Convexity and The Principle of Duality

A gentle and visual introduction to the topic of Convex Optimization (part 2/3). In this video, we give the definition of convex sets, convex functions, and convex optimization problems. We also present a beautiful and extremely useful notion in convexity optimization, which is the principle duality.

This is the second video of the series.

Typos:
- At 7:59, there is an extra minus sign in the right hand side of the equation A^TAx = -A^Tb. The correct equation is A^TAx = A^Tb,which leads to the solution x = (A^TA)^-1 A^T b.

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Timestamps:

0:00 Previously

1:00 Definition of Convex Sets
1:47 Definition of Convex Functions
2:45 Definition of Convex Optimization Problems
3:36 Duality for Convex Sets
6:09 Duality for Convex Functions
8:40 Examples

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Credit:

This video would not have been possible without the help of Gökçe Dayanıklı.

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🎵 Music

[snailscout]

I feel fortunate to live in a time were there are people who teach hard-to-understand concepts for free in a easy to grasp fashion. Hats off to you and thank you a lot

[parahumour]

Amazing video, your channel deserves more views. I would suggest having a section where you ask the viewers questions so they stop and think and end up being onboard with the understanding

[raulsena]

Thanks for the video, I was reading many books to understand this and you explain it plain and simple. Keep it up!

[virmaq]

As a visual learner, this video helped me tremendously. Thank you!

[jiaqi]

Extremely good introduction!!!! It is very hard to imagine how much work put behind the video!! Thanks for your input on this!!
I already worked on convex optimization problem in a research project for a few months but honestly I really don't know what is special about convex optimization. Thanks for giving us the intuition behind it!!

[Sirentuber]

Good God. This is so beautiful and intuitively explained. Can thank you enough for this! you are the savior.

[wexwexexort]

I always like to watch the visual explanations even though I know the topic quite well and to be honest, you do a really good job on both explanations and visuals.

[rubenosmond]

Thank you sir. I'm having a hard time understanding this concept for my machine learning class and you helped me in a beautiful fashion. May you have great things in line.

[cmatiolli]

Your videos are awesome. The right balance of math concepts and intuition to explain complex ideas is the perfect fit for this essential concept.

[vats]

Woah! Thanks a lot sir, for such an intutive explaination of convexity. The best explaination I have seen on the internet so far!

[박시연-mm]

great video! What program did you use to make this fantastic visualization?

[imotvoksim]

The least squares example you show at 7:59 has a wrong sign as far as I can tell!.Otherwise a great video providing the intuition I was looking for, took down some notes, hopefully they finally stick and I understand this dual magic once and for all, thanks!

[colin_hart]

The code samples for linear programming and least squares are swapped at 0:23.

I’ve been enjoying your work. Thanks for sharing!

[parhamzolfaghari]

Wonderful! I always wonder why the professors and teacher follow the worst method possible to teach materials.

[gustavgille]

The least squares error example is beautiful!!!

[jiaqint]

I learnt so much from this video, I love you so much

[phogbinh]

God like overview of the topic. Thank you.

[SonLeTyP]

Awesome :) cant wait to see next episode :D

[thebifrostbridge]

Great video. Fun fact, the autogenerated subtitles at 9:32 says: "to optimization problems with cancer friends"

[werdasize]

Really nice! But one thing I didn’t understand was at 5:02 ish. You say that the intersection of those support planes is the convex set.. but in your example, isn’t the intersection of the planes just a bunch of connected lines? Not sure if I understood correctly.