I'm not a math expert so sorry for possible trivial questions.
I have written this mixed integer nonlinear program (MINLP): $ \begin{align} \min & \sum_{i \in \mathcal{I}}{\left(\alpha_i+\beta_i\sum_{j \in \mathcal{J}}{z_{ij}^{-1}}+\eta_i\left(\sum_{j \in \mathcal{J}}{z_{ij}^{-1}}\right)^{\gamma_i}\right)x_i} + \sum_{i \in \mathcal{I}}{\sum_{j \in \mathcal{J}}{\delta_{ji}y_{ij}}} + \sum_{i \in \mathcal{I}}{\sum_{j \in \mathcal{J}}{\left(\frac{z_{ij}}{\zeta_i}-1\right)}y_{ij}}\\ \text{subject to} & \notag \\ & \sum_{j \in \mathcal{J}} z_{ij} \le Z_i,\quad i \in \mathcal{I} \\ & \sum_{i \in \mathcal{I}} y_{ij} = 1,\quad j \in \mathcal{J} \\ & y_{ij} \le x_i,\quad i \in \mathcal{I},j \in \mathcal{J} \\ & x_i \in \left\{0,1\right\},\quad i \in \mathcal{I} \\ & y_{ij} \in \left\{0,1\right\},\quad i \in \mathcal{I},j \in \mathcal{J} \\ & z_{ij} \in [0,1]\quad i \in \mathcal{I},j \in \mathcal{J} \\ \text{where} & \notag \\ & \alpha_i,\beta_i,\eta_i \in \mathbb{R},\quad i \in \mathcal{I} \\ & \zeta_i \in \mathbb{R}\setminus\{0\},\quad i \in \mathcal{I} \\ & \delta_{ji} \in \mathbb{R},\quad i \in \mathcal{I},j \in \mathcal{J} \\ & \gamma_i \ge 0\quad i \in \mathcal{I} \\ \end{align} $
and now I want to solve. My decision variables are $x_i$, $y_{ij}$, and $z_{ij}$. The other terms are constants.
I really appreciate if someone can guide me in solving it. I've read somewhere that the first step I should perform is a convexity test on objective and constraint functions. I have to compute the Hessian of each function, but how to do it? Then, what next?
Is there any (possibly free) tool that is able to automatically solve this problem for me?
Is a there a good introductory book where I can start?
Thank you very much in advance!