What are the key concepts of Lagrange Multipliers?

In summary, the speaker has just started multi-dimensional calculus and is specifically studying Langrange's multipliers. They have some questions to help them understand the concepts better and are looking for an explanation on the graph of the function rather than its practical usage. They also mention being shaky on the concept of differentiating a plane and are looking for alternative explanations. They have already looked at one resource but are still seeking additional information.
  • #1
mtanti
172
0
I've just started multi dimensional calculus, among which Langrange's Multipliers. I have some questions which will help me grasp the concepts since I'm a very curious guy...

a) What are you finding exactly with this technique?
b) What is the constraint?
c) What does the extra variable represent?
d) Is there another more intuitive but less effiecient way to obtain the same result?

My lecturer is a very poor teacher and just wants to get on with the syllabus I guess... I would like an explanation on the actual graph of the function and not about the practical usage of it such as in economics...

P.S. I understood the reasoning behind partial differentiation but I'm still shakey on the fact that to differentiate a plane you can only find a general gradient for each axis (y and x) and not a single general gradient function f'(x,y).

Thanks!
 
Mathematics news on Phys.org
  • #3
Already been there... I was looking for something different...
 

1. What are Lagrange Multipliers and how are they used in optimization?

Lagrange multipliers are a mathematical technique used in optimization problems to find the maximum or minimum value of a function subject to certain constraints. They involve creating a new function, known as the Lagrangian, which combines the original objective function with the constraints using a set of variables called the multipliers. By setting the partial derivatives of the Lagrangian equal to zero, the optimal values of the variables can be found, providing the optimal solution to the problem.

2. Can Lagrange Multipliers be used for both constrained and unconstrained optimization problems?

Yes, Lagrange multipliers can be used for both constrained and unconstrained optimization problems. In unconstrained problems, there are no constraints to consider and the Lagrangian will simply be the objective function. In constrained problems, the Lagrangian will include the constraints and the multipliers will be used to find the optimal values of the variables subject to those constraints.

3. What is the role of the multipliers in Lagrange Multipliers?

The multipliers in Lagrange multipliers play a crucial role in finding the optimal solution to an optimization problem. They are used to incorporate the constraints into the objective function and to ensure that the optimal solution satisfies all of the constraints. By setting the partial derivatives of the Lagrangian with respect to the multipliers equal to zero, the optimal values of the multipliers can be found, which in turn provide the optimal values of the variables.

4. How do Lagrange Multipliers differ from other optimization techniques?

Lagrange multipliers differ from other optimization techniques in that they can handle both equality and inequality constraints, while many other techniques can only handle one or the other. Additionally, Lagrange multipliers can be used for both single and multi-objective optimization problems, making them a versatile tool in the field of optimization.

5. Are there any limitations or drawbacks to using Lagrange Multipliers?

One limitation of using Lagrange multipliers is that they can only find local extrema, meaning they may not always find the global optimum. Additionally, the method can become computationally expensive for problems with a large number of constraints, as each constraint requires its own multiplier. In some cases, other optimization techniques may be more efficient and effective in finding the optimal solution.

Similar threads

Lagrange multipliers understanding
    • Calculus and Beyond Homework Help
Replies
8
Views
470
B Constrained Optimization with the KKT Approach
    • General Math
Replies
7
Views
1K
MHB Holly's questions at Yahoo Answers regarding Lagrange multipliers
    • General Math
Replies
1
Views
3K
Can't get Lagrange multiplier to work in a single exercise
    • Calculus and Beyond Homework Help
Replies
16
Views
2K
How to model a function of a box's volume using Lagrange multiplier methods
    • Calculus and Beyond Homework Help
Replies
1
Views
818
Constraint force using Lagrangian Multipliers
    • Advanced Physics Homework Help
Replies
2
Views
824
I Lagrange multipliers and critical points
    • Calculus
Replies
7
Views
2K
Lagrange Multipliers in Calculus of Variations
    • General Math
Replies
4
Views
3K
MHB What are the steps for solving a problem using Lagrange Multipliers?
    • Calculus
Replies
9
Views
2K
MHB Finding Extrema with Lagrange Multipliers: A Step-by-Step Guide
    • Calculus
Replies
9
Views
2K

Hot Threads

Recent Insights

Back
Top