We derive closed-form extensions of the sequential and parallel Riccati recursions for solving dual-regularized linear-quadratic regulator (LQR) problems, with $O(N)$ sequential time and $O(\log(N))$ parallel time, respectively. We show that these subproblems arise when using regularized primal-dual interior-point methods to solve smooth, constrained, non-convex, discrete-time optimal control problems via multiple-shooting, even in the presence of stagewise equality or inequality constraints, and without imposing any rank requirements on constraint Jacobians. We prove that, when certain inertia conditions on the Newton-KKT matrix are met, each nonzero primal step is a descent direction of an augmented barrier-Lagrangian merit function. We characterize these inertia conditions in terms of the positive-definiteness of the dual-regularized Riccati pivots (a weaker condition than the standard LQR positive-definiteness requirements), thereby yielding inexpensive certificates of the required inertia. We provide MIT-licensed implementations of our methods in C++ and in JAX, as well as a full formalization of our results in Lean. We benchmark our algorithm against leading optimal control and nonlinear programming solvers on complex trajectory optimization problems, establishing competitive performance on moderate problems and substantial gains as the horizon length, problem dimension, and constraint count increase.

翻译：本文推导了序列和并行Riccati递推的闭式扩展形式，用于求解双正则化线性二次型调节器问题，分别具有$O(N)$的串行时间复杂度和$O(\log(N))$的并行时间复杂度。我们证明，在使用正则化原始-对偶内点法通过多重打靶求解光滑、带约束、非凸的离散时间最优控制问题时，即使存在逐阶段等式或不等式约束且不对约束雅可比矩阵施加任何秩要求，这些子问题仍会出现。当牛顿-KKT矩阵满足某些惯性条件时，我们证明每个非零原始步长是增广障碍-拉格朗日罚函数的下降方向。我们通过双正则化Riccati主元的正定性（比标准LQR正定性要求更弱的条件）来刻画这些惯性条件，从而为所需惯性提供低成本的验证方法。我们在C++和JAX中提供了基于MIT许可证的实现，并在Lean中完整形式化了我们的结论。我们针对复杂轨迹优化问题，将所提算法与领先的最优控制和非线性规划求解器进行基准测试，证明其在中等规模问题上具有竞争性性能，且随着时域长度、问题维度和约束数量的增加，性能提升显著。