I am trying to solve a minimization mathematical model toward global optimality. The objective function is the sum of some ratios. The numerator of each fraction is quadratic, and the denominator is linear as follows
$ Min\ \sum_k\left(\frac{\sum_i\left(P_{ik}^{\left(1\right)}+P_{ik}^{\left(2\right)}\right).\sum_i\left({\tau{}}^{(1)}P_{ik}^{\left(1\right)}+{\tau{}}^{(2)}P_{ik}^{\left(2\right)}\right)\ \ \ }{2h_k\left(h_k-\sum_iP_{ik}^{\left(1\right)}-\sum_iP_{ik}^{\left(2\right)}\right)}\right)$
$\sum_kP_{ik}^{\left(1\right)}=r_i^{\left(1\right)},\forall{}i$
$\sum_kP_{ik}^{\left(2\right)}=r_i^{\left(2\right)},\ \forall{}i$
$\sum_iP_{ik}^{\left(1\right)}+\sum_iP_{ik}^{\left(2\right)}\leq{}h_k,\forall{}k$
$P_{ik}^{\left(1\right)},\ P_{ik}^{\left(2\right)}\geq{}0,\ \forall{},i,k$
All constraints are linear. My perception is that the objective function is not convex as it is the summation of some quasi-convex functions. The number of variables of each type is around 20. Therefore, an optimization software product like LINGO, MAPLE etc. can solve the model fairly quickly but the global optimality is not guaranteed.
Do you have any idea how we can convincingly solve the model to reach the optimal global solution or at least ensure our solutions are close to it?
HINT
Sounds like each term of the sum is independent of the others since it uses different variables. So let's minimize term-by-term. Let $$ f(x,y) = \frac{ax^2+(a+b)x y+b y^2}{2c_k(c_k-x-y)} $$ Can you find the extrema of this 2D function using the usual techniques from multi-variable calculus? e.g., solve $\vec{\nabla} f = \vec{0}$?