AI 中文总结
该研究提出基于rake-compress树收缩的并行Riccati递归求解器，用于场景树MPC，可实现O(log N)跨度，开发了对应JAX包并证明其与KKT系统等价性。
AI 中文摘要
场景树模型预测控制（MPC）通过有根树表示未来信息，并在该树上优化非预见性策略。求解所得非线性规划的数值方法通常通过一系列分支线性二次调节器（LQR）子问题计算搜索方向。标准树Riccati递归虽需线性工作量，但其依赖链与树高成正比。我们提出一种基于rake-compress树收缩的代数精确并行求解器：在独立局部控制凝聚后，其两个操作作用于表示条件二次函数的节点与边数据。rake操作消除一个叶节点及其父边，将其约化贡献添加至父节点数据；compress操作消除一个一元节点，用一条边替代其两条相邻边，采用与链上并行Riccati方法相同的条件值组合。二者共同将任意有根树收缩至根节点，反转收缩过程可恢复所有Riccati系数、状态、控制及乘子。给定可重用拓扑规划，含N个节点且状态与控制维度固定的求解过程，算术工作量与存储量均为O(N)，跨度为O(log N)，与树高、平衡性及最大出度无关。该公式允许半正定对偶正则化（包括无正则化情况），且精确线性提升覆盖标准场景-MPC约定：每个信息节点对应一个控制量。我们证明了收缩恒等式及其与Karush-Kuhn-Tucker（KKT）系统的等价性。三个MIT许可的JAX包分别实现双向收缩、对偶正则化LQR求解器，以及面向用户的树结构最优控制原对偶内点求解器。
英文摘要
Scenario-tree model predictive control (MPC) represents future information by a rooted tree and optimizes a nonanticipative policy over that tree. Numerical methods for solving the resulting nonlinear program typically compute their search directions through a sequence of branched linear-quadratic regulator (LQR) subproblems. The standard tree Riccati recursion requires linear work but has a dependency chain proportional to tree height. We present an algebraically exact parallel solver based on rake-compress tree contraction. After independent local control condensation, its two operations act on node and edge data that represent conditional quadratic functions. A rake eliminates a leaf and its parent edge, adding their reduced contribution to the parent-node data. A compress eliminates a unary node and replaces its two adjacent edges by one edge, using the same conditional-value composition as parallel Riccati methods on a chain. Together they contract an arbitrary rooted tree to its root; reversing the contraction recovers every Riccati coefficient, state, control, and multiplier. Given a reusable topology plan, a solve with $N$ nodes and fixed state and control dimensions has $O(N)$ arithmetic work and storage and $O(\log N)$ span, independently of tree height, balance, and maximum out-degree. The formulation allows positive-semidefinite dual regularization, including the unregularized case, and an exact linear-size lifting covers the standard scenario-MPC convention of one control per information node. We prove the contraction identities and equivalence to the Karush-Kuhn-Tucker (KKT) system. Three MIT-licensed JAX packages implement the bidirectional contraction, the dual-regularized LQR solver, and a user-facing primal-dual interior-point solver for tree-structured optimal control.
Comments16 pages
