  We study a class of stochastic nonconvex optimization in the form of
$\min_{x\in\mathcal{X}} F(x):=\mathbb{E}_\xi [f(\phi(x,\xi))]$, i.e., $F$ is a
composition of a convex function $f$ and a random function $\phi$. Leveraging
an (implicit) convex reformulation via a variable transformation
$u=\mathbb{E}[\phi(x,\xi)]$, we develop stochastic gradient-based algorithms
and establish their sample and gradient complexities for achieving an
$\epsilon$-global optimal solution. Interestingly, our proposed Mirror
Stochastic Gradient (MSG) method operates only in the original $x$-space using
gradient estimators of the original nonconvex objective $F$ and achieves
$\tilde{\mathcal{O}}(\epsilon^{-2})$ complexities, which matches the lower
bounds for solving stochastic convex optimization problems. Under booking
limits control, we formulate the air-cargo network revenue management (NRM)
problem with random two-dimensional capacity, random consumption, and routing
flexibility as a special case of the stochastic nonconvex optimization, where
the random function $\phi(x,\xi)=x\wedge\xi$, i.e., the random demand $\xi$
truncates the booking limit decision $x$. Extensive numerical experiments
demonstrate the superior performance of our proposed MSG algorithm for booking
limit control with higher revenue and lower computation cost than
state-of-the-art bid-price-based control policies, especially when the variance
of random capacity is large.
  KEYWORDS: stochastic nonconvex optimization, hidden convexity, gradient
methods, passenger network revenue management, air-cargo network revenue
management
