A Minimax Optimization Approach to the Spherical Codes Problem

A spherical code with parameter (n, N, γ) is a set of N points on the unit sphere in n dimension for which the dot product of unit vectors from the origin to any two points is less than or equal to γ. If each point on the unit sphere is treated as an electron, one approach to this spherical code problem is to define an electrostatic energy of electrons and to optimize for minimum energy. A configuration with minimum energy corresponds to a useful spherical code, but may not be the global minimum solution. Thus, minimizing the energy function does not always minimize y. A global solution for the spherical code problem for most of parameters (n, N) is unknown, especially for spherical codes in higher dimension. In this paper, a minimax optimization approach for obtaining spherical codes is analyzed. In this paper, four algorithms are proposed to solve the spherical code problem. The first algorithm is implemented using linear programming with trust region. The second is based on the introduction of the energy function. The third algorithm is based on minimizing the maximum of the dot product of any pairs of unit vectors and the last algorithm is based on the Groebner Bases. By running simulation with these algorithms, configurations for many of the best known spherical codes found in literature were obtained.