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-dual Newton-KKT system. As our algorithm is a specialization of NPSQP, 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, ALTRO, GNMS, FATROP, and FDDP. As our algorithm avoids sequential rollouts of the nonlinear dynamics, it can be combined with (S\"arkk\"a and Garc\'ia-Fern\'andez, 2023) to run in $O(\log(N))$ parallel time per iteration (where $N$ is the number of stages), as well as $O(1)$ parallel time per line search iteration. Therefore, this paper provides a practical, theoretically sound, and highly parallelizable (for example, with a GPU) method for solving nonlinear discrete-time optimal control problems.

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，通过将序列二次规划算法与基于$\ell_2$增广拉格朗日原始-对偶价值函数相结合来实现求解。我们利用LQR算法高效求解原始-对偶牛顿-KKT系统。由于该算法是NPSQP的特定实现，它继承了后者的通用性质，包括全局收敛性、快速局部收敛性，且无需二阶校正或维度扩展，从而改进了现有的直接多重打靶方法（如acados、ALTRO、GNMS、FATROP和FDDP）。由于避免了非线性动力学的顺序迭代展开，该算法可与（Särkkä和García-Fernández, 2023）结合，实现每次迭代$O(\log(N))$的并行时间（其中$N$为阶段数），以及每次线性搜索迭代的$O(1)$并行时间。因此，本文提供了一种实用、理论可靠且高度可并行化（例如通过GPU实现）的非线性离散时间最优控制问题求解方法。