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

In summary, the challenge is to find the maximum and minimum values of x for a given level curve, with the constraint equation f(x,y)-c=0. The extreme values of x occur along a ridge where the partial of f with respect to y is zero. The current method uses a combination of Newton's method and Brent's method, but the speaker is wondering if there is a more elegant and mathematically rigorous approach. The suggestion is to use Lagrange multipliers, optimizing the function g(x,y)=x with the constraint equation f(x,y)-c=0. However, after further discussion, it is determined that the original approach of solving two (nonlinear) equations in two unknowns may be the most practical.
  • #1
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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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  • #2
Why not just use Lagrange multipliers, optimizing the function g(x,y)=x with the constraint equation f(x,y)-c=0?
 
  • #3
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.
 

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

What is an Odd Optimization Problem?

An Odd Optimization Problem is a type of problem in which the goal is to find the minimum or maximum value of a function, subject to certain constraints, but the function itself is odd. This means that the function has symmetry about the origin, and therefore, the minimum and maximum values occur at the same point.

What are some common examples of Odd Optimization Problems?

Some common examples of Odd Optimization Problems include finding the minimum or maximum value of a polynomial function, such as a parabola, or finding the shortest path between two points on a graph with odd symmetry.

How do you solve an Odd Optimization Problem?

To solve an Odd Optimization Problem, you must first identify the function and any constraints that it is subject to. Then, you can use mathematical techniques such as calculus or linear programming to find the minimum or maximum value of the function. It is important to also check for symmetry and consider any special cases that may arise.

What are the applications of Odd Optimization Problems?

Odd Optimization Problems have various applications in fields such as engineering, economics, and physics. For example, finding the optimal design for a bridge or maximizing profits for a company can both be framed as Odd Optimization Problems.

What are some challenges of solving Odd Optimization Problems?

One challenge of solving Odd Optimization Problems is that they can be more complex and difficult to solve compared to other types of optimization problems. It is also important to carefully consider the constraints and any special cases, as they can greatly impact the solution. Additionally, some functions may not have a global minimum or maximum, making it challenging to determine the optimal solution.

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