Change of Optimal Values: A Pre-calculated Metric

TL;DR

This paper derives an exact equation to pre-calculate the change in optimal values for linear and certain nonlinear least norm optimization problems, enabling faster decision-making without solving full optimization problems.

Contribution

It introduces a novel metric for efficiently estimating optimal value changes in least norm optimization problems, extending to nonlinear cases via linearization.

Findings

The metric accurately predicts optimal value changes in numerical examples.

Pre-calculation of optimal value change is at least ten times faster than solving full problems.

The approach is validated through trajectory alignment and other experiments.

Abstract

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.