  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.
