Assume we have a vector $p=[p_1,p_2,...,p_n]$, where $0<p_1<p_2<...<p_n$. We now define $\Delta_p$ of size $n-1$ as $\Delta_p=[p_2-p_1,...,p_n-p_{n-1}]$. Note the $\Delta_p>0$.
The solution for this problem: $b\cdot\Delta_p = 1$ is pretty simple: $b=\Delta_p/||{\Delta_p}||^2$.
I would like to solve the same problem, but with the following constraint: $b=[b_1,b_2,...,b_{n-1}]$ is a valid solution if and only if $b\cdot\Delta_p = 1$ and $0<b_1<b_2<...<b_{n-1}$.
For example: Let $p=[1,2,3]$, thus $\Delta_p=[1,1]$. The non-constrained solution is: $b=\Delta_p/||{\Delta_p}||^2=[0.5,0.5]$, however, it is not valid under the constraint. A valid solution would be, for instance, $b=[0.4, 0.6]$ or any other $b_1,b_2$ s.t. $0<b_1<b_2$ and $b_1+b_2=1$.
Any suggestions?
It seems OK to take $v/(v\cdot \Delta_p)$ where $v=[1,2,\ldots,n-1]$ (provided $n\geq2$).