I would like to determine whether there exists a solution (and if so, check uniqueness) to the following system of linear equations (with respect to $\eta = (\eta_1,...,\eta_J))$:

$$\begin{aligned} \sum_{j=1}^J (\psi_i -y_j)a_{ij}\eta_j &= 0, \hspace{1em} \forall i=1,...,I, \\ \sum_{j=1}^J \eta_j &= 1, \\ \eta_j &\geq 0, \hspace{1em} \forall j=1,...,J \end{aligned},$$

where there are the following constraints:

  • $a_{ij} \in [0,1] \hspace{1em} \forall i,j$
  • if we rearrange y's in an increasing order $y_1 < y_2 < ... < y_J$, then the following inequalities hold $y_1 \leq \psi_i \leq y_J, \hspace{1em} \forall i=1,...,I$.

Is there a way to determine whether solution exists in general case for any $I$ and $J$? And if there are solutions - how to determine uniqueness?

I solved the most simple case for $I=J=2$. I "deleted" the $I$th equation and solved the following system:

$$\begin{aligned}(\psi_1 - y_1)a_{11}\eta_1 + (\psi_1-y_2)a_{12}\eta_2 &= 0, \\ \eta_1+\eta_2 &= 1, \\ \eta_1 &\geq 0, \\ \eta_2 &\geq 0. \end{aligned}$$

Assuming without loss of geenrality that $y_1 < y_2$ we get the following result: $$\begin{aligned}\eta_1 &= \frac{a_{12}(y_2-\psi_1)}{a_{11}(\psi_1-y_1) + a_{12}(y_2-\psi_1)}, \\ \eta_2 &= \frac{a_{11}(\psi_1-y_1)}{a_{11}(\psi_1-y_1) + a_{12}(y_2-\psi_1)}. \end{aligned}$$

$\eta$'s must be nonnegative, so this implies that either of the following two conditions must hold:

  • $a_{12}(y_2-\psi_1) \geq 0$, $a_{11}(\psi_1-y_1) \geq 0$, and $a_{11}(\psi_1-y_1) + a_{12}(y_2-\psi_1) \geq 0 ;$

  • $a_{12}(y_2-\psi_1) \leq 0$, $a_{11}(\psi_1-y_1) \leq 0$, and $a_{11}(\psi_1-y_1) + a_{12}(y_2-\psi_1) \leq 0 ;$

which due to the constraints on $a_{ij}$'s lead to the equivalent conditions:

  • $y_1 \leq \psi_1 \leq y_2$ and $a_{11}(\psi_1 - y_1) + a_{12}(y_2-\psi_1) \geq 0$;

  • $y_2 \leq \psi_1 \leq y_1$ and $a_{11}(\psi_1 - y_1) + a_{12}(y_2-\psi_1) \leq 0$.

And now it is easily seen that the first case occurs when $y_1 \leq \psi_1 \leq y_2$ and the second case cannot occur, since $y_2 > y_1$. If this solution does not satisfy the "deleted" equation, then there is no solution. But if it does, then the solution is unique.

  • 1
    $\begingroup$ The dash in “the following inequalities hold - $y_1≤ψ_i≤y_J,\quad∀i=1,\ldots,I$” is somewhat misleading. $\endgroup$
    – Ѕааԁ
    Jul 18 '21 at 14:17
  • $\begingroup$ @Saad Right, I edited the question, it was not a minus sign. Thanks. $\endgroup$
    – treskov
    Jul 18 '21 at 18:18

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