Closed form solution for the following problems?

Question

I would like to know if there is a closed-form solution to these optimization problems:

$$ \min_{x\in \mathbb{R}^n} c^Tx$$ and $$\max_{x\in \mathbb{R}^n} c^Tx$$ and both restricted to the following conditions:

$$x\ge0$$ $$\mathbf{1}^Tx= 1$$ Also $c\in\mathbb R^n$

Answer 1

In general, we have $\max_k c_k = \max_{x \in \Sigma} x^T c$, so the answer is $\max_k c_k$.

With $\Sigma = \{ x | x_k \ge 0, \sum_k x_k = 1 \}$.

A similar analysis applies to $\min$ with the obvious result.