Analytical existence of solutions to a system of nonlinear equations with application

We study the existence of analytical solutions to a system of nonlinear equations under constraints linked to the analysis of a road safety measure without computing second derivatives. We formally demonstrate this existence of solutions by applying a matrix inversion principle through Schur complement to a subsystem of equations derived from the main system. The analytical results thus obtained are used to construct a simple iterative procedure to look for optimal solutions as well as an initial solution adapted to data of each case study. We illustrate our results with simulated accident data obtained from the setting-up of a road safety measure. The numerical solutions thus obtained are then compared to those given through a classic Newton-Raphson type approach directly applied to the main system.