Consider an error minimization task \min \norm{Ax - b} (Ax as the “predictions”, and b is the “data”). Some interpretation— approximation: Ax^{*} is the best approximation of the vector b by linear combinations of columns of A geometric: Ax^{*} is a point in \mathcal{R}\left(A\right) closest to b estimation: linear measurement model y = Ax + v you took a measurement y, A is the theoretical measurement, v is the measurement error implausibility of making v error is \norm{v} given y = b (what you measured), most plausible x is x^{*} optimal design: x are design variables, Ax is the result; x^{*} is the design that best approximates desired b Penalty Function Approximation Suppose you are optimizing some design x with respect to some dynamics A. Suppose your design residual is r = Ax - b. And you have some kind of penalty function to describe how comfortable you are with various errors: \phi\left(r_1\right) + … + \phi\left(r_{n}\right).

Huber Penalty Function A quadratic at a small residuals (chiller about small residuals), and a linear to large functions.

“robust penalty” Least Norm Problem \begin{align} \min_{x}\quad & \norm{x} \ \textrm{s.t.} \quad & Ax = b \end{align} geometric: x^{*} is the smallest point in the solution set estimation: b = Ax are perfect measurements of x, and we want the most plausible x, calling \norm{x} the impossibility of x design: x are design variables and b are required results