In this paper, we investigate the online allocation problem of maximizing the overall revenue subject to both lower and upper bound constraints. Compared to the extensively studied online problems with only resource upper bounds, the two-sided constraints affect the prospects of resource consumption more severely. As a result, only limited violations of constraints or pessimistic competitive bounds could be guaranteed. To tackle the challenge, we define a measure of feasibility $\xi^*$ to evaluate the hardness of this problem, and estimate this measurement by an optimization routine with theoretical guarantees. We propose an online algorithm adopting a constructive framework, where we initialize a threshold price vector using the estimation, then dynamically update the price vector and use it for decision-making at each step. It can be shown that the proposed algorithm is $\big(1-O(\frac{\varepsilon}{\xi^*-\varepsilon})\big)$ or $\big(1-O(\frac{\varepsilon}{\xi^*-\sqrt{\varepsilon}})\big)$ competitive with high probability for $\xi^*$ known or unknown respectively. To the best of our knowledge, this is the first result establishing a nearly optimal competitive algorithm for solving two-sided constrained online allocation problems with a high probability of feasibility.

翻译：本文研究在满足下界和上界约束条件下最大化总收益的在线分配问题。与仅考虑资源上界约束的广泛研究的在线问题相比，双边约束对资源消耗前景的影响更为显著。因此，只能保证约束的有限违反或悲观的竞争比。为应对这一挑战，我们定义了一个可行性度量$\xi^*$来评估该问题的难度，并通过具有理论保证的优化程序估计该度量。我们提出了一种采用构造性框架的在线算法，其中使用估计值初始化阈值价格向量，然后动态更新价格向量并用于每一步的决策。可以证明，对于已知或未知的$\xi^*$，所提出的算法分别以高概率实现$\big(1-O(\frac{\varepsilon}{\xi^*-\varepsilon})\big)$或$\big(1-O(\frac{\varepsilon}{\xi^*-\sqrt{\varepsilon}})\big)$的竞争比。据我们所知，这是第一个在具有高可行性概率的情况下，为求解双边约束在线分配问题建立近最优竞争算法的结果。