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arXiv 2205.01774math.OC
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一类随机隐凸优化的高效算法及其在网络收益管理中的应用

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

Xin Chen, Niao He, Yifan Hu, Zikun Ye

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AI总结：

针对一类复合形式的随机非凸优化问题，利用隐凸重构提出仅在原空间运行的MSG算法，复杂度匹配随机凸优化下界，将其应用于航空货运NRM问题，实验验证其收益更高、计算成本更低。

摘要：

我们研究了一类形如 $\boldsymbol{\text{min}}_{x \text{∈}\boldsymbol{\text{X}}} F(x) := \boldsymbol{\text{E}}_ξ[f(ϕ(x,ξ))]} 的随机非凸优化问题，即F是凸函数f与随机函数ϕ的复合函数。通过变量变换 $u=\boldsymbol{\text{E}}[ϕ(x,ξ)]$ 得到（隐式）凸重构形式，我们开发了基于随机梯度的算法，并确定了其达到ε-全局最优解所需的样本复杂度和梯度复杂度。值得注意的是，我们提出的Mirror Stochastic Gradient（MSG，镜像随机梯度）方法仅在原始x空间中运行，使用原始非凸目标F的梯度估计器，实现了 $\tilde{\boldsymbol{\text{O}}}(ε^{-2})$ 的复杂度，这与求解随机凸优化问题的下界相匹配。在预订限额控制下，我们将具有随机二维容量、随机消耗和路由灵活性的航空货运网络收益管理（NRM）问题建模为该随机非凸优化的特例，其中随机函数 $ϕ(x,ξ)=x \boldsymbol{\text{∧}}ξ$，即随机需求ξ截断预订限额决策x。大量数值实验表明，我们提出的MSG算法在预订限额控制中性能优异，与现有最先进的基于投标价格的控制策略相比，收益更高且计算成本更低，尤其在随机容量方差较大时表现更突出。

英文摘要：

We study a class of stochastic nonconvex optimization in the form of $\min_{x\in\mathcal{X}} F(x):=\mathbb{E}_ξ[f(ϕ(x,ξ))]$, i.e., $F$ is a composition of a convex function $f$ and a random function $ϕ$. Leveraging an (implicit) convex reformulation via a variable transformation $u=\mathbb{E}[ϕ(x,ξ)]$, 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, 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 $ϕ(x,ξ)=x\wedgeξ$, i.e., 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. KEYWORDS: stochastic nonconvex optimization, hidden convexity, gradient methods, passenger network revenue management, air-cargo network revenue management

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