Determine the dual cone $\mathbb{K}^{*}$ of the following cones?

Question

a) $ \mathbb{K} = \mathbb{R}_{+}^{n}$

b) $ \mathbb{K} = \mathbb{S}_{+}^{n}$

c) $ \mathbb{K} = \{(x,t) | \begin{Vmatrix} x \end{Vmatrix}_{2} \leq t\}, x \in \mathbb{R}^{n}, t \in \mathbb{R}^{+}$

This is book convex optimization.

Answer 1

If $\mathbb{R^n}$ is considered as Hilbert space with the standard scalar product $(\cdot,\cdot)$, the dual cone $K^\ast$ to a given cone $K$ is usually defined as $$ K^\ast=\{y \in \mathbb{R^n} \mid \forall x\in K:(x,y)\ge 0\}. $$ For example let $K=\mathbb{R^n}_+$. Then $y \in K^\ast$ means $$ (x,y)=\sum_{k=1}^n x_ky_k \ge 0 ~ for ~ all ~ x_1,\dots ,x_n \ge 0. $$ Clearly $y \in K$ implies $y \in K^\ast$. Reversely, let $\{ e_1,\dots,e_n \}$ be the standard base of $\mathbb{R^n}$. Now $$ y \in K^\ast ~ \Rightarrow ~ y_k=(e_k,y) \ge 0 ~ (k=1,\dots,n) ~ \Rightarrow ~ y \in K. $$ Thus, in this case $K^\ast = K$. Of course, in general $K \not= K^\ast$.