# Formulation and solution of non-linear optimzation problem with inequality constraints

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 $x_1 \le x \le x_n)$ .

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