Set $\{\alpha_i\}$ and $\{\beta_i\}$ as the variables, the optimization problem I need to solve is as follows: \begin{equation} \begin{split} \min\max_{i}\quad &\frac{(a_i^2+b_i^2+c_i^2+d_i^2)^2}{(a_id_i-b_ic_i)^2} \end{split} \end{equation} where \begin{equation} \begin{split} a_i = \sum_{j=1}^{M}l_{ij}\alpha_i,\quad b_i = \sum_{j=1}^{M}h_{ij}\alpha_i,\quad c_i = \sum_{j=1}^{M}l_{ij}\beta_i\quad d_i = \sum_{j=1}^{M}h_{ij}\beta_i \end{split} \end{equation} $\{l_{ij}\}$ and $\{h_{ij}\}$ are the given coefficients. By introducing the auxiliary variable t, we can solve the following equivalent problem: \begin{equation} \begin{split} \min\quad &t\\ \mathrm{s.t.} \quad &(a_i^2+b_i^2+c_i^2+d_i^2)^2-t(a_id_i-b_ic_i)^2\leq 0,\quad i=1,2,\cdots,N \end{split} \end{equation}
I try to solve this problem using the function "fmincon" in MATLAB and select the "active-set" algorithm. But it takes a long time for each iteration and the Max constraint is not zero. I don't know why? Is there any other software to solve this problem?