I am trying to setup an optimization problem with equality and inequality constraints. I want to estimate a specific variable call $\chi$ subject to a minimisation problem -- guidance on the setup and next step to solve it would be helpful. Does it make sense to set the problem up like this? Does a solution exist? Also in case the KKT conditions do not hold how could I remedy that (i.e. could one change the setup)?
\begin{align*} \min_{j,q,b,u} \chi = \frac{1}{u\lambda} * j - 1 \quad \text{s.t.} \\[0.5cm] \text{s.t.}\quad j + q - \epsilon\kappa b &= \alpha u \\ \dfrac{\phi- \theta_x}{\theta_y - \theta_x} b - \dfrac{\theta_x}{\theta_y- \theta_x} (j + q) - \dfrac{1- \theta_x}{\theta_y - \theta_x} &\geq u \\ j+ q - (1 + \kappa)b &\leq \frac{1}{\delta } \\ \dfrac{1}{\tau( \gamma+ \omega\kappa)} \geq b\\ \end{align*}
What I know is that all my greek parameters should be between 0 and 1, $\theta_y < \theta_x$ and that all variables are positive.