Analysis of Optima Set in A Class of Non-Convex Geometric Optimization Problems Using Bifurcation Theory

The current research studies a class of non-convex geometric optimization problems in which 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. This constitutes a class of 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, a critical distance between the two planes of points is determined, 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 $${\mathbb {R}}^2$$ 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.