凸多目标MPC的权重证书：几何表征、$\u2113^1$构造与$\u2113^2$排除
Weight Certificates for Convex Multi-Objective MPC: Geometric Characterization, $\ell^1$ Construction, and $\ell^2$ Foreclosure
AI总结：
针对自动驾驶多目标MPC权重启发式调优的问题，提出权重证书的几何表征与构造方法，验证了其精度提升效果，同时指出逐点有效性局限并推动监控式加权求解方案发展。
AI中文摘要：
自动驾驶规则手册按字典序对规则违规进行排序，模型预测控制（MPC）可通过两种方式强制执行该排序：一是精确方式，每个时间步求解$L{+}1$个顺序规划；二是近似方式，采用由分离启发式规则$w_1\gg w_2\gg\cdots\gg w_L$调优的加权和。我们证明该启发式方法解决的是错误的问题。对于凸优先级有序规划，当权重辅以单位性能系数后，能在字典最优点处支撑成就映射的上像时，加权和可精确复现字典最优解；可容许权重构成外法向锥的单位性能切片。在铰链惩罚下，该切片是通过投影缩放KKT系统得到的多面体，且线性规划可返回带认证裕度的内点权重；在平方铰链惩罚下，只要极限乘子非零，就不存在精确的有限权重，且沿局部极小值分支的违规量以$O(1/w)$的速率衰减。经留存日志校准，在11类符合校准条件的场景中，有9类场景的所得权重各层级分量近乎相等；在包含25条规则的规则手册下的nuPlan闭环实验中，其合法层级事件精度约为匹配启发式权重的两倍。然而，该证书是逐点有效的：没有单个权重在一个回合的所有采样时间步都有效，其中位生存期为一个采样间隔（在原生速率下后续有效时间步数为零），且持久性与活动集稳定性相关。这些发现推动了带选择性级联回退的监控加权求解方法的发展，尽管合规模式监控器仅能检测到部分实测违规。
英文摘要：
Automated-driving rulebooks rank rule violations lexicographically, and model predictive control enforces that ranking either exactly, through $L{+}1$ sequential programs per tick, or approximately, through a weighted sum tuned by the separation heuristic $w_1\gg w_2\gg\cdots\gg w_L$. We show the heuristic answers the wrong question. For a convex priority-ordered program, a weighted sum reproduces the lexicographic optimum precisely when its weight, augmented by a unit performance coefficient, supports the upper image of the achievement map at the lexicographic point; the admissible weights form the unit-performance slice of an outward normal cone. Under hinge penalties this slice is a polyhedron obtained by projecting a scaled-KKT system, and a linear program returns an interior weight with a certified margin; under squared-hinge penalties no finite weight is exact whenever the limiting multiplier is nonzero, with violation along the local minimizer branch decaying as $O(1/w)$. Calibrated on held-out logs, the resulting weights have near-equal tier components in nine of eleven calibration-eligible scenario classes and roughly double legal-tier event precision against a matched heuristic weight in closed-loop nuPlan experiments on a 25-rule rulebook. The certificate is, however, pointwise: no single weight is valid across the sampled ticks of an episode, the median lifetime is one sampling interval (zero subsequent ticks at the native rate), and persistence tracks active-set stability. These findings motivate monitored weighted solves with selective cascade fallback, although the compliance-pattern monitor detects only a subset of measured lapses.
