Convex Analysis
Table of Contents
1. Convex Hull
- Convex Envolope, Convex Closure
1.1. Definition
- A convex hull of a set \(X\), \(\mathrm{co}(X)\), is all convex combinations of the elements in the set \(X\).
1.2. Of Function
- Lower Convex Envelope
- A function whose epigraph is the lower convex hull of the epigraph of \(f\).
1.2.1. Definition
- The lower convex envelope \(\breve{f}\) of a function \(f\) defined on an interval \([a,b]\) is \[ \breve{f} := \sup\{g(x) \mid \text{$g$ convex}, g\le f\}. \]
- May be extended to a family of functions.
- See envelope.
2. Hilbert Projection Theorem
- Famous result of convex analysis.
2.1. Statement
- For every vector \(x\) in a Hilbert space \(H\) and every nonempty closed convex \(C\subseteq H\), there exists a unique vector \(m\in C\) for which \(\Vert c-x \Vert\) is minimized over the vectors \(c\in C\): \(\forall c \in C, \| m - x \| \le \| c - x \|\).
- If \(C\) is also vector subspace , then \(x-m\) is orthogonal to \(C\).