求解线性等式约束LQR问题的高效算法
An efficient algorithm for solving linear equality-constrained LQR problems
AI总结：
本文提出了一种求解线性等式约束LQR问题的高效算法，通过两步方法将问题转化为无约束LQR，实现线性工作量和对数深度的计算效率，并展示了GPU加速和高效实现。
摘要：
我们考虑具有任意分阶段线性等式约束的有限时间线性二次调节器问题。我们提出了一种将问题转化为无约束LQR的两步方法。首先，每个阶段通过其端点状态之间的仿射关系表示。相邻关系的组合消除了共享状态，并且是结合的。因此，一个关联的后缀扫描可以计算出每个节点的状态参数化$x_i=T_i z_i + t_i$，使得剩余时间 horizon 可行。第二，一旦这些参数化已知，每个阶段可以独立转换。原始混合约束和后继处于其可行域的要求导致一个仿射控制映射$u_i=Y_i z_i + Z_i v_i + y_i$。替换产生一个具有更小、可能阶段依赖状态和控制维度的普通无约束LQR。我们证明了两个问题的等价性以及标准凸性假设的保持。该减少具有线性工作量和对数深度的时间 horizon。最后，我们展示通过应用相同的仿射关系收缩和扩展到原始KKT站定性方程，可以恢复乘数。我们实现了并验证了高效顺序和对数深度并行的JAX变种；后者为长时间 horizon 提供了显著的GPU加速。密集KKT比较和原始KKT残差验证了恢复的对偶解。我们以MIT许可证发布JAX包和一个高效的顺序C++实现。
英文摘要：
We consider finite-horizon linear-quadratic regulator (LQR) problems with arbitrary stagewise linear equality constraints. We present a two-part reduction to an unconstrained LQR problem. First, every stage is represented by an affine relation between its endpoint states. Composition of adjacent relations eliminates their shared state and is associative. An associative suffix scan therefore computes, for every node, an affine parameterization $x_i=T_i z_i+t_i$ of the states from which the remaining horizon is feasible. Second, once these parameterizations are known, every stage can be transformed independently. The original mixed constraints and the requirement that the successor lie in its feasible domain yield an affine control map $u_i=Y_i z_i+Z_i v_i+y_i$. Substitution produces an ordinary unconstrained LQR problem in the reduced state and control variables $(z_i,v_i)$, with smaller or equal dimensions, possibly varying by stage. We prove equivalence of the two problems and preservation of the standard convexity assumptions. The reduction has linear work and logarithmic depth in the horizon. Finally, we show that multipliers can be recovered by applying the same affine-relation contraction and expansion to the original KKT stationarity equations. We validate work-efficient sequential and logarithmic-depth parallel implementations against dense KKT solutions and original KKT residuals. An optimized C++/CUDA implementation exploits the varying reduced dimensions and provides a substantial end-to-end GPU speedup over sequential C++ in our experiments. We release the JAX and C++/CUDA packages under the MIT License.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
