Using interior point solvers for optimizing progressive lens models with spherical coordinates

Designing progressive lenses is a complex problem that has been previously solved by formulating an optimization model based on Cartesian coordinates. This work presents a new progressive lens model using spherical coordinates, and interior point solvers are used to solve this new optimization model. Although this results in a highly nonlinear, nonconvex, continuous optimization problem, the new spherical coordinates model exhibits better convexity properties compared to previous ones based on Cartesian coordinates. The real-world instances considered result in nonlinear optimization problems of about 900 variables and 15,000 constraints. Each constraint corresponds to a point on the grid that defines the lens surface. The number of variables depends on the precision of the B-spline basis used for representing the surface; and the number of constraints depends on the shape and quality of the design. We present our results for progressive lenses, which were obtained using the AMPL modeling language and the nonlinear interior point solvers IPOPT, LOQO and KNITRO. The computational results are reported, as well as some examples of real-world progressive lenses that were calculated using this new model. In terms of quality, the progressive lenses obtained by our model are competitive with those of previous models used for commercial eyeglasses.