I'd like to know if the following problem is well formulated and has solutions. I'm very new to the subject of nonlinear optimization with inequality constraints ('teaching myself the Kuhn-Tucker conditions as we speak).
The problem
Given a cubic spline $S(x, p)$ that interpolates any $n$ points $(x_1 , p_1) , \ldots , (x_n , p_n)$, ordered by $x_i$ -- $x$ being an independent variable. I'm looking for new values $(x_i ,p_i^\prime)$ for the set of points such that the gradient of the $S$ does not exceed a preset maximum value $\Delta_{max}$ at any $x$. In other words
$$minimize \; \; \; \sum_{i = 0}^n \; ( p_i - p_i^\prime )^2$$
Subject to the condition
$$\frac{\partial S}{\partial x} \le \Delta_{max}$$
throughout $S$ (or for $x_1 \le x \le x_n)$ .
