## Non-linear optimzation with multipe objectives and multiple inequality constraints

I posted an earlier version of this question on math.stackexchange almost a week ago but have not had any replies. I'm recasting the problem in light of new understanding. I'd like to know if the the problem is well formulated and has solutions/there exist any methods for solving it. I'm very new to the subject of nonlinear optimization with inequality constraints ('teaching myself the Kuhn-Tucker conditions as we speak).

The problem

The $n$ piecewise continuous (Hermite) cubic curves $C_1 , \ldots C_{n}$ interpolate $n+ 1$ consecutive points $(t_1 , p_1) , (t_2 , p_2) , \ldots , (t_{n} , p_{n}) , (t_{n+i} , p_{n+1})$, ordered by increasing $t_i$. I'm looking for the least values of $p_i$, say $p_i^\prime$ such that the gradient of each $C_i$ does not exceed a chosen maximum value $\Delta_{max}$, in addition to other conditions listed below. Otherwise stated:

$$\mbox{Minimize} \; \; \; ( p_i - p_i^\prime )$$

For each $p_i$, subject to the following $3n$ conditions

$$p_i^\prime \le p_i \; \; \; \; \mbox{i.e. }\; p_i \; \mbox{cannot be increased }$$ $$L_i \le p_i\; \; \; \; \mbox{i.e. }\; L_i \; \mbox{is the lower limit of} \; p_i$$ $$\left| \frac{\partial C_i}{\partial t} \right| \le \Delta_{max} \; \; \; \; \mbox{maximum gradient conditions}$$

for each $C_i$.

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