# The Mathematical Basis Of Convex Optimization

2018-02-22

This winter vacation i’m trying to understand this course–Convex Optimization,with current paper.I want make some contribution to some open problems in the field of communicaton optimization.

This will be a series of articles,where i only give the most basic mathematical concepts,the other will be separate article.

Reference material:

• Convex Optimization -by Stephen Boyd
• PKU Course-00136660 Lecture

## Lines and line segements

Suppose $x_1\neq x_2$ are two points in $R^n$,Point of form $y=\theta x_1+(1-\theta)x_2,\theta \in R$ form the line passing through $x_1$ and $x_2$.

Values of the parameter $\theta$ between 0 and 1 correspond to the (closed) line segement between $x_1$ and $x_2$.

## Affine sets

A set $C \subseteq R^n$ is affine if the line through any two distinct points in $C$ lies in $C$.

If for any $x_1$,$x_2 \in C$ and $\theta \in R$,we have $\theta x_1 + (1 - \theta)x_2 \in C$.In other word,$C$ contains the linear combination of any two points in $C$,which sum to one.

The idea can be generalized to more than points.We refer to a point of the form $\theta_1 x_1+\ldots +\theta_kx_k$,where $\theta_1 x_1+\ldots +\theta_kx_k = 1$,as an affine combination of the points $x_1,\ldots,x_k$.

Using induction from the definition of affine set $i.e$ that it contain every affine combination of two points in it,it can be shown that an affine set contain every affine combination of its points.If C is an affine set,$x_1,\ldots,x_k \in C$,and $\theta_1 x_1+\ldots +\theta_kx_k = 1$,then the point $\theta_1 x_1+\ldots +\theta_kx_k$ also belong to $C$.

..etc(i will write it in nextday)