Rake-Compress Riccati Recursions for Parallel Scenario-Tree Model Predictive Control

Abstract

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.

Disclosure

“EREW model,” in Concurrent Computations: Algorithms, Architecture, and Technology, S. K. Tewksbury, B. W. Dickinson, and S. C. Schwartz, OpenAI Codex, using the GPT-5.6 Sol model, assisted with Eds. New York: Plenum Press, 1988, pp. 139–156. drafting and editing this manuscript and implementing some [23] A. L. Delcher, A. J. Grov”

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2026-08-23T19:18:53.300238Z

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