I have the problem: I'd like to generate a uniform sampling of points from a space defined by the bounded constraints and linear inequality described below: $$lb_1 \leq x_1 \leq ub_1$$ $$lb_2 \leq x_2 \leq ub_2$$ $$lb_3 \leq x_1 + x_2 \leq ub_3$$ With two variables and the linear inequality, this results in a 2D surface from which I'd like to sample 50 points uniformly distributed across the surface.
Take the example where $5 \leq x1 \leq 10$, $4 \leq x2 \leq 8$, and $10 \leq x_1 + x_2 \leq 12$. This surface looks like a rhombus. It is easy to create a span of values across $x_1$ {5.0, 5.1, …, 10.0}, then for each point solve for the values of $x_2$ which satisfy the inequality. Is there another way to approach this?
I see similar questions about uniformly sampling points within a cube or sampling of points on a simplex which seem similar to this question. In both of those questions, the solution seemed to be creating a Dirichlet distribution and sampling points from that distribution.
Can this problem be solved through uniformly sampling points on a simplex represented as a Dirichlet distribution?
