L-Infinity Norm of Derivative Objective

We can add a smoothness objective by adding $L_{\infty }$ -norm of the first-order difference to the objective function.

$$\mathrm{minimize}\quad \delta +\eta \left\Vert \Delta h\right\Vert _{\infty }.$$

• $\ensuremath{L_\infty}$ -norm only cares about the maximum difference.
• Large $\eta$ means we put more weight on the smoothness than the side-lobe level.

Let $\sigma\mathrel{\stackrel{\mathrm{\Delta}}{=}}\left\Vert \Delta h\right\Vert _{\infty }$ and set up the inequality constraints

$$-\sigma \leq \Delta h_{i}\leq \sigma \qquad i=1,\ldots ,L-1.$$

In matrix form:

$$\left[\begin{array}{c} -\mathbf{D}\\ \mathbf{D}\end{array}\right]h-\sigma \mathbf1 \le 0.$$

The objective function is then

$$\mathrm{minimize}\quad \delta +\eta \sigma.$$

Chebyshev norm of diff(h) added to the objective function to be minimized ($\eta=1$ ):

Twenty times the Chebyshev norm of diff(h) added to the objective function to be minimized ($\eta=20$ ):