We study stable matching problems under contingent priorities, whereby the clearinghouse prioritizes some agents based on the allocation of others. Using school choice as a motivating example, we first introduce a stylized model of a many-to-one matching market where the clearinghouse aims to prioritize applicants with siblings assigned to the same school and match them together. We provide a series of guidelines to implement these contingent priorities and introduce two novel concepts of stability that account for them. We study some properties of the corresponding mechanisms, including the existence of a stable assignment under contingent priorities, its incentive properties, and the complexity of finding one if it exists. Moreover, we provide mathematical programming formulations to find such stable assignments whenever they exist. Finally, using data from the Chilean school choice system, we show that our framework can significantly increase the number of siblings assigned together while having no large effect on students without siblings.

This study develops mixed-integer quadratically constrained program (MIQCP) and second-order cone program (SOCP) formulations for chance-constrained optimization problems with Gaussian mixture models. By leveraging the convex-concave characteristics of the standard normal cumulative distribution function (CDF), we provide both piecewise-linear inner and outer approximations of the original problem. Optimal objectives from these approximations are shown to lie within an epsilon difference of the true value with an adequate number of linear pieces. Additionally, for the conservative SOCP formulation, an explicit upper bound of the objective value is provided. To expedite computation, we propose a heuristic that selects a significantly lower number of linear pieces than the worst-case complexity, without sacrificing desired approximation accuracy. Through extensive numerical experiments analyzing the trade-off between solution time and approximation quality, we find that our formulations offer a significant computational advantage over classical Sample Average Approximation (SAA) methods across various scenarios, including different problem classes and sizes, mixture components, chance constraint reliability levels, and reference distributions.