Key Sequence

Notation

New Concepts

• Euclidian Geometry Crash Course
  • perspective function
  • linear-fractional function
  • generalized inequality
• convex function
• extended-value extension

Important Results / Claims

• operations that preserve convexity
• seperating hyperplane theorem
• supporting hyperplane theorem
• some convex functions
• some concave fuctions
• convexity preserve line restriction

Questions

Interesting Factoids

\(f: \mathbb{R}^{n} \to \mathbb{R}\) is convex IFF the function \(g: \mathbb{R} \to \mathbb{R}\) is convex:

\begin{equation} g\qty(t) = f\qty(x + tv), \text{dom } g = \qty {t \mid x + tv \in \text{dom }f} \end{equation}

is convex in \(t \in \mathbb{R}\) for any \(x \in \text{dom } f\), \(v \in \mathbb{R}^{n}\).

\begin{equation} f\qty(X) = \log \text{det} X \end{equation}

IF \(C\) and \(D\) are non-empty disjoint \(C \cap D = \emptyset\), and \(C\) and \(D\) are convex sets, \(\exists a \neq 0\) such that:

\begin{equation} a^{T} x \leq b, x \in C \end{equation}

\begin{equation} a^{T}x \geq b, x \in D \end{equation}

The hyperplane:

\begin{equation} \qty {x \mid a^{T} x = b} \end{equation}

separates \(C, D\) such that \(a^{T}x \geq b, a^{T}x \leq b\).

Key Sequence

Notation

New Concepts

• optimization (math)
  • convex optimization
  • non-linear optimization
• convex sets

Important Results / Claims

Questions

Interesting Factoids