Constrained optimization: the basics

This lecture covers the basics of constrained optimization, starting with the definition of a tangent direction and the consideration of a set S. The instructor introduces the problem of finding the minimum subject to constraints, illustrating with examples and trust-region subproblems. The lecture progresses to defining the set of acceptable directions and establishing necessary optimality conditions. The concept of a local minimum is explored through theorems and lemmas, leading to the discussion of Taylor expansions and proofs related to local minima.