I want to solve the optimization problem below numerically using GAMS. Given the formulation, I was wondering if there are suitable ways of transforming and or approximating the given function to simplify/ accelerate the optimization, which is large scale. Furthermore, can one know up front if this problem has a global optimum?
Note: I'm already making straightforward approximations, for instance ignoring terms where $d_{ijm}$ is close to zero.
EDIT: several notation issues and clarifications. The 0 subscript stands for "observed quantity", i.e. its value is known from some data source and constant.
$$\min_{o,b} \sum_{i,m} (\pi_{im} - \pi_{im,0})^2 + \sum_{j,m} (\pi_{jm} - \pi_{jm,0})^2$$
Where
$$\pi_{ijm} = \frac{d_{ijm} \cdot b_{im} \cdot o_{jm}}{\sum_{rst} (d_{rst} \cdot b_{rt} \cdot o_{st})} $$
And
$$\pi_{im} = \sum_j \pi_{ijm} \\ \pi_{jm} = \sum_i \pi_{ijm} $$
Furthermore, for all $i,j,m$
$$ o_{jm},b_{im} \geq 0, 0 < d_{ijm} \leq 1$$
In the formulation above, $\pi_{ijm,0}$ and $d_{ijm}$ are known, constant and nonzero.
Context: this is a transportation problem, i and j are residence and work neighborhoods, m is a mode of transport. $d_{ijm}$ is the travel cost in terms of travel time for the triplet ijm, with d going to zero as time decreases.
There are 1e4 locations, each one doubling as residence (i) and work (j). There are 4 modes m.
$\pi_{ijm,0}$ is the probability of a person choosing ijm observed in reality using data. Across the ijm domain, most $\pi_{ijm,0}$ are zero.
$\sum_i \pi_{ijm}$ represents the probability of living in i and using mode m, similar for j and work. In the optimization, I want to bring these sums as closely as possible to their observed counterparts.
$b_{im}$ represents the attractiveness of choosing i to live and mode m, similarly for $o_{jm}$ and work.