How to Solve Optimization Problems with Multiple Variables and Constraints?

[Inertigratus]

Well, I'm having trouble doing optimization problems (maximizing and/or minimizing a function in more then one variable with/without constraints).

Would be a great help if someone could give me some good links on this topic or some methods generally.

If the domain is compact; where are the points that could possibly maximize/minimize the function?
Is it either points that satisfy the equation [itex]\nabla[/itex][itex]f = 0[/itex] and points on the boundary?
In one problem I did, the point that maximized the function didn't satisfy [itex]\nabla[/itex][itex]f = 0[/itex], how come?

How do I examine the boundary? if the domain is defined by an inequality and the equality corresponds to the boundary, do I just solve for either variable and plug into the original equation? What if it's a three variable function?

If the domain isn't compact, and both x and y go from 0 to infinity, what do I do then?

 

[Inertigratus]

For example if I want to find the optima on the boundary of (if they exist): [itex]f(x, y) = (x^2 + y)e^{-x-y}[/itex] and:
[itex]0 \leq x \leq \infty , 0 \leq y \leq \infty[/itex]
I can check when either variable is 0, what else can I do?

 

[HallsofIvy]

Well, I'm having trouble doing optimization problems (maximizing and/or minimizing a function in more then one variable with/without constraints).

Would be a great help if someone could give me some good links on this topic or some methods generally.

If the domain is compact; where are the points that could possibly maximize/minimize the function?
Is it either points that satisfy the equation [itex]\nabla[/itex][itex]f = 0[/itex] and points on the boundary?

Yes, either a point in the interior such that [itex]\nabla f= 0[/itex] or a point on the boundary.

In one problem I did, the point that maximized the function didn't satisfy [itex]\nabla[/itex][itex]f = 0[/itex], how come?

Then it must have been a point on the boundary.

How do I examine the boundary? if the domain is defined by an inequality and the equality corresponds to the boundary, do I just solve for either variable and plug into the original equation? What if it's a three variable function?

If the original domain is n-dimensional, then its boundary is n-1 dimensional. You should be able to write the boundary in terms of n-1 parameters (possibly by solving the equation for the boundary for one of the variables in terms of the remaining n-1 variables). Then solve the n-1 dimensional problem, including looking at its boundary.

If the domain isn't compact, and both x and y go from 0 to infinity, what do I do then?

Then there may not be a max or min. Go ahead and find what local max and min you have, compare to what happens as x and y go to infinity.

For example if I want to find the optima on the boundary of (if they exist): [itex]f(x, y) = (x^2 + y)e^{-x-y}[/itex] and:
[itex]0 \leq x \leq \infty , 0 \leq y \leq \infty[/itex]
I can check when either variable is 0, what else can I do?

Yes, the boundary consists of the lines x= 0 and y= 0. On x= 0, [/itex]f(0, y)= ye^{-y}[itex]. [itex]f'= e^{-y}- ye^{-y}= 0[/itex] when y= 1. Similarly, on y= 0, [itex]f(x, 0)= x^2e^{-x}[/itex]. [itex]f'= 2xe^{-x}- x^2e^{-x}= 0[/itex] when x= 0 or x= 2. Possible max and min are at (0, 1), (0, 0), and (2, 0). To determine if they are global max or min, compare the value of the function at those points with points where [itex]\nabla f= 0[/itex] and the limits as x and y go to infinity.