  In this paper, we propose the first continuous optimization algorithms that
achieve a constant factor approximation guarantee for the problem of monotone
continuous submodular maximization subject to a linear constraint. We first
prove that a simple variant of the vanilla coordinate ascent, called
Coordinate-Ascent+, achieves a $(\frac{e-1}{2e-1}-\varepsilon)$-approximation
guarantee while performing $O(n/\varepsilon)$ iterations, where the
computational complexity of each iteration is roughly
$O(n/\sqrt{\varepsilon}+n\log n)$ (here, $n$ denotes the dimension of the
optimization problem). We then propose Coordinate-Ascent++, that achieves the
tight $(1-1/e-\varepsilon)$-approximation guarantee while performing the same
number of iterations, but at a higher computational complexity of roughly
$O(n^3/\varepsilon^{2.5} + n^3 \log n / \varepsilon^2)$ per iteration. However,
the computation of each round of Coordinate-Ascent++ can be easily parallelized
so that the computational cost per machine scales as
$O(n/\sqrt{\varepsilon}+n\log n)$.
