Is there any way of reformulating the following problem so that it can be solved by means of e.g. Matlab's fmincon?
$\min f(x_1,\dots,x_n)$ subject to $c(x_1,\dots,x_n,t)\le0$, $t\in[t_{\min},t_{\max}]$
Here $f:\mathbb{R}^n\rightarrow\mathbb{R}$ is a convex function. The constraint function $c:\mathbb{R}^{n+1} \rightarrow \mathbb{R}$ is nonlinear and varies smoothly with the parameter $t$. A solution $(x^*_1,\dots,x^*_n)$ must satisfy the constraint $c\le0$ for all $t\in[t_{\min},t_{\max}]$.
EDIT:
I believe it can be assumed that the constraint function $c$ is convex for each $t$.
EDIT 2:
I want to design a propeller for an axial pump with a prescribed "sweep" $s(r)$, where $r$ is the radial coordinate in its cylindrical coordinate system $(r,\theta,z)$. The sweep is defined as $\pi/2-\phi(r)$, where $\phi$ is the angle between the tangent to the leading edge stacking curve $\gamma$ and the tangent to the meanline at the leading edge, which i call $u=u(r)$ from now on. The sweep $s$ must satisfy $s\ge a_0 + \frac{r}{r_{\max}}a_1$.
If I constrain the shape of the stacking curve $\gamma$ to be polynomial in the $r-\theta$ and the $r-z$ planes respectively, e.g. $\theta(r)=p_1(r)$ and $z(r)=p_2(r)$ with coefficients $x_1,\dots,x_k$ for $p_1$ and $x_{k+1},\dots,x_n$ for $p_2$, I will have a tangent $v=v(x_1,\dots,x_n)$ to $\gamma$.
Thus, since $\cos\phi=\frac{(u\cdot v)}{\|u\|\cdot\|v\|}$, we have $\frac{(u\cdot v)}{\|u\|\cdot\|v\|}\le\cos(\pi/2 -a_0 - \frac{r}{r_{\max}}a_1)$, which, in essence, constitutes $c(x_1,\dots,x_n,r)$. Note that $u=u(r)$ and $v=v(x_1,\dots,x_n)$.
The objective function $f$ is the arc length of $\gamma$, i.e. $f=\int \sqrt{\|v\|^2+1}dr$.
Hope this made sense...