Lagrange multipliers, using tangency to solve constrained optimization
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- Опубликовано: 12 сен 2024
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The Lagrange multiplier technique is how we take advantage of the observation made in the last video, that the solution to a constrained optimization problem occurs when the contour lines of the function being maximized are tangent to the constraint curve.
Somehow you've managed to compress a 1 hour long lecture into 9 minutes long video with better explanations than my lecturer, thanks a lot! :)
8.42 minutes , not 9
@@BROWNKEY Well, all the better
Aye a 3 yo comment just got replied 2 days ago. Plus he's the man and the legend Grant 3Blue1Brown himself o7
It is the case. Lecturer in my university explain these concepts for 3 hours but still leave us confused
@@BROWNKEY I'd say 8.7 minutes
the new guy for khan academy is so mathematical ... I love his explanations so much they are so deep instead of just giving a set of techniques and methods on how to solve exams he gets in the core of things... that's what we always for in Khan Academy
Hoodar Rock look for his own RUclips channel, 3blue1brown. Amazing explanations and great videos.
@@luffy5246 hii what is the name of that channel?
@@jipuragi6483 3Blue1Brown.
@@astradrian thanks a ton
@@jipuragi6483 bruh
Is that 3blue1brown? OMG!
yeah i think so because he have worked for sal khan (khan academy)..
Oh. I was thinking this voice is so not Khan and somehow very familiar. Then I saw this comment. Interesting to know ;)
his iconic voice
I was about to comment the same thing !
Lol, same thought. I was like, Grant?!?
this is divine. This just cleared my mind up 😭😭 your explanations are so clear and mathematical, yet intuitive! Thanks a lot 😊
4:27 Though his name may sound French, Lagrange was actually Italian. Actually he was born Italian, his birth name beeing Lagrangia, then migrated to France and changed his name.
ok nerd
Grant hits that yeet again. What a boss
Khan Academy has really revolutionized learning. Today we have so many online learning platforms and all of these are in a way off-springs of Khan Academy. Topic wise learning makes the hour long lecture approach of colleges redundant. Most professors at universities are very knowledgable no doubt but not so great educators. To be able to impart the knowledge you hold is an art. Cheers to Khan Academy.
3blue1brown Congratulation, I love the fact you are working with Khan Academy, thats great...
With math it's always the same way: When you don't understand it, it's hell but when you got it, it's pretty cool. :)
Thank you for such a nice explanation!
thats what makes mathematics beautiful
This guy is just maths bae. Best maths channel on RUclips and best Khan Academy videos for maths. what a beast.
This was so well explained that i'd call it a masterpiece.
I´ve just contributed pt-br subtitles, please accept them so that this great material is available to a larger audience!
Thank you very much!! this explanation is life-saving. I'm trying to understand Lagrange duality for support vector machines and I've watched many videos but I'm still stuck. Now I have a better taste of what it is about.
3 blue 1 brown?????
I havent watched the video yet and have no idea what Lagrange multipliers are, but here is how I'd do it:
1=x^2 +y^2
x=sqrt(1-y^2)
f(x,y)=x^2y
f(y)= (1-y^2)y= y - y^3
f'(y)=1- 3y^2 = 0
y = +-sqrt(1/3)
x = +-sqrt(2/3)
f(+ - sqrt(2/3),+ - sqrt(1/3))= + - 2*sqrt(1/3)/3
And I was right. But I understand the need for a more general method to solve these since its not always this easy to express one variable explicitly from another. But this method can serve as a great shortcut.
Lagrange was Italian. I don't know why, but we know him by his French name "Joseph Louis Lagrange" rather than his Italian name: "Giuseppe Luigi Lagrangia".
I thought you were joking but you're not lol, I just looked it up and it looks like he was naturalized French.
It's complicated. Lagrange was born in Piedmont, Italy. However, he later moved to France, and in an unrelated series of events, Piedmont was annexed by France. As a result, he gained French citizenship and French and Italians both claimed him as their own.
As for his parentage, he actually comes from a family that is both French AND Italian, and he spent more of his life in Paris than in Piedmont.
On a plaque that was placed on the Eiffel Tower when it opened he was listed as a "prominent French scientist", but today his place of birth still lies in Italy.
I think if you had asked him whether he was French or Italian he would have either expounded on his indifference to nationalism, or explained that citizenship is more complicated than one's place of birth.
It certainly doesn't seem incorrect for Grant to refer to him as French though.
thank you for doing this. I liked that they are put into small pieces instead of a long lecture.
Hi.. Nice video... Can anyone share which playlist it is part of.. I want to watch the whole course and somehow suggestions that youtube gives for next video is kind of random...
Thank you. I understood the concept quite easily but probably not as completely as I would like. What could happen if the two surfaces have more than a point with the gradients being proportional but not touching each other? it can't happen when using the constraint itself as an equation right? but could the equations touch each other in different points?
The explanation is perfect. I wonder which program do you use to visualize it ? Or anyone know what program is this
While the lagrange method with lambda is great to learn, it is actually a lot less gruel in examples such as these to solve the equations without involving lambda. Take the requirement grad f || grad g and write it as a determinant, det(fx fy; gx gy) == 0 grad f and grad g are parallel; that's one equation and the constraint is another equation -> two equations and two unknowns. :)
I noticed this, but my professor mentioned that there are some equations where lambda plays a role. I'm not sure what they could be though.
I am literally Watching this the day before my final, and this is way better than how my textbook went about this.
Very nicely done! I haven't done anything with math like this for 40+ years, and I was able to follow along very well. Thank you.
Wow, was not expecting to get an explanation from Grant when I clicked on a Khan Academy video. Very cool!
f = lambda*g is super. I learned that in university, but his explanation is really insightful.
Thankyou!! It was tremendously helpful. You are saving lives here.
Why do we assume that the gradients of f and g at a point would have exactly same direction?
I think even though they touch each other at the point, there is no way that the direction of gradient would exactly same??
And never have found the answer yet..
i couldn't find "the next video" . could you please link it somewhere here? thank you :)
for future viewers: there is a playlist called "multivariable calculus" that contains all these lectures. you can find the playlist from the description!
6:26 pullin out that Sal impression
Genius, real mathematics......
Thanks for saving my life, Grant. You are the best.❤
That was SO clear I cannot thank you enough.
You legit just saved my test grade tomorrow. Cheers
This video helped me visualize everything about lagrange multipliers! thank you for posting
(A problem in an Earl W. Swokowski calculus book) "Find the points on the graph of 1/x + 2/y + 3/z = 1 which are closest to the origin." Answer: (a, 2^(1/3)a, 3^(1/3)a), as a = 1 + 2^(2/3) + 3^(2/3), approx. (4.667, 5.881, 6.732). The shortest distance is approx. 10.084. Why is this so; as x=1, y=-2, z=3 is used; which makes the equation equal to 1; and the distance from the origin is sqrt (1^2 + (-2)^2 + 3^2) = sqrt (14) which is approx. 3.742; which is less than 10.084?? Is this problem restricted only to the octant where x, y, and z are all positive??
If you are interested, this was found by Joseph-Louis Lagrange, author of Mécanique analytique matching Newton's Principia in comprehensiveness over mechanics. If you have taken physics and are familiar with Newtonian mechanics, then read "The Lazy Universe" by Jennifer Coopersmith, where she gives an introductory view into the Principle of Stationary Action and Lagrange was key in defining it. Remember: most beautiful and useful mathematics come from understanding nature, and this method you are learning does just that, it maximizes/minimizes some "thing" which is what nature loves to do.
Mate you nailed it, excellent explanation
This video's example makes sense. The problems that pop up on the test are a different story.
3Blue1Brown you are awesome bro, love it! Great teaching, and teaching voice, makes learning simpler, faster, more enjoyable, and the visuals help so much.
Nice video. Which tool do you use to generate the graph from equation ?
thank you! the animation and explanation are awesome, it helps a lot
Beautifully explained.
Finally!! at 4:15 it all makes sense! THANK YOU
I feel so stupide for not having watched these videos when I was strugeling to understand multivariable calculus, but it still feels good to watch them in my free time :)
OMG this guy is pure genious!
I recognize this voice ! I'm pretty sure it's Grant from the 3 Blue 1 Brown channel !!
Excellent explanation, as always !
Excellent geometric intuition!
Question: Consider we have a continuously decreasing function i.e. the value of the function decreases as we move away from the origin in the x-y plane. In such a case, the point that maximizes the function whilst satisfying the constraint won't be at the tanget, right (in the words of the video - where the two curves just kiss each other)?
I believe it will, but only on one side
World-class teaching...
What tool do you use to have an interactive 3d graphics in the presentation?
Gradient is a vector with the partial derivative for x and partial derivative for y
Beautiful explanation
Clear explanation ! Thanks for all your effort!
Im in love with this dude
I love how the visual makes it clear that Lagrange Multipliers are eigenvalues
Very helpful. Thanks for all your videos!
thanks!!! very good and exhaustive explanation
Thank you so much for this epic! Worth watching.
3blue1brown deserves a Nobel Prize in math education
there's no nobel prize for math or education
Why can we set the function to a constant and it is still a function? It should be a single point right?
For example:
x^2 + y = 10
=> x = some value
and y is some value
I'm wondering hard why use a lambda constant to express proportionality, one could have used a determinant. Is it because of simpler computations ? because lambda has a meaning ? or is it purely historical that this approach has been preferred ?
"Lagrange one of those famous french mathematicians...".... Italians getting triggered! :D
Quasnt Hered naturalized French , so French.
I know I did! :-)
Other than that, nice explanation!
He did end up finishing his life in France ;)
Huh maybe some nerds are getting triggered. As an Italian, I feel like we have enough mathematicians and scientists to claim already B-)
@@OfficialAnarchyz Yeah you have enough! Give some to us Austrians xD
Amazing explanation!
very well explained and nice quality. thanks!
Literally have an exam on this in 4 hours :) cheeeers
Ben lol.
hope you made it through bro.
How did it go??
I got 56% haha. P's get degrees right?
Where are you now?
Khan crushing it as usual
"shot ourselves in the foot by giving ourselves a new variable to deal with" :-)
How can I find the first video of this series, please?
This is fantastic. Thank you
Thats nice, but how would we visualize it graphically if it was a minimization problem? So for maximization, it's when both graphs are tangent, what about minimization?
Wow. excellent explanation.
excellent presentation.
if we eliminate y in f(x,y), using the circle equation, and then differentiate f(x,y(x)), won't that work?
If it doesn't ask to maximize (or minimize), how can we know that it indeed maximizes (or minimizes) the given expression?
how are these videos made? like which software is used?
Could you also do this but take the inner product of the gradient of f with g and setting it equal to zero?
@Ar'Khan _ Khizarkhajul No. I think I'll post it on the math stack exchange. They usually answer quick. I'll post a link if they answer. I guess I could try it myself to see if I can get the same result but I'm too lazy
Hi. If we imagine f(x,y) to be such that the contour lines of f(x,y) are lines parallel to the y-axis such that the contour line corresponding to the max f(x,y) is x=0. In that case, would this method apply all the same? g(x,y) and the constraint g(x,y) = 1 is assumed to be the same. Thanks!
Did you get an answer?
I'm struggling with the same question.
Which playlist this video is part of?
What program are you using for those graphs????
i can see why Lagrange Multipliers works here because of tangency. What about if f(x,y)=3-y^2 ... then we know the maximum is on the line y=0 but this contour is NOT tangent to the constraint. (although you do still get the right answer if you apply the method). Why is this? Are there some functions this method won't work for? If so what is the condition?
Often time the light modifier is in the frame or the background is uneven. I wonder how the finela pictures turn out.
Great explanation, thanks for the efforts. For the interpretation(insight) on ∇f(x)=λ∇g(x) where x=[x1,x2,...,xn] is the solution for the extreme, is it because that such extreme only exist when the pulling force of the gradients are proportional to each other because they have the same tangent line? for example, if we expand the size of the circle g(x) in the original example, the original f(x) overlaps with g(x) at points where they have different tangent lines, which implies gradients on different directions on f and g correspondingly, which means that there is a space for improvement for f(x)? Can anyone help?
But here we are lucky because the two curve are tangent, what if it is
not the case? I do not understand how we can generalize this for all
constrained optimizations, though I know it is possible. For instance what if we want to optimize f on
the set x²+(y-1)²=1? Then there are no tangency of the curves f(x,y)=c and
x²+(y-1)²=1but still the langrangian method works. Some argument is missing
here...
What if f got bigger as the contour lines got closer though? Then wouldn't the tangent point be where it is at its minimum?
I don't get the part where two gradients are proportional, i do understand that they will be in same direction, but why they should be proportional to each other.
Little non-mathematical correction: Joseph-Louis Lagrange was Italian. Born in the Italian city of Turin with the name of Giuseppe Luigi Lagrangia and later naturalized as Fench.
Lambda is the same thing as lagrange multiplier. Is lambda just a scalar for vectors?
I think it's just a tool to help us establish a relation in this case. However, you could definitely think about it as a vector multiplier if that makes more sense. After all, we do use the gradient vector to establish the relation.
Which software is that?
Thank you! I don't understand my prof but I can understand this
is this Grand Sandorson?
If only this was posted 2 weeks ago when we had our test on it :(
math1052??
randomdude135 No I'm in high school :(
Daveed 78 dammn. You're doing this in hs??? I'm doing this in university hahaha
randomdude135 I lucked out,my high school does a dual credit with a local college
doing on the 2nd year of university... Lagrange multipliers... MATH251
Is it possible to use f(x, y) = 1 as the third equation instead of the constraint, x² + y² = 1?
edit: would this not give you 1/lambda?
No. The contour line of f tangent to the unit circle is not of value 1. You may be tricked by the fact that the clip shows the contour lines of f extending in the same XY plane as the circle, but in fact they are extending upwards, in the Z direction. What you see are projections in the XY plane.
You saved me for my Micro Econ test
why tf would this be on micro econ
Very helpful! Thank you! Do you have a video on least squares constrained by inequalities? I am struggling with it.
What happens if you have more than one constraint equation?
You have fewer possible points to consider for being a maxima.
Simply beautiful...
I am wandering why the direction of the gradient in the half below of the plan goes in the opposite direction? when you draw the vector gradient for g(x,y)=x^2+y^2 all the directions for the vectors of the gradient were going outward vector? why is that?
Because the function has a local minimum at the origin on the x-y plane. All paths of steepest ascent lead away from this point. Thus, the gradient diverges at this point. The gradient diverges at every point on this particular function of g(x)=x^2+y^2 .
awesome video!
Isn't g(x,y)=x²+y²-1 (which will Lead to the same gradient)
I almost forgot. 3B1B used to work for Khan Academy