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.
翻译:各种优化问题的形式是最低规范优化。 在本文中, 我们研究两个递增构建的最小规范优化问题之间的最佳值变化, 并在第二个问题中包含新的测量结果。 我们证明在线性最低规范优化问题中计算最佳值的变化是一个精确的方程式。 由于本文件的结果, 最佳值的变化可以预先计算为指导在线决策的衡量标准, 而不解决第二个优化问题, 只要第一个优化问题的解决办法和一致性存在, 其结果可以扩大到线性最小的最远优化问题, 以及通过线性化( 非线性)平等限制的非线性最短距离优化。 本文的这一推论为RA- L 2018 Bai 等人 所显示的经验性观测提供了理论上合理的解释。 作为额外的贡献, 我们提出另一个优化问题, 即调整给出的两个轨迹, 以进一步证明如何使用该计量标准。 测量的准确性得到验证, 数字性实例在总体上相当令人满意( 见 RA- L 2018 bai 等人 等人 等人 的实验, 和 非线性最短的最短的距离优化, ), 除非在最接近于最接近于最接近于最差的模型中, 。