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 violation 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 high probability of feasibility.

翻译：在本文中,我们调查了在受下限和上限限制的情况下最大限度地增加总收入的在线分配问题。 与经过广泛研究的仅有资源上限的在线问题相比, 双向限制对资源消费前景的影响更为严重。 因此, 只有有限的违反限制或悲观的竞争界限才能得到保证。 为了应对这一挑战, 我们定义了衡量可行性的尺度 $xi $, 以评估这一问题的难度, 并通过理论保证优化常规来估计这一计量。 我们提议采用一个建设性的计算法框架, 即我们使用估算开始启用一个阈值价格矢量, 然后动态更新价格矢量, 并在每一步骤上使用它来进行决策。 由此可以证明, 拟议的算法是 $( 1- O ) (\ frac)\ varepsilon_ \ \ \ varepsilon}\\ varepsilon} big, 或者 $( 1- O) i- O (frac) exprecial six) presslational commission a big) exfective extizeal comlistal real resmissal presmission the the the the the the the the firstal firstal firstal firstal firstal proal presolgal presolgal presmalmalmluttalmluttalmlational presmlational pres.