I have a convex optimization problem of the form: $$ \begin{aligned} \operatorname*{minimize}_{\mathrm{x} = (\mathrm{x}_1, \dots, \mathrm{x}_m) \in \mathbb{R}^{nm}} &\quad f(\mathrm{x}) = \mathrm{x^T A x + c^T x + ||Bx||} \\ \text{subject to} &\quad \mathrm{x}_j \in \mathcal{P} \end{aligned} $$ In the problem above $\mathrm{x}$ is partitioned to $m$ sub-vectors, $A$ is a diagonal positive-definite matrix, $\mathcal{P}$ is a polyhedron and $||\cdot||$ is some norm.
I am looking for results which connect properties of $f$ to the existance of a minimizer $\mathrm{x}^*$ for which some (or all) of the $\mathrm{x}_j$ are extreme points of $\mathcal{P}$.