# convex functions

(6 hours to learn)

## Summary

Intuitively, convex functions are bowl-shaped. They are significant in optimization, because it is often possible to efficiently find the global optimum of a convex function.

## Context

This concept has the prerequisites:

- convex sets (The domain of a convex function must be a convex set.)
- vectors (Convex functions are defined in terms of vectors.)
- linear approximation (The first-order condition for convexity is that the linear approximation lie below the function.)
- gradient (The first-order condition for convexity is that the linear approximation lie below the function.)
- higher-order partial derivatives (The second-order condition for convexity is that the second-derivative matrix be positive semidefinite.)
- positive definite matrices (The second-order condition for convexity is that the second-derivative matrix be positive semidefinite.)

## Goals

- Know the definition of a convex function (in multiple dimensions)

- Know and be able to apply alternative characterizations of convex functions
- first-order condition (linear approximation lies below the function)
- second-order condition (second derivative matrix is positive semidefinite)

- Know some examples of convex functions

- Why can the value of the function outside its domain be taken to be infinity?

## Core resources (read/watch one of the following)

## -Free-

→ Convex Optimization

A graduate-level textbook on convex optimization.

Location:
Section 3.1, "Basic properties and examples" of Chapter 3, "Convex functions," pages 67-78

Other notes:

- Some of the examples may use concepts you haven't seen. If so, just try to get the gist.

## See also

- Convex functions are part of the definition of convex optimization problems , a broad framework for optimization where the global optimum can often be found.
- Some important examples of convex functions:
- vector norms
- utility functions are typically assumed to be concave
- regularization penalties in machine learning are often convex
- likelihood functions for exponential families