Multivariable Optimization - Closest point on surface

In summary, the student was trying to solve a problem involving derivatives, but was having difficulty because they were trying to do it in terms of variables that were in terms of other variables. After some confusion and self-doubt, the student finally got it and solved the problem.
  • #1
theWapiti
15
1

Homework Statement



Find the first-octant point P(x,y,z) on the surface closest to the given fixed point Q (0,0,0).
The surface x2y2z=4

Homework Equations



gif.latex?d%3D%5Csqrt%7Bx%5E2%2By%5E2%2B(%5Cfrac%7B4%7D%7Bx%5E2y%5E2%7D)%5E2%7D.gif
is the distance along PQ.

EDIT:

2By%5E2%2B(%5Cfrac%7B4%7D%7Bx%5E2y%5E2%7D)%5E2%3Dx%5E2%2By%5E2%2B%5Cfrac%7B16%7D%7Bx%5E4y%5E4%7D.gif


The Attempt at a Solution



gif.latex?z%3D%5Cfrac%7B4%7D%7Bx%5E2y%5E2%7D.gif


frac%7B64%7D%7Bx%5E5y%5E4%7D%3D0%5C%5C%0A%5C%5Cx%3D%5Cfrac%7B2%7D%7By%5E%5Cfrac%7B4%7D%7B6%7D%7D.gif


I get stuck here every time. I feel like I'm just selling myself short here, but I don't know how to resolve the situation for when the critical point has a variable in it.
 
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  • #2
theWapiti said:

Homework Statement



Find the first-octant point P(x,y,z) on the surface closest to the given fixed point Q (0,0,0).
The surface x2y2z=4

Homework Equations



gif.latex?d%3D%5Csqrt%7Bx%5E2%2By%5E2%2B(%5Cfrac%7B4%7D%7Bx%5E2y%5E2%7D)%5E2%7D.gif
is the distance along PQ.

The Attempt at a Solution



gif.latex?z%3D%5Cfrac%7B4%7D%7Bx%5E2y%5E2%7D.gif


frac%7B64%7D%7Bx%5E5y%5E4%7D%3D0%5C%5C%0A%5C%5Cx%3D%5Cfrac%7B2%7D%7By%5E%5Cfrac%7B4%7D%7B6%7D%7D.gif


I get stuck here every time. I feel like I'm just selling myself short here, but I don't know how to resolve the situation for when the critical point has a variable in it.

Your derivative is incorrect. Try again, more slowly this time.

When you've done that, then take the partial with respect to y (this should be easy because the problem is symmetric, so you just need to swap the x and y variables in your partial with respect to x) and set that equal to 0 as well. You then have two equations in two unknowns.
 
  • #3
Mentallic said:
Your derivative is incorrect. Try again, more slowly this time.

When you've done that, then take the partial with respect to y (this should be easy because the problem is symmetric, so you just need to swap the x and y variables in your partial with respect to x) and set that equal to 0 as well. You then have two equations in two unknowns.

I feel like I'm probably just making myself problems here, but I can't for the life of me get past this still.

I was minimizing for the distance squared, which gave the two partials. But how can I solve if I have y in terms of x and x in terms of y?!
 
  • #4
Well I feel like a dummy. Got it now, obviously. Sometimes the problem is made so much more difficult in your own mind!
 

FAQ: Multivariable Optimization - Closest point on surface

1. What is multivariable optimization?

Multivariable optimization is a mathematical process used to find the optimal solution for a problem that involves multiple variables. It involves finding the values of the variables that maximize or minimize a given objective function.

2. How is multivariable optimization used in finding the closest point on a surface?

In the context of finding the closest point on a surface, multivariable optimization can be used to determine the coordinates of the point that minimize the distance between the given point and the surface. This involves defining an objective function that represents the distance between the point and the surface, and then using multivariable optimization techniques to find the values of the variables that minimize this function.

3. What are some applications of multivariable optimization in finding the closest point on a surface?

Multivariable optimization can be used in a variety of fields such as computer graphics, computer vision, and robotics. It is commonly used in 3D modeling and animation to determine the closest point on a surface for collision detection, surface reconstruction, and other tasks.

4. What are some common multivariable optimization techniques used for finding the closest point on a surface?

Some common techniques used for multivariable optimization in this context include gradient descent, Newton's method, and the Levenberg-Marquardt algorithm. These methods involve iteratively updating the values of the variables until an optimal solution is reached.

5. What are the limitations of using multivariable optimization for finding the closest point on a surface?

One limitation is that the optimal solution found by multivariable optimization may not always be the global minimum or maximum, but rather a local one. Additionally, the complexity of the objective function and the number of variables can greatly affect the computational time required to find the optimal solution.

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