L-One Norm of Derivative Objective

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

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

This can be formulated as an LP by adding a vector of optimization parameters $\tau$ which bound derivatives.

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

In matrix form,

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

The objective function becomes

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

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

**Figure 3.34:**
$\includegraphics[width=\twidth,height=6.5in]{eps/print_lone_chebwin_1}$

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

**Figure 3.35:**
$\includegraphics[width=\twidth,height=6.5in]{eps/print_lone_chebwin_2}$

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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