Generating numbers to be averaged efficiently

Question

During my research, I had encountered the following issue. Consider the equation $$\sum_{k=1}^K\alpha_k P_k = P$$

where $\sum_{k=1}^K\alpha_k=1$, $\alpha_k > 0$, $P_k > 0, P>0$.

So you are given $\{\alpha_k\}$, $K$ and $P$. The task is to generate randomly $\{P_k\}$ such that the above equation is satisfied. I am looking for an efficient way to generate these numbers.

My strategy

I have one method. Basically, we generate $K$ distinct uniform numbers between $(0,1)$. Call them $\{b_k\}$. Then we compute $\sum_{k=1}^K\alpha_k b_k = B$. Now the sequence $\{\frac{b_kP}{B}\}$ is the desired sequence.

However, if you observe, using this method, $P_k$ can never exceed $P$ and in my work, I require some of the values to exceed $P$. E.g. if $\alpha_1=0.5, \alpha_2=0.5$, even $(2P,0)$ is a valid solution.

Is there a way to generate these numbers so as to circumvent this issue?

Update: Given $\{\alpha_k\}$, it follows that $P_k \leq P/\alpha_{\min}$. Note: The above counter example is not valid. Apparently the algorithm works fine.

Answer 1

A reasonable solution is to first generate a random point in the simplex $$ x_1 + \cdots + x_K = 1, \qquad x_1,\ldots,x_n \geq 0. $$ Check out this question for several ways to do this. Then take $$ P_i = \frac{P}{\alpha_i} x_i. $$ This should give you a uniformly random point in the geometric body you are interested in $$ \sum_{i=1}^K \alpha_i P_i = P. $$