Lagrange multipliers, using tangency to solve constrained optimization

Рет қаралды 688,788

7 жыл бұрын

Courses on Khan Academy are always 100% free. Start practicing-and saving your progress-now: www.khanacademy.org/math/mult...
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.

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@Burneynator 7 жыл бұрын

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 3 жыл бұрын

8.42 minutes , not 9

@Burneynator 3 жыл бұрын

@@BROWNKEY Well, all the better

@rhn122 3 жыл бұрын

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

@garyjia7703 3 жыл бұрын

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

@mintylemon66 2 ай бұрын

@@BROWNKEY I'd say 8.7 minutes

@jonaqpetla_ 7 жыл бұрын

Is that 3blue1brown? OMG!

@asadullahfarooqi254 6 жыл бұрын

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

@perfumedsea 5 жыл бұрын

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

@jinanlife 5 жыл бұрын

his iconic voice

@muhammadjoshua7464 5 жыл бұрын

I was about to comment the same thing !

@BlackRose4MyDeath 5 жыл бұрын

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

@hoodarrock2453 7 жыл бұрын

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 7 жыл бұрын

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

@jipuragi6483 Жыл бұрын

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

@astradrian Жыл бұрын

@@jipuragi6483 3Blue1Brown.

@jipuragi6483 Жыл бұрын

@@astradrian thanks a ton

@bofa722 Жыл бұрын

@@jipuragi6483 bruh

@ca94305 2 жыл бұрын

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

@saahilnayyer6865 2 жыл бұрын

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.

@MrSkizzzy 6 жыл бұрын

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

@spencertaylor6910 5 жыл бұрын

Grant hits that yeet again. What a boss

@Cyrusislikeawsome 7 жыл бұрын

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

@tinayang7351 6 жыл бұрын

thank you for doing this. I liked that they are put into small pieces instead of a long lecture.

@hellelo.5840 6 жыл бұрын

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

@ednaT1991 5 жыл бұрын

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 3 жыл бұрын

thats what makes mathematics beautiful

@rfolks92 5 жыл бұрын

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 4 жыл бұрын

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

@joshuaflackua 2 жыл бұрын

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.

@franciscorivas4036 4 жыл бұрын

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.

@diannebanal1650 7 жыл бұрын

This video helped me visualize everything about lagrange multipliers! thank you for posting

@robertcohn8858 4 жыл бұрын

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.

@Johncowk 4 жыл бұрын

That was SO clear I cannot thank you enough.

@kylewolfe_ Жыл бұрын

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

@gigglification 5 жыл бұрын

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

@fjgozzi 4 жыл бұрын

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

@charliethatcher404 6 жыл бұрын

You legit just saved my test grade tomorrow. Cheers

@mantacid1221 26 күн бұрын

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

@NicolasSchmidMusic 3 жыл бұрын

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 :)

@foadabodahood9509 6 жыл бұрын

Finally!! at 4:15 it all makes sense! THANK YOU

@dennishuang3498 4 жыл бұрын

Clear explanation ! Thanks for all your effort!

@sdpto 7 ай бұрын

Thanks for saving my life, Grant. You are the best.❤

@learningindia6733 Жыл бұрын

Genius, real mathematics......

@maxbardelang6097 2 жыл бұрын

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 9 ай бұрын

ok nerd

@AAA-uv1ny 9 ай бұрын

thank you! the animation and explanation are awesome, it helps a lot

@alias40anon 6 жыл бұрын

Mate you nailed it, excellent explanation

@jadedjimmy 5 жыл бұрын

6:26 pullin out that Sal impression

@Revetice 7 жыл бұрын

very well explained and nice quality. thanks!

@real_john_doe 4 жыл бұрын

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

@adrianpabloalvarez2523 Жыл бұрын

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?

@turbopotato4575 7 жыл бұрын

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 7 жыл бұрын

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.

@GOPALS1967 4 жыл бұрын

Beautifully explained.

@christopherandrewmartin494 2 жыл бұрын

Very helpful. Thanks for all your videos!

@goclbert 2 жыл бұрын

I love how the visual makes it clear that Lagrange Multipliers are eigenvalues

@poiuwnwang7109 4 жыл бұрын

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

@guillermo._._ 3 жыл бұрын

Excellent geometric intuition!

@jigneshrathod3714 6 жыл бұрын

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...

@nahblue 2 жыл бұрын

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 2 жыл бұрын

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.

@giorgossartzetakis8771 3 жыл бұрын

OMG this guy is pure genious!

@scoffer2150 9 ай бұрын

Thank you so much for this epic! Worth watching.

@mertbeser9837 2 жыл бұрын

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

@danawen555 3 жыл бұрын

thanks!!! very good and exhaustive explanation

@ND-kl8lo 4 ай бұрын

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

@williamcaldbeck 4 жыл бұрын

This is fantastic. Thank you

@tsrevo1 6 жыл бұрын

Wow. excellent explanation.

@samuelvaldezgil 2 жыл бұрын

Im in love with this dude

@AngeloArrifano 2 жыл бұрын

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

@speedracer1702 2 жыл бұрын

Amazing explanation!

@thatsharma1066 Ай бұрын

The explanation was so much intuitive and awesome. is this guy grant sanderson from 3b1b?

@matlabmalayalam3288 3 жыл бұрын

World-class teaching...

@mermaid6380 5 жыл бұрын

Thank you! I don't understand my prof but I can understand this

@yavarjn2055 3 жыл бұрын

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

@doctorb9264 3 жыл бұрын

excellent presentation.

@ddos87 4 жыл бұрын

Khan crushing it as usual

@queenstrategy904 3 жыл бұрын

Gradient is a vector with the partial derivative for x and partial derivative for y

@ruralmetropolitan 7 жыл бұрын

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

@GreyEyedAthena 7 жыл бұрын

Quasnt Hered naturalized French , so French.

@ihbarddx 6 жыл бұрын

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

@philippelaferriere2661 4 жыл бұрын

He did end up finishing his life in France ;)

@OfficialAnarchyz 4 жыл бұрын

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

@Labroidas 3 жыл бұрын

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

@justadude8716 Жыл бұрын

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.

@benisbuff 7 жыл бұрын

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

@ibrahimalkhorsani2533 7 жыл бұрын

Ben lol. hope you made it through bro.

@user-vb4eq4vx1q 7 жыл бұрын

How did it go??

@benisbuff 7 жыл бұрын

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

@FsimulatorX 6 жыл бұрын

Where are you now?

@user-zk4bt7oy8n Жыл бұрын

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?

@ikhwanjeon7370 3 жыл бұрын

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..

@PedroLucas-fs8mc 7 жыл бұрын

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

@AmitDotAcademy 2 ай бұрын

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

@sarfarazmemon2429 6 жыл бұрын

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

@TheAryedemented 4 жыл бұрын

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

@jeatig 6 жыл бұрын

(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??

@ericbischoff9444 6 жыл бұрын

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 ?

@arpitbahety5643 2 жыл бұрын

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 2 жыл бұрын

I believe it will, but only on one side

@Siam2233 5 жыл бұрын

THANK YOU

@karthik-ex4dm 5 жыл бұрын

awesome video!

@focker0000 5 жыл бұрын

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

@devkunjadia3792 9 ай бұрын

awesome video

@saurabhsingh-ow7ue 4 жыл бұрын

thank you sir...

@chandreshmaurya8102 3 жыл бұрын

Anyone knows which software is used to draw the counter and gradients?

@DennyMapleSyrup 7 жыл бұрын

If only this was posted 2 weeks ago when we had our test on it :(

@randomdude135 7 жыл бұрын

math1052??

@DennyMapleSyrup 7 жыл бұрын

randomdude135 No I'm in high school :(

@randomdude135 7 жыл бұрын

Daveed 78 dammn. You're doing this in hs??? I'm doing this in university hahaha

@DennyMapleSyrup 7 жыл бұрын

randomdude135 I lucked out,my high school does a dual credit with a local college

@amidg4x4 7 жыл бұрын

doing on the 2nd year of university... Lagrange multipliers... MATH251

@studyselection2881 Жыл бұрын

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

@worldrecordegg5638 6 жыл бұрын

it's really great

@andreasstolten9179 11 ай бұрын

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

@Hecklit 6 жыл бұрын

Thank you so much :D

@mustafakartal3403 7 жыл бұрын

thank you

@luciafresno3179 3 жыл бұрын

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

@ahmednesartahsinchoudhury2628 26 күн бұрын

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

@shahabansari5201 3 жыл бұрын

Simply beautiful...

@arslanhojiyev5996 3 жыл бұрын

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

@laraeldabet6299 3 жыл бұрын

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?

@Shadowfax2 5 жыл бұрын

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!

@apoorvmishra6992 2 жыл бұрын

Did you get an answer? I'm struggling with the same question.

@carlossal04 3 жыл бұрын

What program are you using for those graphs????

@bfedkjwerfegregfrerg 2 жыл бұрын

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.

@user-iv9sz8dx1g 2 жыл бұрын

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 Жыл бұрын

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 .

@swarnavasinharoy7023 2 жыл бұрын

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

@ahnafinqiyadarko6841 3 жыл бұрын

Which playlist this video is part of?

@safooraranjbaran1466 2 жыл бұрын

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

@pacchutubu 3 жыл бұрын

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

@praneelmadhuvanesh3770 9 ай бұрын

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?

@jacobhm7429 5 жыл бұрын

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?

@valeriuok 4 жыл бұрын

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.

@zixiaoxu977 6 жыл бұрын

superb

@michaelc.4321 5 жыл бұрын

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

@thebullybuffalo 5 жыл бұрын

Could you also do this but take the inner product of the gradient of f with g and setting it equal to zero?

@thebullybuffalo 4 жыл бұрын

@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