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