Lagrange multipliers, using tangency to solve constrained optimization

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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.

[@Burneynator]

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! :)

[@BROWNKEY]

8.42 minutes , not 9

[@Burneynator]

@@BROWNKEY Well, all the better

[@rhn122]

Aye a 3 yo comment just got replied 2 days ago. Plus he's the man and the legend Grant 3Blue1Brown himself o7

[@garyjia7703]

It is the case. Lecturer in my university explain these concepts for 3 hours but still leave us confused

[@mintylemon66]

@@BROWNKEY I'd say 8.7 minutes

[@jonaqpetla_]

Is that 3blue1brown? OMG!

[@asadullahfarooqi254]

yeah i think so because he have worked for sal khan (khan academy)..

[@perfumedsea]

Oh. I was thinking this voice is so not Khan and somehow very familiar. Then I saw this comment. Interesting to know ;)

[@jinanlife]

his iconic voice

[@muhammadjoshua7464]

I was about to comment the same thing !

[@BlackRose4MyDeath]

Lol, same thought. I was like, Grant?!?

[@maxbardelang6097]

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.

[@woodychelton5590]

ok nerd

[@Cyrusislikeawsome]

This guy is just maths bae. Best maths channel on KZbin and best Khan Academy videos for maths. what a beast.

[@ednaT1991]

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!

[@missghani8646]

thats what makes mathematics beautiful

[@MrSkizzzy]

This was so well explained that i'd call it a masterpiece.

[@fjgozzi]

I´ve just contributed pt-br subtitles, please accept them so that this great material is available to a larger audience!

[@robertcohn8858]

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.

[@real_john_doe]

This video's example makes sense. The problems that pop up on the test are a different story.

[@gigglification]

Thankyou!! It was tremendously helpful. You are saving lives here.

[@giorgossartzetakis8771]

OMG this guy is pure genious!

[@AmitDotAcademy]

Nice video. Which tool do you use to generate the graph from equation ?

[@swarnavasinharoy7023]

I almost forgot. 3B1B used to work for Khan Academy

[@doctorb9264]

excellent presentation.

[@williamcaldbeck]

This is fantastic. Thank you

[@Revetice]

very well explained and nice quality. thanks!

[@saurabhsingh-ow7ue]

thank you sir...

[@Siam2233]

THANK YOU

[@drennal9516]

why do you use vector and not sum?

[@张梓良-f5e]

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?

[@Hecklit]

Thank you so much :D

[@mustafakartal3403]

thank you

[@TheAryedemented]

Very helpful! Thank you! Do you have a video on least squares constrained by inequalities? I am struggling with it.

[@oyindenyafatarakeme]

God bless you!

[@mithsaradasanayake3211]

Finds 3blue1brown Surprise to be sure but a welcome one

[@cristianomarinelli2551]

Lagrange is not French, he was italian, from Turin

[@coolvibrations8817]

wait a minute this isnt ZZ Top

[@seahyx120]

Wait... isn't this the voice of Grant from 3blue1brown?

[@bladerunner924]

yes, he is!

[@louisvanhunsel1932]

sheeeesh this some good math sht

[@70ME3E]

Lagrange pops up a lot. not enough for you to pronounce his name right though

[@joshuaflackua]

What? He has an American accent, but he definitely pronounces it correctly. That's exactly how we say it in France. See "Château Lagrange"

[@gabrieledragotto4224]

Lagrange is from Torino, Italy -- not France!

[@joshuaflackua]

He didn't say Lagrange was from France. He said he was French. 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.

[@aashsyed1277]

3 blue 1 brown?????

[@ruralmetropolitan]

"Lagrange one of those famous french mathematicians...".... Italians getting triggered! :D

[@GreyEyedAthena]

Quasnt Hered naturalized French , so French.

[@ihbarddx]

I know I did! :-) Other than that, nice explanation!

[@philippelaferriere2661]

He did end up finishing his life in France ;)

[@OfficialAnarchyz]

Huh maybe some nerds are getting triggered. As an Italian, I feel like we have enough mathematicians and scientists to claim already B-)

[@Labroidas]

@@OfficialAnarchyz Yeah you have enough! Give some to us Austrians xD

[@hoodarrock2453]

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

[@themax1234521]

Hoodar Rock look for his own KZbin channel, 3blue1brown. Amazing explanations and great videos.

[@jipuragi6483]

@@luffy5246 hii what is the name of that channel?

[@astradrian]

@@jipuragi6483 3Blue1Brown.

[@jipuragi6483]

@@astradrian thanks a ton

[@bofa722]

​@@jipuragi6483 bruh

[@spencertaylor6910]

Grant hits that yeet again. What a boss

[@rfolks92]

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".

[@liammckenna1479]

I thought you were joking but you're not lol, I just looked it up and it looks like he was naturalized French.

[@joshuaflackua]

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.

[@hellelo.5840]

3blue1brown Congratulation, I love the fact you are working with Khan Academy, thats great...

[@eliasminkim]

this is divine. This just cleared my mind up 😭😭 your explanations are so clear and mathematical, yet intuitive! Thanks a lot 😊

[@ikhwanjeon7370]

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..

[@benisbuff]

Literally have an exam on this in 4 hours :) cheeeers

[@ibrahimalkhorsani2533]

Ben lol. hope you made it through bro.

[@T33-q9c]

How did it go??

[@benisbuff]

I got 56% haha. P's get degrees right?

[@FsimulatorX]

Where are you now?

[@luciafresnopm]

i couldn't find "the next video" . could you please link it somewhere here? thank you :)

[@ahmednesartahsinchoudhury2628]

for future viewers: there is a playlist called "multivariable calculus" that contains all these lectures. you can find the playlist from the description!

[@jigneshrathod3714]

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...

[@kwameoduro1241]

why do you have to get yourself confused with lagrange and laplace?Ha! they are not the same.

[@turbopotato4575]

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

[@turbopotato4575]

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.

[@sarfarazmemon2429]

"shot ourselves in the foot by giving ourselves a new variable to deal with" :-)

[@Lets_MakeItSimple]

This lecture is created by our own Strand from 2 blue 1 brown

[@aloo_explains]

Does he sound like 3BLUE1BROWN?.....@3Blue1Brown

[@julesthomas3335]

Lagraaaaangian ! Suis-je le seul français ici ?

[@semiawesomatic6064]

Is this... This sounds like... Oh god.

[@muhammadjoshua7464]

And his style also

[@studyselection2881]

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

[@adrianpabloalvarez2523]

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?

[@ambresipahimalani4198]

but what if the maximum is "in the circle", like a montain that would have its summit above the center of the circle, the two curves would'nt be tangent, would they ?

[@mertbeser9837]

The explanation is perfect. I wonder which program do you use to visualize it ? Or anyone know what program is this

[@ravipratapmishra7013]

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.

[@Wayk123]

Fun fact lagrange developed this method when he was 19 years old

[@firefoxyouth]

Fun fact two: I played with Lego back then

[@anonymousreviewer3816]

Ofcourse he did, why wouldn't he! (-_-) _Talk about setting frigging high expectations_

[@divyamanify]

Is this the voice of the 3blue1brown guy? He used to work in Khan academy!

[@velevki]

Instead of this method, can't we simply use the equations, make them equal, and solve for the tangent point?

[@poiuwnwang7109]

f = lambda*g is super. I learned that in university, but his explanation is really insightful.

[@MoguMogu818]

Hi 3b1b

[@andreasstolten9179]

Often time the light modifier is in the frame or the background is uneven. I wonder how the finela pictures turn out.

[@nahblue]

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. :)

[@joshuaflackua]

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.

[@praneelmadhuvanesh3770]

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?

[@michaelc.4321]

I’m sure that this would make way less sense to me if I didn’t learn about electromagnetic fields

[@محمدالشهري-ظ2ك]

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?

[@carultch]

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 .

[@franciscorivas4036]

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.

[@HakanTheUltimateHoca]

Voice crack at 7:32

[@justadude8716]

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.

[@focker0000]

wow your voice is so familiar to me, and i just realize that it's you 3blue1brown!!!!

[@ND-kl8lo]

3Blue1Brown you are awesome bro, love it! Great teaching, and teaching voice, makes learning simpler, faster, more enjoyable, and the visuals help so much.

[@yavarjn2055]

What tool do you use to have an interactive 3d graphics in the presentation?

[@jadedjimmy]

6:26 pullin out that Sal impression

[@blopotchok]

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...

[@safooraranjbaran1466]

How can I find the first video of this series, please?

[@laraeldabet6299]

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?

[@jeatig]

(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??

[@arpitbahety5643]

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)?

[@bendaniels7346]

I believe it will, but only on one side

[@bfedkjwerfegregfrerg]

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.

[@richardfredlund3802]

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?

[@AngeloArrifano]

I recognize this voice ! I'm pretty sure it's Grant from the 3 Blue 1 Brown channel !! Excellent explanation, as always !

[@gajrajsingh51]

how are these videos made? like which software is used?

[@queenstrategy904]

Lambda is the same thing as lagrange multiplier. Is lambda just a scalar for vectors?

[@joshuaflackua]

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.

[@prejudice8901]

I have an exam in 5.5 hours..😂

[@matlabmalayalam3288]

World-class teaching...

[@mantacid1221]

I am literally Watching this the day before my final, and this is way better than how my textbook went about this.

[@reshovroy7799]

could someone tell me which playlist this is?

[@soumitrajoy7660]

Just love, sanderson the 3b1b.

[@learningindia6733]

Genius, real mathematics......

[@arsilvyfish11]

Is this the voice of Grant or Sal?

[@pacchutubu]

if we eliminate y in f(x,y), using the circle equation, and then differentiate f(x,y(x)), won't that work?

[@ahnafinqiyadarko6841]

Which playlist this video is part of?

[@kylewolfe_]

Wow, was not expecting to get an explanation from Grant when I clicked on a Khan Academy video. Very cool!

[@leosin5767]

3blue1brown deserves a Nobel Prize in math education

[@skrgrnd]

there's no nobel prize for math or education

[@PedroLucas-fs8mc]

what if my function to, say, maximize is x^3 - x and my variety is -2

[@ribhav99]

Grant sanders I love you

[@alokj84]

i can recognize Grant's voice here

[@denizyuksel2309]

Am I the only one thinking about ZZ Top's La Grange?

[@Zeqweery]

Deniz Yüksel ahaw haw

[@BKAM78]

where is the next video please??!!

[@arslanhojiyev5996]

If it doesn't ask to maximize (or minimize), how can we know that it indeed maximizes (or minimizes) the given expression?

[@AshutoshPatidarda24s006]

is this Grand Sandorson?

[@randomforrest9251]

Isn't g(x,y)=x²+y²-1 (which will Lead to the same gradient)