Lecture 2 | Convex Sets | Convex Optimization by Dr. Ahmad Bazzi

[@AhmadBazzi]

Thank you all so much for the kind support. It means a lot. Please make sure you subscribe to the channel to keep it going !! Let's make it to 100K subscribers.

[@Nihit-n5n]

thanks a lot Ahmad for making them public.They are much much better than stanford convex optimization course.

[@aishahoura2619]

00:00:00 Affine Combination 00:01:33 Affine Set 00:08:21 Convex Combination 00:09:25 Convex Set 00:13:45 Convex Hull 00:16:28 Example 1-Convex Cones 00:16:55 Conic Combination 00:20:47 Example 2-Hyperplanes 00:24:22 Example 3-Euclidean Ball 00:26:37 Example 4-Ellipsoid 00:30:40 Norms 00:35:51 Example 5-Polyhedra 00:41:18 Example 6-Positive Semidefinite cone 00:54:31 Operations preserving convexity 01:15:10 Closed & Open set 01:21:10 Solid sets 01:26:25 Pointed set 01:26:57 Proper cones 01:27:28 Generalized Inequalities 01:34:12 Minimum & Minimal Elements 01:46:28 Partial Order 01:48:53 Properties of Generalized Inequalities 01:53:09 Dual Cones 02:04:31 Dual Inequalities

[@matthewygf]

Dude ... You teach things way better than other lecturers, Amazing.

[@aliyahkilback7489]

i am so blessed i came across this video, you have such a great way of simplifying and breaking down hard topics .. God bless you !!

[@ruthsindie2660]

am saying!! I substituted this with actual studying

[@houdazahmoul3348]

Keep those videos coming! What a great lecturer. Thank you!

[@AhmadBazzi]

Thank you for your support, Houda 😊

[@houdazahmoul3348]

@@AhmadBazzi could you please cover the topics of lagrange solution, and lambda iteration methods?

[@AhmadBazzi]

​@@houdazahmoul3348 yes this topic in particular is coming really soon. Please. make sure you subscribe to the channel to receive the latest updates :)

[@houdazahmoul3348]

@@AhmadBazzi thank you.

[@evastellato3531]

Literally the best professor in history

[@swatityagi222]

Best lectures so far

[@aboneolanaaboneolcam9592]

People say your phones Rot your brain but I’m learning more about these calculus problems from this guy than my actual teacher, thank you for existing my guy.

[@maximilianthompson3726]

bazzi plays with sets like jordan plays ball, jumpman jump

[@cemiluces6727]

never thought I'd be watching a course on convex optimization.

[@franciscorivas4036]

Thank you Ahmad. Very clear explanation.

[@sajeelkhan4500]

all these animations make understanding it so easy!

[@AhmadBazzi]

Thank you, Sajeel. I’m glad you found the lecture beneficial!!

[@daddyruby1824]

That explanation at the beginning of constraint equation and objective equation was SO clarifying!

[@ylkevin]

I found this video very substantial with great graphical drawings !!! So attractive!

[@AhmadBazzi]

thank you sir, i’ll try my best to keep it clear and relevant, make sure you subscribe to the channel to receive the latest updates.

[@ritvikdhupkar5861]

Bookmark 23:00 Bookmark 46:00 Bookmark 1:25:00

[@sitdownl584]

Outstanding. Thank you sir.

[@Vic-de6ud]

20:08 mind blowing, first time ever to realize that this is indeed the case, thanks to the amazing graphical illustrations!!

[@AhmadBazzi]

Very glad you found the illustrations useful. I’m actually investing time to produce better illustrations. Stay tuned for future videos where i’ll be emphasizing way more on animations and illustrations.

[@asknpala7802]

example: question 8 34:00. you don't need the derative for this you just take the Full amt of sqft and divide by the number of sides. div 1 is the larger amt. divide again but division 1 by the full amt of sqft that will be the minuim.

[@muradhesenov5245]

You don't understand how helpful these are. Life saver fr you're my goat

[@oyunburada6694]

Ahmad Bazzi is going places.
15:42 makes this complicated topic way more easier. Thank you so much for putting animated efforts in Ahmad.

[@juwanyirenda3457]

The best I have seen on this subject so far. Thanks, Ahmed.

[@xielin9404]

Wow, awesome lectures, thank you very much, your explanations are professional and easier to understand, I'm looking forward to your subsequent videos! 👏👏👏

[@AhmadBazzi]

thank you, i'm glad you found them useful. Make sure you subscribe to the channel to receive the latest updates :)

[@ksaksaseyler6207]

1:39:47 *Convex Straight Savage* should be your motto. Damn man, I’ve never seen such a lecture.

[@kathryngriffin8705]

give this guy a noble prize

[@mariahdubose1305]

and it still isn't enough

[@pauldacus4590]

Sooo... notebook says "Ahmed's Notebook", and your channel says "Ahmad", just wondering which it is. Love this Convex Opt series! Thanks for taking it slow & simple. And I DO appreciate the meticulous editing...

[@AhmadBazzi]

Yea it’s an Ahmad, but both could be used interchangeably.. thank you for your support. I am glad someone out there is finding the videos useful 😊

[@Merkeztayfa46]

I have my calculus final tomorrow and I didn’t understand my teacher’s notes and the first 25 minutes of this video was all I needed. Thank you!

[@putrafishing3979]

Optimization problems nearly killed me back in Calc 1. I’m glad you’re doing more on this subject.

[@kadirtapur34]

Here's a trick at 37:00. You don't have to find the derivative of the whole radical because the minimum of what's inside the radical is the same thing as the whole function. So you can just find the derivative of x^2+(3x+5)^2 which is 20x+30, and you get -3/2 for x as well

[@Scott_Raynor]

48:30 principle minors positive semidefinite

[@josephsylvan6511]

1:39:56 - 1:40:04 I like that pause you gave before mentioning "therefore..the proof is done!"...felt it funny 😂😎 Thanks a lot sir for such wonderful explanations.

[@AhmadBazzi]

Yeah haha just trying to see if the proof makes sense!! You are welcome, Joseph!! Please make sure you subscribe to the channel and hit the notification button to receive future notifications about the channel :)

[@andywang4189]

So genius way to explain norm cone, thanks!

[@esinyerlikaya670]

Wow I actually understand now some convex sets. Never thought I would be able to.

[@No1Ringtones]

For 40:00, do you have to graph it to know if (7, 6) are the x2 and y2? Or are there ways to know without graphing? Asking for the more complicated functions that will show up.

[@worse4086]

My biggest problem in my calculus 1 course is the algebra, and finding ways to simplify or rationalize problems so that the formulas and methods given apply. My professor always shortcuts her algebra by two or three steps, making her work hard to follow. Thank you for showing every step along the way to the solution. Your videos mean the world to me this semester, and I'm sure I'll be back next semester for calculus 2!

[@emirefeaktas5200]

When you find the first derivative and set it equal to zero, sometimes you can get 1 critical point, other times you may get 2 or more critical points. You can use the first derivative test to determine if the critical point is a maximum or a minimum. If f' changes from - to +, it's a minimum. If f' changes from + to -, it's a maximum. You definitely need to do this if you have 2 or more critical points. If there's only one critical point, the problem usually tells you what kind of critical point you're dealing with. For this, if the question says find the dimensions of a rectangle that maximizes the area, the critical point is most likely a maximum.

[@umutbeytc914]

I love you so much i have a test tomorrow and ur the only one whos explanation i can actually understand

[@hakanaleksandrow6707]

Thanks! I'm starting to get it! And seeing the power of the derivative.

[@ama_awm1970]

Thank you for everything you do my guy, we are so lucky to have people like you in the world ong.

[@thearithyt4957]

Jump to 20:46 if you're on the perimeter of a fence problem and try to maximize the area.

[@sukhdeepsingh1921]

This video really covers everything regarding the optimization problems. Thanks a lot.

[@chick3n71]

1) When a vector is drawn, it should be from the origin. 2) The norm cone example should be {(x,r): ||x|| ⩽ r} where r is the positive value in the additional dimension the cone is supposed to scale in. 3) In the polyhedra example drawn for Ax = b, there is only solution for every pair of subequations, but no solution overall (as all have to be met). 4) In the positive semi-definite cone example, it should be (xy)^1/2 ≥ |z|, which is (xy)^1/2 ≥ z if z≥0 & (xy)^1/2 ≥ -z if z

[@brookscruickshank6367]

55:24 i once spent 10 hours proving a set to be convex using the definition, however it turns out much easier using the notion of intersection of convex sets.

[@AhmadBazzi]

That is correct.

[@ziadabdulla719]

شكرا جزيلا على الشرح المبسط والواضح

[@Ra7aePubg]

wollah! And he drops the bombshell just when I have an assignment on this!!!!

[@_.jasinveliu]

This video popped up at the right time.. we re having calculus this semester

[@nubfighterpweef6235]

No one knows how much i love this guy

[@yunussk3669]

AP calc exam in 3 days. Posted right on time. Thanks.

[@olaylaryt5840]

Just what I needed with my AP Calc exam next week!

[@musti4297]

this man has covered every unit of my 4 years of math, Chemistry, Biology, and physics. how

[@ahmadmoussa3771]

Hey, awesome lecture! Suggestion: for the cone example, maybe you could use the word spotlight or flashlight instead of projector, because that makes more sense to me when I imagine it. Just my two cents!

[@AhmadBazzi]

thanks, Ahmad and thank you for the tip ..Coming to think about it, you're right. It does look like a flashlight !! I'll keep that in mind for future lectures :) thank you once again.

[@tugcekandemirfanhersey8697]

Thank you so so much! I don't think I'd be doing this well without your help!

[@youtubetricks3569]

THANKS!! Learned this not so long ago, this is definitely gonna come in handy 👍

[@belediyelerishutdownyapane6416]

I have learned more about calc in the past hour then all year

[@sevketozbey6805]

This guy is awesome, not from this earth.

[@quocphongpham6257]

I just want to say: thank you so so so much!

[@neptune6k]

Doing god's work....Respect!!

[@balkanuzivo9455]

Super lecture Ahmad. Please proceed with convex optimization lectures. Best wishes from the USA.

[@ggames5770]

I just wanted to let you know that this video is really helpfull for my Linear Algebra course in university! Thank you so much!

[@dogukan8595]

I have my final this morning and this video is helping a lot

[@mustafasezer8383]

Wow, exactly what I was looking for

[@emiryasa5577]

Thanks! About to use this for my calc final soon 😎

[@smithlawrence465]

When you do these right, it makes calculus seem kind of fun.

[@Berkow009]

You're a real life hero

[@elissathiel4948]

53:17 I LOVE CHURROS !!

[@AhmadBazzi]

who doesn't ?

[@jnnl_hnd]

right on time, you absolute beast

[@azhartecheditz644]

at 14:00 you divided x into both sides in order to solve for y but could you have subtracted x like you did before and plugged in the 400-x into the S=x+y problem thus still coming up with the 20 as Yand then solved from there?

[@MustafaCan-oq4tb]

So the first derivative shows the highest and lowest values of slope. In order to find the max of that graph (first derivative graph), you find the second derivative. The second derivative does show the POI, but that's the steepest part of graph (highest slope value).

[@ibrahimcankaradag5898]

You sir, are amazing and I am so thankful! Going to buy your merch!

[@yunusemre1643]

For question 12. A = r^2 for max area of rectangle inscribed in circle. Does that only apply to a semicircle? Or also a full circle? Or would the full circle have a max area of 2r^2 ? Or D^2 diameter squared?

[@jeetendradhall76]

To understand a course from a Stanford video, I landed at this video. I think I got really lucky.

[@ramazankizilyildirim3476]

Really it helped a lot. Thanks for this great work!!

[@AhmadBazzi]

You're welcome, Ramazan ! Glad you found it useful !! For the meantime, make sure you subscribe to the channel here kzbin.info

[@AA71_PUBG1]

You made it simpler than what my teacher discussed about this lesson.

[@happysoph023]

thank you so much for this video! I was so confused reading chapter 2 of boyd and vandenberghe

[@rakeshkumarbiswal9489]

You are a great teacher luv from india 👍

[@AhmadBazzi]

Thank you so much from France, Rakesh :))

[@whoisbella8]

Wooow ahmad bazzi you are genius !!!!!!!!!!!!!!!

[@hotgirlsupdatestv5]

In question 11, I think the critical points found as zero of the first derivative should be evaluated with the second derivative to classify the points where the curve will have local minimum and a local maximum and not the zeros of the second derivative function rather(those are used for checking intervals of concavity). Thank you sir

[@jocelynwehner2163]

10:42 is the croissant a chocolate one ? does it matter for convex sets ?

[@AhmadBazzi]

No it's a plain croissant ha ha. Chocolate ones are convex (i'm joking).

[@oscarfrausto4502]

Awesome lectures, I am trying to understand and implement this paper "Multi-target Tracking in Multiple Non-overlapping Cameras Using Fast-Constrained Dominant Sets" and I think your videos are useful for that purpose. Greetings from Mexico "

[@atessonmez1142]

thank you with all my heart man

[@xiaokewang2549]

Thank you very much for your videos. Have you made some documents or materials about convex optimization? I would like to read such things.

[@infonejo2737]

TY SO MUCH!!! You are a LIFESAVER!!

[@m.y.s4260]

Awesome lecture, thanks!!!

[@AhmadBazzi]

my pleasure

[@xeyalytb4194]

Wow perfect timing

[@erenkurt9068]

If no one else got my back, I know FOR SURE that The Organic Chemistry Tutor got my back.

[@htetnaing007]

Thank you so much for these videos.

[@AhmadBazzi]

You are so welcome!

[@tuongnguyen9391]

Thank you so so much :)) but I hope that this video could be split into 15 minutes session because my attention span could not keep up. However I feel graceful when you include the time stamp :))) it really help me to plan my study.

[@doriaconry8598]

He is the best yes !

[@razee685]

How does the dimensionality of a set affect its convexity, and what considerations should be made when working with high-dimensional convex sets?

[@JIHANEKIKOU]

You are my favorite human being.

[@piyushvarshney9227]

Best Lecture. Thanks Prof.

[@nicolagnecco]

Thanks for the great video! I have a remark at minute 7:49. You say that the set {x} is a particular case of the set {x: Ax = b} when A = identity matrix and b = x. However, in that case, we have that the second set writes {x: x = x} = Real numbers. So I guess that {x} is not a special case of the second set. Am I missing something?

[@AhmadBazzi]

You are correct. Mathematically speaking, it is not a subset of {x: Ax = b}. I was just trying to say that if the set {x: Ax = b} is affine, then so is {x}.

[@nicolagnecco]

@@AhmadBazzi Thanks a lot! It makes complete sense :) Thanks again for the fantastic Convex Optimization series!

[@AhmadBazzi]

@@nicolagnecco oh, my pleasure. I'm glad you found it useful : )

[@rabelz6620]

Never thought convex sets were that easy to understand

[@bos6251]

I like the way of your teaching.

[@tibbivideolar4059]

28:13 is there any significance to dividing by the variable for the side opposite of the river, rather than the variable that makes up two sides? I get a different answer when i divide 10000 by the same variable with two sides.

[@hizirmurat44]

I love your way of explanation .. way to go.

[@dariunjoefield2444]

at 51:00, why are you using the second derivative to find a maximum? Doesn't the second derivative tell you the POI, and first derivative tell you local max and mins?

[@mrbird50]

What are some common examples of convex sets in mathematics and real-world applications, and how are they used?

[@kxnii_kr]

Thank you , my greatest tutor, you're much appreciated sir

[@peopeo9406]

our best teacher

[@dollajitt]

I believe it would be (2r)^2 or d^2 (d being the diameter like you said) since you would technically have two semi-circles in a full circle. So you are right! If you have any other calculus questions, I would be happy to help you out. So feel free to stop by my channel sometime, I have some videos on these topics as well that may help you! Have a great day :)

[@sallydubuque1763]

1:16:46 closed and open at the SAME TIME .. yup that's what topology is like. AND IF YOU DON'T GET IT, YOU'RE THE PROBLEM !!

[@AhmadBazzi]

I agree that topology is not one of the easiest math topics.