I have constants $\sigma$, $\mu$, and $q_0$; a boolean matrix $\alpha$; and vectors $A, \beta, r, d,$ and $Q$, all quantities non-negative. In the following equation,

$$0=\sum_j \alpha_{kj} \Bigg[ \beta_j r_j \frac{\ln (A_j - r_j(\alpha_{kj}Q_k + \sum_{i\neq k} \alpha_{ij} Q_i)) + \sigma^2 - \mu}{\sigma^2 (A_j - r_j(\alpha_{kj}Q_k + \sum_{i\neq k} \alpha_{ij} Q_i))} + (1-\beta_j)\frac{\ln(1-d_j)}{q_0} \Bigg]$$

how do I solve for $Q_k$? Alternatively I want to solve for the vector entire $Q$ vector, but I assume this is the easier path.

If it is not possible to solve for $Q$, why not? Is there a way to numerically approximate it?

I'm trying to obtain the maximum likelihood estimate of the parameters for a model I'm building. I have constants $\sigma$, $\mu$, and $q_0$; a boolean matrix $\alpha$; and vectors $A, \beta, r, d,$ and $Q$, all quantities non-negative. I set the derivative of the log likelihood function equal to zero with

$$0=\sum_j \alpha_{kj} \Bigg[ \beta_j r_j \frac{\ln (A_j - r_j\sum_i \alpha_{ij} Q_i) + \sigma^2 - \mu}{\sigma^2 (A_j - r_j\sum_i \alpha_{ij} Q_i)} + (1-\beta_j)\frac{\ln(1-d_j)}{q_0} \Bigg]$$

How can I solve for $Q$?

I have constants $\sigma$, $\mu$, and $q_0$; a matrix $\alpha$; and vectors $A, \beta, r, d,$ and $Q$. In the following equation,

$$0=\sum_j \alpha_{kj} \Bigg[ \beta_j r_j \frac{\ln (A_j - r_j(\alpha_{kj}Q_k + \sum_{i\neq k} \alpha_{ij} Q_i)) + \sigma^2 - \mu}{\sigma^2 (A_j - r_j(\alpha_{kj}Q_k + \sum_{i\neq k} \alpha_{ij} Q_i))} + (1-\beta_j)\frac{\ln(1-d_j)}{q_0} \Bigg]$$

how do I solve for $Q_k$? Alternatively I want to solve for the vector entire $Q$ vector, but I assume this is the easier path.

If it is not possible to solve for $Q$, why not? Is there a way to numerically approximate it?

I have constants $\sigma$, $\mu$, and $q_0$; a boolean matrix $\alpha$; and vectors $A, \beta, r, d,$ and $Q$, all quantities non-negative. In the following equation,

$$0=\sum_j \alpha_{kj} \Bigg[ \beta_j r_j \frac{\ln (A_j - r_j(\alpha_{kj}Q_k + \sum_{i\neq k} \alpha_{ij} Q_i)) + \sigma^2 - \mu}{\sigma^2 (A_j - r_j(\alpha_{kj}Q_k + \sum_{i\neq k} \alpha_{ij} Q_i))} + (1-\beta_j)\frac{\ln(1-d_j)}{q_0} \Bigg]$$

how do I solve for $Q_k$? Alternatively I want to solve for the vector entire $Q$ vector, but I assume this is the easier path.

If it is not possible to solve for $Q$, why not? Is there a way to numerically approximate it?