Can I Optimize X for a Given Level Curve with Partial Derivatives?

[hotvette]

I have a function z=f(x,y) that is reasonably well behaved (single global maximum). I can readily compute the value of z as well as partials of z with respect to x and y. I can also quite easily find the maximum.

The challenge is to find the maximum and minimum values of x where c = constant = f(x,y). In other words, I'm trying to find the extreme values of x for a given level curve. I also know that the extreme values of x occur along a ridge (i.e. where the partial of f with respect to y is zero). In a way, this is a backwards constrained optimization problem, where the value of the function is contrained and the goal is to optimize the values of the variables.

Currently, I'm using a combination of Newton's method and Brent's method to successfully solve the problem, but I'm wondering if there might be a more elegant and mathematically rigorous approach. Any suggestions?

 

[Erland]

Why not just use Lagrange multipliers, optimizing the function g(x,y)=x with the constraint equation f(x,y)-c=0?

 

[hotvette]

Wow, I didn't realize it was really that simple. Turns out the optimization exercise gets me right back to what I already knew (partial f / partial x = 0 and f - c = 0). Two (nonlinear in this case) equations in two unknowns. Straightforward on the surface. Then it comes to a choice of how to solve the two equations. Newton's method reveals a very badly conditioned system that I can't seem to get around. Thus, it appears my original approach may indeed be the most practical. Thanks for helping get the entire picture more clear.