  We investigate the continuous non-monotone DR-submodular maximization problem
subject to a down-closed convex solvable constraint. Our first contribution is
to construct an example to demonstrate that (first-order) stationary points can
have arbitrarily bad approximation ratios, and they are usually on the boundary
of the feasible domain. These findings are in contrast with the monotone case
where any stationary point yields a $1/2$-approximation (Hassani et al.
(2017)). Moreover, this example offers insights on how to design improved
algorithms by avoiding bad stationary points, such as the restricted continuous
local search algorithm (Chekuri et al. (2014)) and the aided measured
continuous greedy (Buchbinder and Feldman (2019)). However, the analyses in the
last two algorithms only work for the discrete domain because both need to
invoke the inequality that the multilinear extension of any submodular set
function is bounded from below by its Lovasz extension. Our second
contribution, therefore, is to remove this restriction and show that both
algorithms can be extended to the continuous domain while retaining the same
approximation ratios, and hence offering improved approximation ratios over
those in Bian et al. (2017a). for the same problem. At last, we also include
numerical experiments to demonstrate our algorithms on problems arising from
machine learning and artificial intelligence.
