Convex Optimization - L2 Convex sets

Convex Optimization - L2 Convex sets

1. Affine and convex sets

2.1 Affine set

Line: through \(\boldsymbol{x}_1\), \(\boldsymbol{x}_2\) all points

\[\boldsymbol{x} = \theta \boldsymbol{x}_1 + (1-\theta) \boldsymbol{x}_2, \quad \forall \theta \in \mathbb{R} \]

Affine set: contains the line through any two distinct points in the set

Example: solution set of linear equations \(\{ \boldsymbol{x} | \mathbf{A} \boldsymbol{x} = \boldsymbol{b}\}\)

Conversely, every affine set can be expressed as solution set of system of linear equations)

2.2 Convex set

Line segment: Between \(\boldsymbol{x}_1\), \(\boldsymbol{x}_2\) all points

\[\boldsymbol{x} = \theta \boldsymbol{x}_1 + (1-\theta) \boldsymbol{x}_2, \quad \forall \theta \in [0,1] \]

Convex set: contains line segment between any two points in the set

\[\boldsymbol{x}_1, \boldsymbol{x}_2 \in C, \quad \forall \theta \in [0,1] \quad \Rightarrow \quad \theta \boldsymbol{x}_1 + (1-\theta) \boldsymbol{x}_2 \in C \]

2. Some important examples

2.1 Convex combination and convex hull

Convex combination:

Convex hull \(\text{conv }S\): set of all convex combinations of points in \(S\)

2.2 Convex cone

Conic (nonnegative) combination:

Conic cone:

2.3 Hyperplanes and halfspaces

2.4 Euclidean balls and ellipsoids

2.5 Norm balls and norm cones

2.6 Polyhedra

2.7 Positive semidefinite cone

3. Operations that preserve convexity

3.1 Intersection

3.2 Affine function

3.3 Perspective and linear-fractional function

4. Generalized inequalities