  This research studies a non-convex geometric optimization problem arising
from the field of optical wireless power transfer. In the considered
optimization problem, the cost function is a sum of negatively and fractionally
powered distances from given points arbitrarily located in a plane to another
point belonging to a different plane. Therefore, it is a strongly nonlinear and
non-convex programming, hence posing a challenge on the characterization of its
optimizer set, especially its set of global optimizers. To tackle this
challenge, the bifurcation theory is employed to investigate the continuation
and bifurcation structures of the Hessian matrix of the cost function. As such,
two main results are derived. First, there is a critical distance between the
two considered planes such that beyond which a unique global optimizer exists.
Second, the exact number of maximizers is locally derived by the number of
bifurcation branches determined via one-dimensional isotropic subgroups of a
Lie group acting on $\mathbb{R}^2$, when the inter-plane distance is smaller
than the above-mentioned critical distance. Consequently, numerical simulations
and computations of bifurcation points are carried out for various
configurations of the given points, whose results confirm the derived
theoretical outcomes.
