Convex Optimization : some definitions

Linear relation: Ax

Affine relation: Ax+b

Affine Set

1. contains the line through any two distinct points in the set:
2. can be expressed as the solution set of a system of linear equations

Convex Set

1. If two points are in the set, then the line segment as well.
2. Affine set with 0

Convex Combinations

1. Consider any number of points, with their coefficient being positive and summing up to 1

Convex Hull

1. All convex combinations of any points from the set.

Conic (non-negative) combination

1. Like convex combinations but without the constraint of "summing up to 1"
2. Formed with any points together with the origin.

Conic Hull

1. All conic combinations of any points from the set.

Hyperplane and halfspace

1. hyperplanes are affine and convex: 
2. halfspaces are convex

Polyhedron/polytope

1. solution space of a system of inequalities like in a linear program
2. intersections of a finit set of hyperplanes and halfspaces

Operations that preserves covexity

1. S is convex set -> f(S) too, f^-1(S) as well. with f an affine function
2. scale, translation, projection...
3. perspective (projection) function and its inverse: . 
4. linear fractional function: a fraction of linear functions. ,with the denominator >0.
5. can be seen as a streching of vision, or the projection of a 3d object  onto a camera. x=[x1,x2]. If the object is convex then the result is convex too.

 

 

 

 

Generalized Inequalities

A proper cone is closed (contains boundary), solid (not empty not a ray) and pointed(not a line).

define  , reads "x less than y with respect to proper cone K". Remember K somehow define a 'positive' space, consider an orthant. The coordinates y-x must still in the same orthant.

1. componentwise inequality: , x
2. matrix inequality:,(w.r.t positive semidefinite)
3. , but the latter defines also a "linear ordering", while the former doesn't, where comes the difference between Minimum (less than anyone else) and Minimal(less than anyone else that are comparable).

TO BE CONTINUED