we initiate the study of the
maximization of functions of the form $F(x) = G(x) +C(x)$ over a solvable
convex body $P$, where $G$ is a smooth DR-submodular function and $C$ is a
smooth concave function. This class of functions is a strict extension of both
concave and continuous DR-submodular functions for which no theoretical
guarantee is known. We provide a suite of Frank-Wolfe style algorithms, which,
depending on the nature of the objective function (i.e., if $G$ and $C$ are
monotone or not, and non-negative or not) and on the nature of the set $P$
(i.e., whether it is downward closed or not), provide $1-1/e$, $1/e$, or $1/2$
approximation guarantees. We then use our algorithms to get a framework to
smoothly interpolate between choosing a diverse set of elements from a given
ground set (corresponding to the mode of a determinantal point process) and
choosing a clustered set of elements (corresponding to the maxima of a suitable
concave function). Additionally, we apply our algorithms to various functions
in the above class (DR-submodular + concave) in both constrained and
unconstrained settings, and show that our algorithms consistently outperform
natural baselines.
