# Overdetermined system of simple nonlinear equations

Let $$I$$, $$J$$ and $$K$$ denote index sets. There are two kinds of variables $$x_{j,k} \in [0, 1]$$ and $$y_{i,k} \in [0, 8760]$$ and two kinds of constants $$a_{i,j,k} \geq 0$$ and $$b_j \geq 0$$. I'm considering a system of $$(|I| \cdot |K| + 1) \cdot |J|$$ equations \begin{align} &0 = x_{j,k} \cdot b_j \cdot y_{i,k} - a_{i,j,k}\\[2mm] &0 = \sum_{k \in K}{x_{j,k}} - 1 \end{align} in $$(|I| + |J|) \cdot |K|$$ unknowns. I assume $$|I|, |J| \geq 2$$ such that the system is always overdetermined for all $$|K|$$. As the system is rather simple I was wondering whether

1. there exists a unique solution
2. the system is analytically tractable

For the first part, for the case when $$b_j =1, a_{i,j,k} = 0 \,\, \forall i,j,k$$, we can have multiple solutions. Moreover, when $$\exists i,j,k \,\,\,\, b_j = 0, a_{i,j,k} > 0$$, there are no solutions.

As for tractability, we need to do a few cases

Case 1: If there exists a $$j$$ such that $$b_j =0$$ and $$a_{i,k,j} > 0$$ for any $$i,k$$ then there is no solution that satisfies all the constraints.

Case 2: In case $$b_j =0$$, we have $$a_{i,j,k} = 0$$ for all $$i,k$$. In this case, neglect the equations with $$b_j=0$$.

If there is a $$j$$ such that $$\exists i,k \,\, a_{i,j,k}/b_j > 8760$$, there is no solution.

Let $$c_{i,j,k} = a_{i,j,k}/b_j$$

Hence, we have that $$x_{j,k} = c_{i,j,k}/y_{i,k}$$ from the first set of constraints.

Now, we can use the second set of constraints to solve for $$y_{i,k}$$. The problem can be converted to to solving set of linear equations with variables in the range $$[1, \infty)$$

• Regarding your second to last sentence I would have come up with $y_{i,k} = c_{i,j,k} / (1- \sum_{\ell \in K \setminus \{k\}}{x_{j,\ell}}) = c_{i,j,k} / (1- \sum_{\ell \in K \setminus \{k\}}{c_{i,j,\ell} / y_{i,\ell}})$. But I don't know about the last sentence. Would you mind stating the set of linear equations? Commented Sep 9, 2022 at 7:20
• Let $z_{i,k} = 8760/y_{i,k}$. First set is, $c_{i,j,k} - c_{i,j',k} = 0$. Second is, $\sum_{k\in K} c_{i,j,k}z_{i,k} = 8760$ Commented Sep 9, 2022 at 15:51