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Odd Optimization Problem

  1. Oct 8, 2013 #1

    hotvette

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    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?
     
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  3. Oct 8, 2013 #2

    Erland

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    Why not just use Lagrange multipliers, optimizing the function g(x,y)=x with the constraint equation f(x,y)-c=0?
     
  4. Oct 9, 2013 #3

    hotvette

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    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.
     
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