# Uniform Sampling over Convex Polytope (not full-dimensional)

I want to simulate a uniform distribution on a convex polytope that is not full-dimensional for optimization purposes (to generate random points on the set I want to minimize over). The polytope is defined as follows:

$L_{83}:=\{(x_1,\cdots,x_{83})\in\mathbb{R}^{83}: 0 \leq x_i \leq o_i \ \forall i=1..83, \sum_{i=1}^{83} a_ix_i = C_1, \sum_{i=1}^{83} b_ix_i = C_2\} \neq \emptyset$ whereas $o_i$, $C_i$, $a_i$ and $b_i$ are given (non-zero) constants.

Does anyone have an idea how I can do this?