Convexity and The Principle of Duality

Описание: 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.

Part 1: What is (Mathematical) Optimization? (   • What Is Mathematical Optimization?  )
Part 2: Convexity and the Principle of (Lagrangian) Duality (   • Convexity and The Principle of Duality  )
Part 3: Algorithms for Convex Optimization (Interior Point Methods). (   • The Karush–Kuhn–Tucker (KKT)  Conditions a...  )



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:
🐍 Manim and Python : https://github.com/3b1b/manim
🐵 Blender3D: https://www.blender.org/
🗒️ Emacs: https://www.gnu.org/software/emacs/


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

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🎵 Music

Vincent Rubinetti (https://vincerubinetti.bandcamp.com/)
Carefree by Kevin MacLeod (   • Thinking Music  )