The Mathematical Basis Of Convex Optimization


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:

Lines and line segements

Suppose are two points in ,Point of form form the line passing through and .

Values of the parameter between 0 and 1 correspond to the (closed) line segement between and .


Affine sets

A set is affine if the line through any two distinct points in lies in .

If for any , and ,we have .In other word, contains the linear combination of any two points in ,which sum to one.

The idea can be generalized to more than points.We refer to a point of the form ,where ,as an affine combination of the points .

Using induction from the definition of affine set 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,,and ,then the point also belong to .

..etc(i will write it in nextday)