Convex Sets and Functions

This lecture covers the concepts of convex sets and functions, defining convex sets as those where a linear combination of two points remains in the set, and convex functions as those satisfying a specific inequality. It also discusses minimizers of convex functions, optimality conditions, and the characterization of convexity using first and second-order conditions. Examples of affine and quadratic functions are provided, along with the Cauchy-Schwarz inequality and the Descent Lemma for continuously differentiable functions with Lipschitz gradient.