Efficient Algorithms for a Class of Stochastic Hidden Convex Optimization and Its Applications in Network Revenue Management

We study a class of stochastic nonconvex optimization in the form of ${\text{min}}_{x\in \mathcal{X}}F(x)≔{\mathbb{E}}_{\xi }[f(\varphi (x,\xi ))]$, that is, F is a composition of a convex function f and a random function $\varphi$. Leveraging an (implicit) convex reformulation via a variable transformation $u=\mathbb{E}[\varphi (x,\xi )]$, we develop stochastic gradient-based algorithms and establish their sample and gradient complexities for achieving an $ϵ$-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 $\tilde{\mathcal{O}}({ϵ}^{-2})$ complexities, matching 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 $\varphi (x,\xi )=x\wedge \xi$, that is, the random demand ξ 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.

Funding: This work was partly supported by the National Science Foundation [Grants CMMI-1761699, CRII-1755829], the ZJU-UIUC Institute Research Program, and NCCR Automation in Switzerland.

Supplemental Material: The computer code and data that support the findings of this study are available within this article’s supplemental material at https://doi.org/10.1287/opre.2022.0216.