2 of 5
edited title

# How to find the vertices of the set $\{v_i\in \mathbb{R}:a_1\ge v_1\ge v_2\ge \cdots\ge v_n\ge 0,\ q_2\le \sum_{i=1}^n p_iv_i\le q_1\}$?

I am given a set of inequalities $$v_1\ge v_2\ge \cdots\ge v_n\ge 0$$, $$q_2\le \sum_{i=1}^n p_iv_i\le q_1$$, with $$q_1,q_2$$ positive reals, and only one bound for the coordinates: $$v_1\le a_1$$, where $$a_1\ge 0$$. My question is:

What are the vertices of the polytope created by these set of constraints?

I could easily find the vertices when $$a_1=\infty$$, by equating $$k$$ of the variables, at a time, for $$1\le k\le n$$, and finding the solution from the second inequality. However, I am not sure how to incorporate the bound on $$v_1$$ to find the vertices. I suspect there will be more vertices than the ones found when there is no upper bound on $$v_1$$, similar to this problem here. Can anyone kindly give some pointers regarding how to find the vertices? Also, please refer to some relevant literature. Thanks in advance.