# Spline fit with bounded derivations

How can I do a Spline Fit with bounds on some derivations?

Problem

Given:

• Set of data points $t_k, x_k$
• Set of nodes $n_i$
• order $D$ of the spline (in my case $D=5$)
• lower and upper bounds $m_d$,$M_d$ on the $d$-th derivation for $1\le d \le r$ (in my case $r=3$)

Wanted:

The spline $f$ (of order $D$) with nodes $n_i$, that fits the data point best in terms of square errors (ie. $f$ minimizes $\sum_k{(f(t_k)-x_k)^2}$) under the condition that $m_d \le f \le M_d$ for $1 \le d \le r$

Discussion

Without the restriction on the derivations this is a well known procedure, refer for example to: http://www.geometrictools.com/Documentation/BSplineCurveLeastSquaresFit.pdf

Maybe this problem can be dealt with convex optimization, since the set of splines, that meet the restrictions, is convex and the function to be minimized is a quadratic form on the coefficients of the spline. But I don't know much about convex optimization, so I would be grateful for a reference on a text that fits best to the problem at hand.

For my purposes, it doesn't need to be a least square fit. A fit, that minimizes the maximum distance would be as useful or even better. Possible advantage: the maximum is a convex function. Disadvantage: the maximum is not partially differentiable in the coefficients.

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