We introduce a new algorithm for solving unconstrained discrete-time optimal control problems. Our method follows a direct multiple shooting approach, and consists of applying the SQP method together with an $\ell_2$ augmented Lagrangian primal-dual merit function. We use the LQR algorithm to efficiently solve the primal component of the Newton-KKT system, and use a dual LQR backward pass to solve its dual component. We also present a new parallel algorithm for solving the dual component of the Newton-KKT system in $O(\log(N))$ parallel time, where $N$ is the number of stages. Combining it with (S\"{a}rkk\"{a} and Garc\'{i}a-Fern\'{a}ndez, 2023), we are able to solve the full Newton-KKT system in $O(\log(N))$ parallel time. The remaining parts of our method have constant parallel time complexity per iteration. Therefore, this paper provides, for the first time, a practical, highly parallelizable (for example, with a GPU) method for solving nonlinear discrete-time optimal control problems. As our algorithm is a specialization of NPSQP (Gill et al. 1992), it inherits its generic properties, including global convergence, fast local convergence, and the lack of need for second order corrections or dimension expansions, improving on existing direct multiple shooting approaches such as acados (Verschueren et al. 2022), ALTRO (Howell et al. 2019), GNMS (Giftthaler et al. 2018), FATROP (Vanroye et al. 2023), and FDDP (Mastalli et al. 2020).

翻译：我们提出了一种用于求解无约束离散时间最优控制问题的新算法。该方法采用直接多重打靶策略，通过将SQP方法与$\ell_2$增广拉格朗日原对偶罚函数相结合来实现。我们利用LQR算法高效求解牛顿-KKT系统的原分量，并通过反向对偶LQR回代求解其对偶分量。同时，我们提出了一种新的并行算法，可在$O(\log(N))$并行时间内求解牛顿-KKT系统的对偶分量，其中$N$为阶段数。结合(Särkkä and García-Fernández, 2023)的工作，我们能够在$O(\log(N))$并行时间内求解完整的牛顿-KKT系统。该方法的其余部分每次迭代具有恒定的并行时间复杂度。因此，本文首次提供了一种实用且高度可并行化（例如，使用GPU）的非线性离散时间最优控制问题求解方法。由于本算法是NPSQP (Gill et al. 1992)的特化形式，因此继承了其全局收敛性、快速局部收敛性，且无需二阶修正或维度扩展，从而改进了acados (Verschueren et al. 2022)、ALTRO (Howell et al. 2019)、GNMS (Giftthaler et al. 2018)、FATROP (Vanroye et al. 2023)和FDDP (Mastalli et al. 2020)等现有直接多重打靶方法。