I have a row vector as $P=(p_1,p_2,p_3)$ which should be obtained from the following constraints:
- $a_1p_1 + a_2p_2 + a_3p_3 = b_1p_1 + b_2p_2 + b_3p_3$ ($a_i$'s and $b_i$'s are known)
- $p_1 + p_2 + p_3 = 1$
- $p_i\geq 0,\ i=1,2,3.$
Furthermore, there may be more than one P vector satisfying the above constraints. Then, I want to use each P vector in the following equations to plot (x,y) in a 2-dimensional space.
- $x = p_1x_1 + p_2x_2 + p_3x_3$
- $y = p_1y_1 + p_2y_2 + p_3y_3$ (Both $x_i$'s and $y_i$'s are known)
I would be very thankful if anyone has any idea!
Thank you very much for your help! Your codes gave so many ideas to me since I'm a beginner in Mathematica.
But the point is that putting $p_i=0$ at every step and finding the corresponding 2-dimensional line is not always correct because there may be some solutions whose $p_i$'s are not equal to 0.
Maybe, I did not explain properly about my problem. Here is what I'm looking for: I have the following set of constraints:
- $P.C>=0$, where $P=(p_1,p_2,p_3)$ and $C=(c_1,c_2,c_3)$ and $C$ is known.
- $P.D>=0$, where $P=(p_1,p_2,p_3)$ and $D=(d_1,d_2,d_3)$ and $D$ is known.
- $P.E>=0$, where $P=(p_1,p_2,p_3)$ and $E=(e_1,e_2,e_3)$ and $E$ is known.
- $p_1+p_2+p_3=1$ (* edited *)
- $p_i>=0$ for all $i$
As I said before, there may be several $P$ vectors satisfying the above constraints. Furthermore, there may be non-negative $P$ vectors, e.g. $P=(1/3,1/3,1/3)$ that satisfies the above constraints. After obtaining all such $P$ vectors, based on them, I want to plot the following x-y variables in 2 dimensions:
Furthermore, the way my problem is modeled, I am sure that I have a line in 2D. I tried so many different methods found in internet, but unfortunately, none of them worked.
Again, I would be very grateful if anyone help me in this case!