  Multivariate time series forecasting with hierarchical structure is widely
used in real-world applications, e.g., sales predictions for the geographical
hierarchy formed by cities, states, and countries. The hierarchical time series
(HTS) forecasting includes two sub-tasks, i.e., forecasting and reconciliation.
In the previous works, hierarchical information is only integrated in the
reconciliation step to maintain coherency, but not in forecasting step for
accuracy improvement. In this paper, we propose two novel tree-based feature
integration mechanisms, i.e., top-down convolution and bottom-up attention to
leverage the information of the hierarchical structure to improve the
forecasting performance. Moreover, unlike most previous reconciliation methods
which either rely on strong assumptions or focus on coherent constraints
only,we utilize deep neural optimization networks, which not only achieve
coherency without any assumptions, but also allow more flexible and realistic
constraints to achieve task-based targets, e.g., lower under-estimation penalty
and meaningful decision-making loss to facilitate the subsequent downstream
tasks. Experiments on real-world datasets demonstrate that our tree-based
feature integration mechanism achieves superior performances on hierarchical
forecasting tasks compared to the state-of-the-art methods, and our neural
optimization networks can be applied to real-world tasks effectively without
any additional effort under coherence and task-based constraints
