L-One Norm of Derivative Objective

Another way to add smoothness constraint is to add -norm of the derivative to the objective:

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

(4.82)

Note that the 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:

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

(4.83)

In matrix form,

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

(4.84)

The objective function becomes

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

(4.85)

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

**Figure:** norm of `diff(h)` added to the objective function ( **$\eta =1$** )
$\includegraphics[width=\twidth,height=6.5in]{eps/print_lone_chebwin_1}$

**Figure:** Six times the norm of `diff(h)` added to the objective function ( **$\eta =6$** )
$\includegraphics[width=\twidth,height=6.5in]{eps/print_lone_chebwin_2}$

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