Let $x = (3,4,5)$ and let $A = \{z = (z_1,z_2,z_3) \in \mathbb{R}^3 : z_1+z_2+z_3 = 1\}$. Prove the existence and uniqueness of solution to the nonlinear optimization problem
$$min_{z \in A} \|z-x\|_2^2$$
I would say we should try to take care of the side condition $z \in A$ first. We can do this by setting $z_3 = 1 - z_1 -z_2$ and obtain:
$$ \|z-x\|_2^2 = \langle z-x,z-x \rangle_2 = (z_1-x_1)^2 + (z_2-x_2)^2 + (z_3-x_3)^2$$ $$=(z_1-x_1)^2 + (z_2-x_2)^2 + (1 - z_1 -z_2 - x_3)^2$$
$$=(z_1-3)^2 + (z_2-4)^2 + (-4 - z_1 -z_2)^2$$
So I guess we should proceed by calculating the Hessian and the gradient of the above term, we call it $f$, and we obtain:
$$\nabla f(x) = (4z_1+4,4z_2+2)^t$$
$$\nabla^2 f(x) = \begin{pmatrix} 4 & 0\\ 0 & 4 \end{pmatrix}$$
It can thus be easily seen that $z^* = (-1, -0.5, 2.5)$ is the unique minimum of $f$.
Is this correct?