The problem of combining multiple forecasts of related quantities that obey expected equality and additivity constraints, often referred to a hierarchical forecast reconciliation, is naturally stated as a simple optimization problem. In this paper we explore optimization-based point forecast reconciliation at scales faced by large retailers. We implement and benchmark several algorithms to solve the forecast reconciliation problem, showing efficacy when the dimension of the problem exceeds four billion forecasted values. To the best of our knowledge, this is the largest forecast reconciliation problem, and perhaps on-par with the largest constrained least-squares-problem ever solved. We also make several theoretical contributions. We show that for a restricted class of problems and when the loss function is weighted appropriately, least-squares forecast reconciliation is equivalent to share-based forecast reconciliation. This formalizes how the optimization based approach can be thought of as a generalization of share-based reconciliation, applicable to multiple, overlapping data hierarchies.

翻译：将多个相关量的预测结果进行组合，使其满足预期的等式与可加性约束，这一问题通常被称为层次化预测协调，自然可表述为一个简单的优化问题。本文探讨了大型零售商所面临规模下的基于优化的点预测协调方法。我们实现并基准测试了多种求解预测协调问题的算法，证明了在问题维度超过四十亿个预测值时算法的有效性。据我们所知，这是迄今为止规模最大的预测协调问题，其规模或许可与已解决的最大规模约束最小二乘问题相媲美。本文还提出了若干理论贡献。我们证明，对于一类受限问题且损失函数权重设置恰当时，最小二乘预测协调等价于基于份额的预测协调。这从形式上说明了基于优化的方法可视为基于份额协调方法的推广，适用于多个重叠的数据层次结构。