constrained optimization including sum of two upper incomplete gamma function in both fitness function and constraint

68 Views Asked by At

i'm trying to solve this constrained optimization problem
the constraint is $$\zeta=\frac {\Gamma \left(M,\frac {\lambda}{\left |\sum_{i=1}^Nw_i*h_{ei} \right|^2 + \sum_{i=1}^N\left |w_i \right|^2} \right)}{\Gamma(M)}*\beta +\frac {\Gamma \left(M,\frac {\lambda}{\sum_{i=1}^N \left |w_i \right|^2} \right)}{\Gamma(M)}*(1-\beta)$$
and the fitness function is
$$P_d=\frac {\Gamma \left(M,\frac {\lambda}{\left |\sum_{i=1}^Nw_i*h_{pi} \right|^2+\left |\sum_{i=1}^Nw_i*h_{ei} \right|^2 + \sum_{i=1}^N\left |w_i \right|^2} \right)}{\Gamma(M)}*\alpha +\frac {\Gamma \left(M,\frac {\lambda}{\left |\sum_{i=1}^Nw_i*h_{pi} \right|^2 + \sum_{i=1}^N\left |w_i \right|^2} \right)}{\Gamma(M)}*(1-\alpha)$$
where $\Gamma(M,...)$ is upper incomplete gamma function. it is notable that $h_{pi}$ and $w_i$ are complex variables
in the above equations $\zeta$, $M$, $N$, $\beta$, $\alpha$, $\lambda$, $h_{pi}$ for(i=1,...,N) and $h_{ei}$ for(i=1,...,N) are known.
the parameters of our optimization problem are $\lambda$ and $w_i$ for (i=1,...,N) and the fitness function is $P_d$
this problem is from my telecommunication model
pleas guide me to solve this constrained optimization problem