Change of Optimal Values: A Pre-calculated Metric

by   Fang Bai, et al.

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.


Standard Dual Quaternion Optimization and Its Application in Hand-Eye Calibration

Several common dual quaternion functions, such as the power function, th...

General Framework for Metric Optimization Problems with Delay or with Deadlines

In this paper, we present a framework used to construct and analyze algo...

Edge convex smooth interpolation curve networks with minimum L_∞-norm of the second derivative

We consider the extremal problem of interpolation of convex scattered da...

A Penetration Metric for Deforming Tetrahedra using Object Norm

In this paper, we propose a novel penetration metric, called deformable ...

Good and Bad Optimization Models: Insights from Rockafellians

A basic requirement for a mathematical model is often that its solution ...

On the representativeness of approximate solutions of discrete optimization problems with interval cost function

We consider discrete optimization problems with interval uncertainty of ...

Automatically Learning Compact Quality-aware Surrogates for Optimization Problems

Solving optimization problems with unknown parameters often requires lea...

Please sign up or login with your details

Forgot password? Click here to reset