Lagrange Multipliers: Advantages & Necessity?

In summary, the conversation discusses the use of Lagrange multipliers in solving minimization/maximization problems, specifically in finding the shortest distance from a point to a plane. It is mentioned that this can be done without using Lagrange multipliers by setting partial derivatives equal to zero and substituting for z. However, it is noted that using Lagrange multipliers is the only way for constrained maximum/minimum. The conversation also touches on when it is necessary to use Lagrange multipliers and how to determine when it is better to use them. Overall, the main takeaway is that the steps required by Lagrange multipliers can be done without calling it by its name.
  • #1
DR13
150
0
I do not have one specific question that needs answering. Rather, it is about Lagrange multipliers in general.

So for certain minimization/maximization questions (ie find the shortest distance from some point to some plane) it seems that one could solve the question using lagrange multipliers and not using lagrange multipliers (I have done it both ways). One could set both partials equal to 0 and solve for x and y without using lagrange (as long as a function of x and y is substitued for z). Or, one could not substitute for z and use lagrange multipliers to find the distange.

Is this right? Or am I missing something? Also, is it ever 100% necessary to use lagrange multipliers? How can one tell when it is better to use lagrange multipliers and when it is better not to?

Thanks
DR13
 
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  • #2
Ok, you do the same steps required by Lagrange multipliers, but you don't call it Lagrange multipliers. :)

Using Lagrange is the only way for constrained maximum/minimum.
 
  • #3
Oh I get it. By substituting for z before taking the partials you are just doing the step earlier rather than later. Is that right?
 
  • #4
I'm not sure I follow you completely.
Do you have an example ?
 
  • #5
Its fine. I worked it out myself and get it now. Thanks!
 

Related to Lagrange Multipliers: Advantages & Necessity?

What are Lagrange multipliers and how are they used in optimization?

Lagrange multipliers are a mathematical tool used in optimization problems to find the maximum or minimum value of a function subject to constraints. They involve creating a new function called the Lagrangian, which incorporates the original function to be optimized and the constraints, and then finding the critical points of this function.

What are the advantages of using Lagrange multipliers?

One of the main advantages of using Lagrange multipliers is that they allow for the optimization of a function subject to multiple constraints at the same time. This can be more efficient than solving each constraint separately. Additionally, Lagrange multipliers provide a systematic approach to solving optimization problems, making it easier to find an optimal solution.

When is it necessary to use Lagrange multipliers?

Lagrange multipliers are necessary when solving optimization problems with constraints that cannot be easily expressed or solved using other methods. They are particularly useful when the constraints are non-linear or when the optimization problem involves multiple variables.

Can Lagrange multipliers be used for both unconstrained and constrained optimization?

Yes, Lagrange multipliers can be used for both unconstrained and constrained optimization problems. In the case of unconstrained optimization, the Lagrange multiplier is set to zero and the original function is optimized. In constrained optimization, the Lagrange multiplier is used to incorporate the constraints into the optimization process.

Are there any limitations or drawbacks to using Lagrange multipliers?

One limitation of using Lagrange multipliers is that they may not always provide the global optimum solution, but rather a local optimum solution. Additionally, the Lagrangian function can become very complicated for problems with many constraints, making it difficult to solve analytically. In these cases, numerical methods may be necessary.

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