I have a 3D (multivariate) function f(x,y) which can be represented as a surface with constraints.
When the surface is viewed from the side (as below), such that the Y axis is not visible, there is a line A to B, which is a 2D function of x, which represents the minimum values of f(x,y) across the full range of x and y within the constraints.
For this simple example it is possible to determine the equation of this 2D function through inspection of the second chart, but I am looking for a more elegant method, such as a derivation or transform, which will allow me to arrive at the 2D equation through mathematical operations rather than methods such as curve fitting. Does such a method exist?
What you are looking for is the function
$$p(x) = \min_y f(x,y)$$
Assuming there are only minima and no maxima, we can simply say that
$$p(x) = f\left(x,\left( \frac{\partial}{\partial y} f(x,y) = 0 \right)\right)$$
which evaluates to an expression that is only dependent on $x$.