Intuition and Examples for Lagrange Multipliers (Animated)

We discuss the idea behind Lagrange Multipliers, why they work, as well as why and when they are useful.

External Images Used:

Further Reading:
1. The Variational Principles of Mechanics, Cornelius Lanczos, 2.1-2.6

Timestamps:
0:00 - Introduction
0:18 - Free Optimization in 1 Dimension
1:30 - Constrained Optimization
2:25 - Auxiliary Equations
3:05 - Solving with Substitution
4:25 - Where Substitutions break down
7:17 - Lagrange Multipliers
9:23 - Types of Extrema
12:04 - Lagrange Multiplier Example
13:42 - Final Notes

[poouhaha]

Nice video - I appreciate the 3B1B-esque animations. This video was shared on an oxford university undergraduate physics group chat so you are helping a lot of people!

[HuLi-iota]

Really helps, not only your video did makes me know it works, but why it works

[Kaepsele]

Nice! I've convinced myself before that Lagrangian multipliers work, but only at a symbolic level. The intuition that it just means that the gradients of f and g align was new to me, and gives me a much better understanding of why it works :)

And should I ever forget how to do it, this will allow me to rederive it quickly, which is always useful. So, Thanks! I'll check out your other videos.

[monsieur]

Wow this is a great video! I remember studying this in undergrad (engineering) and when I wanted to write my phd (physics) I decided to optimize numerically because I didn't want to be questioned about the method.

[mauritzwiechmamnn]

Thank you, a big help for my bachelors thesis!

[fabriai]

Very nice video... thanks a lot. I was struggling to wrap my head around plugging L into the Grad resolution. Your explanation is easy to follow and sensible. Brilliant!

[algorithminc.]

Thanks. Great video. I look forward to scoping your other videos. Subscribed. Cheers

[Dhruvbala]

Brilliant insight! I wish you hadn't glossed over the part at 9:40 so quickly -- as that's kind of the crux of the entire video. I appreciate the thought you put into the explanation, though

[youssefabsi]

thank you a lot. wish you had a series solely for optimization

[maxyazhbin]

I like your content, it is education, not entertainment that has a bunch of music

[orandanon]

Nice video.
Few remarks:
1. Lagrange Multipliers Methods is a necessary condition for extrema, but not sufficient. Indeed one can give examples where "just" running the algorithm display yields a wrong solution. One way to solve this problem one needs to show that the domain is compact (i.e., closed and bounded), and then by Weierstrass, one knows that the function has a minimum and a maximum in the domain.
2. In the final notes section the generalization for several constraints requires that g_1, ..., g_k are linearly independent.

[hannahnelson]

Thank you! This helped me understand what the Lagrange equation means!

[tranhoanglong]

This really helped me a lot, thank you for your explanation. I dont know why it does not have more views by now, maybe cause the video is still new. Great work ✨🇻🇳❤

[francoparnetti]

I always wondered how to deduce the Lagrange function. Is there a way to prove (in an "elegant" way) that the function does what it does? Or did Lagrange just say "this just works and thats it"?

[griffinbur]

Is there a mistake or two around 7:30? I follow most of this, but “there’s a constrained optimization in n dimensions which is the same as a free optimization problem in n-1 dimensions” seems to match my understanding more than what’s said. But I might be missing something.

[FFTD]

Is there a reference to the statement made at 7:22?

[aashsyed]

How about maximizing a multivariable function with the constraint x²+y²< 1 ??

[hosz]

nice geometric interpretation of extra dimensions! We live in a 5D space with a constraint x5=0.

[muuubiee]

Eh... I had some slight understanding of Lagrange multipliers (do note though, Gradients are NOT covered in the suggested course literature, not sure what they were thinking), but none of this really made sense.

If you want to explain something you have to explain it on a level lower than the current one. If the person understands concepts like Gradients and Constraints they'll probably not have any problems with understanding Lagrange. Therefore there's no point in assuming that the people watching understands gradients and constraints. Your video is essentially trying to teach Lagrange multipliers to people who already understand Lagrange multipliers (and may be revisiting this to understand a concept that builds on Lagrange multipliers).

[yt-]

@2:57 n-1 degree of freedom or 1 degree of freedom