# Minimisation of Concave Function

$f\left(\mathbf{x}\right):\mathbb{R}_+^n\rightarrow\mathbb{R}_+$ is a concave monotonically increasing function to be minimised over the feasible region $\sum_{i=1}^n x_i=1$ and $x_i\geq 0\quad\forall1\leq i\leq n$.

Given that the feasible region is a convex polytope, is it possible to say anything about the optimal $\mathbf{x}^*$? I have a hunch that at $\mathbf{x}^*$, at least one of the inequality constraints will be satisfied with equality. In other words $\mathbf{x}^*$ will be right on the edge of the feasible region. But can't prove it. Am I right there?

If it helps to consider a special case, the function is

$f\left(\mathbf{x}\right)=\log|\mathbf{I}_k+\sum_{i=1}^{n}x_i\mathbf{A}_i|$

where $\mathbf{A}_i\in\mathbb{C}^{k\times k}$ are all positive semidefinite Hermitian matrices.

• What do you mean by "monotonically increasing"? – littleO Aug 15 '13 at 5:24
• $f\left(\mathbf{x}\right)$ is monotonically increasing with respect to each of $x_1, x_2, \dots, x_n$ while the others are held constant. – Della Aug 15 '13 at 5:58