I am not a mathematica user. I need to find the solution to a given problem, which we already described there for theoretical analysis. I'll rewrite here the problem:
Let $\mathbf a = (a_1,...,a_d)$ is a constant in $\mathbb R_{+}^d \setminus \{\mathbf 0\}$ (some $a_i$ can be $0$ but not all, all are $\ge 0$).
Define the set $$S(\mathbf a) = \left\{\mathbf x \in \mathbb C^d:\; \sum_{i=1}^d \frac{x_i +1}{x_i -1}a_i = 1 \right\}.$$
Define the distance of the set to zero as :
$$d(\mathbf a) = \min\limits_{\mathbf x \in S(a)} \sum_{i=1}^d \lvert x_i \rvert$$
We found out that this distance is always greater than $1$, and I am wandering if there could be a simple function that computes it from $\mathbf a$.
When $d = 1$, we already solved the problem by hand.
If $d=2$ and for particular values of $a_1,a_2$, wolfram alpha finds a theoretical minimum with the following command:
Minimize[{Abs[x] + Abs[y], (3 (1 + x))/(-1 + x) + (2 (1 + y))/(-1 + y) == 1}, {x, y}]
wich produces the same output in a mathematica kernel.
How can i generalize this command to tell mathematica to find the function $d(\mathbf a)$ for me ? (let's say for a given dimension $d$, i can run it once for each dimension if needed). I want it to consider $\mathbf a$ as a nuisance parameter and give me a minimum that depends on $\mathbf a$.
Minimize
works over the reals. So do you want to minimize over the real or the complex numbers? I supposeMinimize
accidentally works in the example, because the solution turns out to be real. $\endgroup$Minimize
code was not good, i know. $\endgroup$