Find the support function of $\{x\in\mathbb{R}^{n}|x^{T}Qx+2b^{T}x+c\leq0\}$

Question

I need to find the support function of $S=\{x\in\mathbb{R}^{n}|x^{T}Qx+2b^{T}x+c\leq0\}$ where we assume Q is symmetric and definitely positive and we know that $S\neq\phi$. I thought using KKT method but it is not working for me. any ideas?

Answer 1

Hint:

Minimise $x^TQx+2 b^Tx$ to get $Qx^*+b = 0$. Let $x=x^*+\delta$. Then $S = \{x^*+\delta | \delta^T Q \delta \le -x^*Qx^* - 2 b^T x^* -c\}$. (You should check my arithmetic.)

And so $\sigma_S(h) = \max_{x \in S} h^T x = \max_{\delta^T Q \delta \le -x^*Qx^* - 2 b^T x^* -c} h^T(x^*+\delta) $.