Constrained Extrema and Lagrange Multipliers

In summary, Lagrange multipliers are a mathematical tool used to find the extreme values of a function subject to one or more constraints. They work by setting up a system of equations, known as the Lagrange equations, and solving for the values of the variables that yield the optimal solution. Constraints in constrained extrema problems are conditions that the variables must satisfy, and Lagrange multipliers can be used in higher dimensions as well. However, they have limitations such as only being applicable to continuous functions and potentially not finding all extreme values.
  • #1
throneoo
126
2
Suppose I have a function f(x,y) I would like to optimize, subject to constraint g(x,y)=0.

Let H=f+λg,

The extrema occurs at (x,y) which satisfy
Hy=0
Hx=0
g(x,y)=0

Suppose the solutions are (a,b) and (c,d).

If f(a,b)=f(c,d) , how do I determine whether they are maxima or minima?
 
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  • #2
One way is brute force. Evaluate the function slightly away from the test point.
 

Related to Constrained Extrema and Lagrange Multipliers

1. What is the purpose of Lagrange multipliers in constrained extrema problems?

Lagrange multipliers are used to find the extreme values of a function subject to one or more constraints. They allow us to incorporate the constraints into our optimization problem and find the optimal solution.

2. How do Lagrange multipliers work?

In order to find the extreme values of a function subject to a constraint, we set up a system of equations known as the Lagrange equations. The solution to this system of equations gives us the values of the variables that will yield the extreme values of the function under the given constraint.

3. What are the constraints in constrained extrema problems?

Constraints in constrained extrema problems are conditions that the variables must satisfy in order to be considered a valid solution. These can be equations, inequalities, or other mathematical expressions.

4. Can Lagrange multipliers be used in higher dimensions?

Yes, Lagrange multipliers can be used to find constrained extrema in higher dimensions. The same principles and equations apply, but the system of equations will have more variables and equations to solve.

5. Are there any limitations to using Lagrange multipliers?

One limitation to using Lagrange multipliers is that they can only be used for finding constrained extrema of continuous functions. Additionally, they may not be able to find all the extreme values of a function, as there may be multiple local extrema or no extrema at all.

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