Optimisation Masterclass - Convex Function: Basic Properties

Optimization Masterclass - Convex Functions: Basic Properties - Ep 9 Smart Handout: https://colab.research.google.com/dri... Welcome to the next lecture in our Optimization Masterclass! In this session, we introduce Convex Functions. In this episode we tackle the foundational theory of convex functions. We start with the core definitions, exploring Jensen's inequality and the differences between convexity, concavity, and their strict forms. From there, we break down how to evaluate these functions using first-order (global underestimators) and second-order (Hessian matrix) characterizations. To ground the theory, we work through proofs for a variety of standard functions, including quadratics, norms, the max function, and the log-sum-exp. Finally, we bridge the gap between functions and sets by examining alpha-sublevel sets and the epigraph. Don't forget to check the smart handouts from the link above, and let me know in the comments what you'd like to see next! ➡️Subscribe to follow the entire Optimization Masterclass series    • Optimization Masterclass   and hit the Notification Bell 🔔 so you don't miss future episodes covering linear programming, duality, gradient methods, and more! 👍 Like this video if you find it helpful! 💬 Comment below with your questions or what you'd like to see next! Resources: My Research Lab page: https://giordanoscarciotti.com/ Smart Handout: https://colab.research.google.com/dri... Recommended Textbooks: https://web.stanford.edu/~boyd/cvxboo... Connect: LinkedIn:   / giordano-scarciotti-70167115   #Optimization #ConvexOptimization #MathematicalOptimization #ImperialCollegeLondon #OptimizationMasterclass #MachineLearning #DataScience #Engineering #OperationsResearch #lecture **CHAPTERS** 00:00 Introduction & Course Overview 00:42 Chapter Overview: Convex Functions 01:44 Basic Definition of a Convex Function 03:18 Jensen's Inequality 04:08 Strict Convexity & Concavity 05:30 Affine Functions & Line Restriction 08:18 Extended-Value Functions 13:28 First-Order Characterization 18:04 Second-Order Characterization 20:45 Basic Examples (Quadratic & 1/x²) 22:50 Common Convex Functions (Exponentials, Powers, Logarithms) 26:55 Norms & The Max Function 30:40 Quadratic-over-Linear Function 31:56 Log-Sum-Exp Function 34:29 Geometric Mean & Log-Determinant 39:52 Alpha-Sublevel & Superlevel Sets 44:12 The Epigraph