This paper studies the online stochastic resource allocation problem (RAP) with chance constraints. The online RAP is a 0-1 integer linear programming problem where the resource consumption coefficients are revealed column by column along with the corresponding revenue coefficients. When a column is revealed, the corresponding decision variables are determined instantaneously without future information. Moreover, in online applications, the resource consumption coefficients are often obtained by prediction. To model their uncertainties, we take the chance constraints into the consideration. To the best of our knowledge, this is the first time chance constraints are introduced in the online RAP problem. Assuming that the uncertain variables have known Gaussian distributions, the stochastic RAP can be transformed into a deterministic but nonlinear problem with integer second-order cone constraints. Next, we linearize this nonlinear problem and analyze the performance of vanilla online primal-dual algorithm for solving the linearized stochastic RAP. Under mild technical assumptions, the optimality gap and constraint violation are both on the order of $\sqrt{n}$. Then, to further improve the performance of the algorithm, several modified online primal-dual algorithms with heuristic corrections are proposed. Finally, extensive numerical experiments on both synthetic and real data demonstrate the applicability and effectiveness of our methods.

翻译：本文研究了带有机会约束的在线随机资源分配问题（RAP）。在线RAP是一个0-1整数线性规划问题，其中资源消耗系数与对应的收益系数按列逐步揭示。当一列数据被揭示时，相应的决策变量需即时确定，且无法获知未来信息。此外，在线应用中，资源消耗系数通常通过预测获得。为刻画其不确定性，本文引入了机会约束。据我们所知，这是首次将机会约束纳入在线RAP问题中。假设不确定变量服从已知的高斯分布，则该随机RAP可转化为一个带有整数二阶锥约束的确定性非线性问题。随后，我们对此非线性问题进行线性化处理，并分析原始在线对偶算法在求解线性化随机RAP时的性能。在温和的技术假设下，最优性间隙与约束违反量均达到$\sqrt{n}$量级。为进一步提升算法性能，我们提出了若干带有启发式修正的改进型在线对偶算法。最后，基于合成数据与真实数据的大量数值实验验证了我们方法的适用性与有效性。