The context of my problem:
I'm trying to implement an optimization problem in MATLAB using YALMIP. The objective function computes the integral cost of the squared norm of the derivative of my NURBS-parametrized curve to find the control points that minimize it's length. For semantics reasons I can't implement my objective function in MATLAB, as it is a rational function, so I came up with an alternative: solve the optimization problem using the homogenous form of the curve, which is a B-spline and I can implement.
My problem:
I need to have a mathematical proof that there is or isn't an equivalence between solving the optimization problem in 4D and then using the NURBS definition to go back to 3D, or solving in directly in 3D. I have tried to calculate both norms, but I don't see an equivalence which is of use to me. Here are the functions:
$$ \pmb{Homogenous\ form\ of\ the\ curve}: \mathbf{\overline{c}}(t) = \sum_{i=0}^n B_{i,d}(t) \overline{P}_i, \ where\ \overline{P}_i= \begin{bmatrix} x_i w_i\\ y_i w_i\\ z_i w_i\\ w_i \end{bmatrix} $$
$$ \pmb{Rational\ form\ of\ the\ curve:\ } \mathbf{c}(t)= \frac{\sum_{i=0}^n B_{i,d}(t) w_i P_i}{\sum_{j=0}^n B_{j,d}(t) w_j},\ where P_i = \begin{bmatrix} x_i\\ y_i\\ z_i \end{bmatrix} $$
$$ \pmb{Objective function:\ } \mathbf{P} =arg \min_{P} \int_{t_{min}}^{t_{max}} {\Vert \mathbf{\dot{x}}(t)\Vert}^2 dt,\ with\ t_{min},\ t_{max}\ some\ time\ instants\ and\ \mathbf{x}(t)\ either\ \mathbf{c}(t)\ or\ \overline{\mathbf{c}}(t) $$
$$ \pmb{Constraints:}\ \mathbf{c}(t_k) = \sum_{i=0}^n B_{i,d}(t_k) P_i = w_k\ \forall k = 0\dots N,$$ where $w_k$ some waypoint the curve has to pass through at the time instant $t_k$. $B_{i,d}(t)$ are b-spline basis functions of order d (degree d-1).
My question:
How does one prove or disprove that these two functions have the same global minima, or in other words the optimization problem can be applied on either function, the end result being the same. I don't want an explicit answer as this question relates to my bachelor thesis. Instead, can someone give me some starting point and direct me towards the mathematical field or theories that can help me solve this question?
Thank you!