definition
relationship to convex sets
operations preserve convexity
global optimality (necessary $\equiv$ sufficient)
definition
relationship to convex sets
operations preserve convexity
global optimality (necessary $\equiv$ sufficient)
$$ \Delta_n = \{x\in \R^n \mid \sum x_i\le1, x\ge0\} $$
$$ \bar{\Delta}_n=\{x\in\R^n\mid\sum x_j=1, x\ge0\} $$
$\text{Conv}(S)=\{\sum_{i=1}^k\lambda_ix_i\mid\lambda \in\bar{\Delta}_k,x_i\in S,k\in \N\}$, where $\N$ denotes non-negative
$$ \Delta_2=H^-{\begin{pmatrix}0\\-1\end{pmatrix},0}\cap H^-{\begin{pmatrix}-1\\0\end{pmatrix},0}\cap H^-_{\begin{pmatrix}1\\1\end{pmatrix},1} $$
Caratheodory’s Theorem (Chp6, Beck)
Hyperplane: $H_{a,\beta}=\{x\in \R^n\mid a^Tx=\beta\}$
Halfspace: $H^{-1}_{a,\beta}$
$$ H^-_{a,\beta}=\{x\in\R^n\mid a^Tx\le\beta\}, (a\ne 0) $$
Definition. A function $f:C\to \R$, $C\subseteq \R^n$ (convex set), is convex if