constituents
- set \(C\)
- \(\theta \in \mathbb{R}\)
requirements
A set \(C\) is a convex set if:
\begin{equation} x_1, x_2 \in C, 0 \leq \theta \leq 1 \implies \theta x_{1} + \qty(1-\theta) x_{2} \in C \end{equation}
definitions
additional information
operations that preserve convexity
Convex sets is a calculus! Methods to showing complexity:
Anything in Euclidian Geometry Crash Course
- apply definition: show \(x_1, x_2 \in C, 0 \leq \theta \leq 1 \implies \theta x_1 + \qty(1-\theta) x_2 \in C\)
- use convex functions
- show that \(C\) is obtained from convex sets via the following operations
intersection
Intersections of any number of convex sets, include infinite, are convex.
affine mapping
Suppose \(f : \mathbb{R}^{n} \to \mathbb{R}^{m}\) is affine, that is, \(f\qty(x) = Ax + b\) for \(A \in \mathbb{R}^{m \times n}\) and \(b \in \mathbb{R}^{m}\).
The image of a convex set under affine \(f\) is convex:
\begin{equation} S \subseteq \mathbb{R}^{n} \text{ is cvx } \implies f\qty(S) = \qty {f\qty(x) \mid x \in S} \text{ is cvx } \end{equation}
The inverse image of \(f^{-1}\qty( C)\) of a convex set \(f\) is convex:
\begin{equation} C \subseteq \mathbb{R}^{m} \text{ is cvx } \implies f^{-1}\qty( C) = \qty {x \in \mathbb{R}^{n} \mid f\qty(x) \in C} \text{ is cvx} \end{equation}
perspective mappings
perspective function and its inverse image preserves convexity.
linear-fractional function
for \(f: \mathbb{R}^{n} \to \mathbb{R}^{m}\):
\begin{equation} f\qty(x) = \frac{Ax + b}{ c^{T}x + d} \end{equation}
where:
\begin{equation} \text{dom} f = \qty {x \mid c^{T} x + d > 0} \end{equation}
linear-fractional function and its inverse image preserves convexity.