Optimization
Optimization Problem
Problem Defination
A problem of the form
\[\min f(x) \\ \text{subject to } x \in C\] $f(x)$ is the objective function
 C is called the constrained set
 find $x$ s.t. $f(x)$ is minimized
Problem Formulation
 Model the problem as optimization (Modelling)
 objective function
 constraints
 Solve using optimization (Solver)
 Problem class
 linear/nonlinear
 smooth/nonsmooth
 convex/nonconvex
 use suitable solver based on problem class
 Problem class
Convexity
 Indication function relates convex functions and convex sets
 Epigraph of $f$ is a set of all points above the function.
 if a fucntion is convex, then its epigraph is convex and viceversa
Convex sets
Convex functions

Max functions
\[f(x) = \text{ max } \{x_1, x_2, ..., x_n\} \text{ is convex on } \mathbb{R}^n\]