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

TL;DR

This paper introduces a stochastic gradient algorithm for a class of nonconvex optimization problems with applications in network revenue management, achieving optimal complexity and outperforming existing policies in numerical tests.

Contribution

It develops a Mirror Stochastic Gradient method that operates directly in the original space, with proven optimal complexity for stochastic nonconvex optimization.

Findings

The MSG algorithm achieves $ ilde{O}(rac{1}{ ext{epsilon}^2})$ complexity, matching lower bounds.

Numerical experiments show higher revenue and lower computation cost compared to bid-price policies.

The method effectively handles stochastic nonconvex problems with hidden convexity in network revenue management.

Abstract

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