(ch:cvx-opt)=
# Convex Optimization

This chapter provides general introduction to convex optimization problems.
Convex optimization focuses on a special class of mathematical 
optimization problems where:

* The real valued function being optimized (maximized or minimized)
  is convex.
* The feasible set of values for the function is a closed convex set.

```{figure} ../images/convex_function.png
---
name: convex_function
---
For a convex function, the line segment between any two
points on the graph of the function does not lie below
the graph between the two points. Its epigraph is a 
convex set. A local minimum of a convex function 
is also a global minimum.
``` 

Convex optimization problems are usually further classified into

* Least squares
* Linear programming
* Quadratic minimization with linear constraints
* Quadratic minimization with convex quadratic constraints
* Conic optimization
* Geometric programming
* Second order cone programming 
* Semidefinite programming 

There are specialized algorithms available for each of these
classes.

Some of the applications of convex optimization include:

* Portfolio optimization
* Worst case risk analysis
* Compressive sensing
* Statistical regression
* Model fitting
* Combinatorial Optimization