Strong convexity and smoothness of solutions to geometric optics problems via dynamic programming

In this letter, a formulation of Fermat's principle as an optimization problem over a finite number of stages is used to prove strong convexity and smoothness of the solutions to certain geometric optics problems. The class of problems considered in this letter consists in the determination of the trajectory followed by a ray between a light source and a final fixed point, separated by a finite number of layered homogeneous media, characterized by their refractive indices. To obtain the theoretical results, a dynamic programming argument is used in the analysis. Then, strong convexity and smoothness are exploited to get an error bound valid when an exact solution is replaced by an approximate solution. Numerical results are provided to validate the theoretical achievements.