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.

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14 ноя 2016

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Комментарии : 241

@Burneynator 7 лет назад

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 года назад

8.42 minutes , not 9

@Burneynator 3 года назад

@@BROWNKEY Well, all the better

@rhn122 3 года назад

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 года назад

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

@mintylemon66 3 месяца назад

@@BROWNKEY I'd say 8.7 minutes

@jonaqpetla_ 7 лет назад

Is that 3blue1brown? OMG!

@asadullahfarooqi254 6 лет назад

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

@perfumedsea 5 лет назад

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

@jinanlife 5 лет назад

his iconic voice

@muhammadjoshua7464 5 лет назад

I was about to comment the same thing !

@BlackRose4MyDeath 5 лет назад

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

@ca94305 2 года назад

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

@hoodarrock2453 7 лет назад

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 лет назад

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

@jipuragi6483 Год назад

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

@astradrian Год назад

@@jipuragi6483 3Blue1Brown.

@jipuragi6483 Год назад

@@astradrian thanks a ton

@bofa722 Год назад

@@jipuragi6483 bruh

@saahilnayyer6865 2 года назад

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.

@spencertaylor6910 5 лет назад

Grant hits that yeet again. What a boss

@MrSkizzzy 6 лет назад

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

@Cyrusislikeawsome 7 лет назад

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

@tinayang7351 6 лет назад

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

@hellelo.5840 6 лет назад

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

@franciscorivas4036 4 года назад

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 лет назад

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

@robertcohn8858 4 года назад

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.

@fjgozzi 4 года назад

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

@gigglification 5 лет назад

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

@ednaT1991 6 лет назад

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 года назад

thats what makes mathematics beautiful

@Johncowk 4 года назад

That was SO clear I cannot thank you enough.

@kylewolfe_ Год назад

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

@sdpto 8 месяцев назад

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

@charliethatcher404 6 лет назад

You legit just saved my test grade tomorrow. Cheers

@AAA-uv1ny 10 месяцев назад

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

@dennishuang3498 4 года назад

Clear explanation ! Thanks for all your effort!

@foadabodahood9509 6 лет назад

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

@NicolasSchmidMusic 3 года назад

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 :)

@maxbardelang6097 2 года назад

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 10 месяцев назад

ok nerd

@Revetice 7 лет назад

very well explained and nice quality. thanks!

@christopherandrewmartin494 2 года назад

Very helpful. Thanks for all your videos!

@alias40anon 6 лет назад

Mate you nailed it, excellent explanation

@learningindia6733 Год назад

Genius, real mathematics......

@mantacid1221 Месяц назад

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

@GOPALS1967 4 года назад

Beautifully explained.

@guillermo._._ 3 года назад

Excellent geometric intuition!

@adrianpabloalvarez2523 Год назад

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?

@ND-kl8lo 5 месяцев назад

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

@real_john_doe 4 года назад

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

@jigneshrathod3714 6 лет назад

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...

@scoffer2150 10 месяцев назад

Thank you so much for this epic! Worth watching.

@annang.3176 14 дней назад

Beautiful explanation

@mertbeser9837 2 года назад

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

@williamcaldbeck 4 года назад

This is fantastic. Thank you

@giorgossartzetakis8771 4 года назад

OMG this guy is pure genious!

@tsrevo1 7 лет назад

Wow. excellent explanation.

@danawen555 3 года назад

thanks!!! very good and exhaustive explanation

@speedracer1702 2 года назад

Amazing explanation!

@poiuwnwang7109 4 года назад

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

@samuelvaldezgil 2 года назад

Im in love with this dude

@rfolks92 5 лет назад

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 года назад

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 года назад

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.

@matlabmalayalam3288 3 года назад

World-class teaching...

@goclbert 2 года назад

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

@jadedjimmy 5 лет назад

6:26 pullin out that Sal impression

@AngeloArrifano 2 года назад

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

@doctorb9264 3 года назад

excellent presentation.

@yavarjn2055 3 года назад

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

@thatsharma1066 2 месяца назад

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

@turbopotato4575 7 лет назад

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 лет назад

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.

@ddos87 4 года назад

Khan crushing it as usual

@mermaid6380 5 лет назад

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

@karthik-ex4dm 5 лет назад

awesome video!

@PedroLucas-fs8mc 7 лет назад

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

@Siam2233 5 лет назад

THANK YOU

@devkunjadia3792 10 месяцев назад

awesome video

@AmitDotAcademy 3 месяца назад

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

@Hecklit 6 лет назад

Thank you so much :D

@worldrecordegg5638 6 лет назад

it's really great

@mustafakartal3403 7 лет назад

thank you

@user-zk4bt7oy8n Год назад

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?

@saurabhsingh-ow7ue 4 года назад

thank you sir...

@shahabansari5201 3 года назад

Simply beautiful...

@queenstrategy904 4 года назад

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

@nahblue 2 года назад

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 года назад

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.

@carlossal04 3 года назад

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

@AlyssaCuriel 7 лет назад

thanks!

@ericbischoff9444 6 лет назад

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 ?

@luciafresnopm 3 года назад

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

@ahmednesartahsinchoudhury2628 Месяц назад

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

@ikhwanjeon7370 3 года назад

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..

@aashsyed1277 3 года назад

3 blue 1 brown?????

@oyindenyafatarakeme 5 лет назад

God bless you!

@arpitbahety5643 2 года назад

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 года назад

I believe it will, but only on one side

@sarfarazmemon2429 6 лет назад

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

@khanhdovanit 3 года назад

A bunch of thanks

@jeatig 6 лет назад

(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??

@scottabbas3335 4 года назад

very ok，thanks

@chandreshmaurya8102 3 года назад

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

@zixiaoxu977 6 лет назад

superb

@gajrajsingh51 3 года назад

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

@TheAryedemented 4 года назад

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

@laraeldabet6299 3 года назад

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?

@studyselection2881 Год назад

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

@benisbuff 7 лет назад

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

@ibrahimalkhorsani2533 7 лет назад

Ben lol. hope you made it through bro.

@user-vb4eq4vx1q 7 лет назад

How did it go??

@benisbuff 7 лет назад

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

@FsimulatorX 6 лет назад

Where are you now?

@leosin5767 Год назад

3blue1brown deserves a Nobel Prize in math education

@bokuwautsu 11 месяцев назад

there's no nobel prize for math or education

@arslanhojiyev5996 3 года назад

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

@justadude8716 Год назад

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.

@safooraranjbaran1466 2 года назад

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

@focker0000 5 лет назад

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

@YashPatel-vt8or 2 года назад

Big Fan Grant Sanderson !!

@Shadowfax2 5 лет назад

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!