  In this paper, we study the problem of maximizing a monotone normalized
one-sided $\sigma$-smooth ($OSS$ for short) function $F(x)$, subject to a
convex polytope. This problem was first introduced by Mehrdad et al.
\cite{GSS2021} to characterize the multilinear extension of some set functions.
Different with the serial algorithm with name Jump-Start Continuous Greedy
Algorithm by Mehrdad et al. \cite{GSS2021}, we propose Jump-Start Parallel
Greedy (JSPG for short) algorithm, the first parallel algorithm, for this
problem. The approximation ratio of JSPG algorithm is proved to be
$((1-e^{-\left(\frac{\alpha}{\alpha+1}\right)^{2\sigma}}) \epsilon)$ for any
any number $\alpha\in(0,1]$ and $\epsilon>0$. We also prove that our JSPG
algorithm runs in $(O(\log n/\epsilon^{2}))$ adaptive rounds and consumes $O(n
\log n/\epsilon^{2})$ queries. In addition, we study the stochastic version of
maximizing monotone normalized $OSS$ function, in which the objective function
$F(x)$ is defined as $F(x)=\mathbb{E}_{y\sim T}f(x,y)$. Here $f$ is a
stochastic function with respect to the random variable $Y$, and $y$ is the
realization of $Y$ drawn from a probability distribution $T$. For this
stochastic version, we design Stochastic Parallel-Greedy (SPG) algorithm, which
achieves a result of $F(x)\geq(1
-e^{-\left(\frac{\alpha}{\alpha+1}\right)^{2\sigma}}-\epsilon)OPT-O(\kappa^{1/2})$,
with the same time complexity of JSPG algorithm. Here $\kappa=\frac{\max
\{5\|\nabla F(x_{0})-d_{o}\|^{2}, 16\sigma^{2}+2L^{2}D^{2}\}}{(t+9)^{2/3}}$ is
related to the preset parameters $\sigma, L, D$ and time $t$.
