L-One Norm of Derivative Objective

Another way to add a smoothness constraint is to add the $L_1$ - norm of the derivative to the objective.

$$\mathrm{minimize}\quad \delta +\eta \left\Vert \Delta h\right\Vert _1.$$

• The $L_1$ norm is sensitive to all the derivatives, not just the largest.

In LP, set up the inequality constraints

$$-\tau _{i}\leq \Delta h_{i}\leq \tau _{i}\qquad i=1,\ldots ,L-1,$$

or, in matrix form,

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

The objective function becomes

$$\mathrm{minimize}\quad \delta +\eta \mathbf1^{T}\underline{\tau}.$$

$L_1$ norm of diff(h) added to the objective function ($\eta=1$ ):

Six times the $L_1$ norm of diff(h) added to the objective function ($\eta=6$ ):