Equivalent optimization problem over simplex

Question

I have an linear program defined for variables over the non-negative orthant of the type: \begin{align*} \min_{x\in \mathbb{R}_{\ge 0}^K} &x^\top a \\ \text{subject to }& Ax = b \end{align*}

Is there any way I can formulate this problem as an equivalent problem over the simplex $\Sigma = \{x\in \mathbb{R}_{\ge 0}^{K}: x^\top \mathbf{1} = 1\}$?

Answer 1

Assuming your original feasible region is bounded, you can enumerate the extreme points and express every feasible solution as a convex combination of the extreme points. This idea is the basis for Dantzig-Wolfe decomposition.