论文标题
最佳值的更改：预计的度量
Change of Optimal Values: A Pre-calculated Metric
论文作者
论文摘要
各种优化问题采用最低规范优化的形式。在本文中，我们研究了两个增量构建的最小规范优化问题之间最佳值的变化，第二个测量值包括新的测量。我们证明了一个精确的方程式来计算线性最小规范优化问题中最佳值的变化。通过本文的结果，可以将最佳值的更改预先计算为指导在线决策的指标，而无需解决第二个优化问题，只要解决了第一个优化问题的解决方案和协方差。该结果可以扩展到线性最小距离优化问题，并通过线性化的（非线性）相等性约束，非线性最小距离优化。本文中的这一推论为RA-L 2018 Bai等人所示的经验观察提供了理论上的自我解释。作为另一个贡献，我们提出了另一个优化问题，即以给定姿势对齐两个轨迹，以进一步演示如何使用度量标准。用数值示例验证了度量的准确性，这通常是令人满意的（请参见RA-L 2018 Bai等人}中的实验），除非在某些极不利的情况下。最后但并非最不重要的一点是，通过提议的度量计算最佳值的速度至少要比直接解决相应的优化问题快一点。
A variety of optimization problems takes the form of a minimum norm optimization. In this paper, we study the change of optimal values between two incrementally constructed least norm optimization problems, with new measurements included in the second one. We prove an exact equation to calculate the change of optimal values in the linear least norm optimization problem. With the result in this paper, the change of the optimal values can be pre-calculated as a metric to guide online decision makings, without solving the second optimization problem as long the solution and covariance of the first optimization problem are available. The result can be extended to linear least distance optimization problems, and nonlinear least distance optimization with (nonlinear) equality constraints through linearizations. This derivation in this paper provides a theoretically sound explanation to the empirical observations shown in RA-L 2018 bai et al. As an additional contribution, we propose another optimization problem, i.e. aligning two trajectories at given poses, to further demonstrate how to use the metric. The accuracy of the metric is validated with numerical examples, which is quite satisfactory in general (see the experiments in RA-L 2018 bai et al.} as well), unless in some extremely adverse scenarios. Last but not least, calculating the optimal value by the proposed metric is at least one magnitude faster than solving the corresponding optimization problems directly.
