L-One Norm of Derivative Objective

Another way to add smoothness constraint is to add $\ensuremath {L_1}$ -norm of the derivative to the objective:

$$\mathrm{minimize}\quad \delta +\eta \left\Vert \Delta h\right\Vert _1$$ (4.82)

Note that the $\ensuremath {L_1}$ norm is sensitive to all the derivatives, not just the largest.

We can formulate an LP problem by adding a vector of optimization parameters $\tau$ which bound derivatives:

$$-\tau _{i}\leq \Delta h_{i}\leq \tau _{i}\qquad i=1,\ldots ,L-1.$$ (4.83)

In matrix form,

$$\left[\begin{array}{r} -\mathbf{D}\\ \mathbf{D}\end{array} \right]h-\left[\begin{array}{c} -\tau \\ -\tau \end{array} \right]\le 0.$$ (4.84)

The objective function becomes

$$\mathrm{minimize}\quad \delta +\eta \mathbf1^{T}\tau .$$ (4.85)

See Fig.3.41 and Fig.3.42 for example results.

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

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