Designing progressive lenses is a complex problem that has been previously solved by formulating an optimization model based on Cartesian coordinates. In this work a new progressive lens model using spherical coordinates is presented, 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 gave rise to nonlinear optimization problems of about 900 variables and 15000 constraints. Each constraint corresponds to a point of the grid used to define the lens surface. The number of variables depends on the precision of a B-spline basis used for the representation of the surface, and the number of constraints depends on the shape and quality of the design. We present results of progressive lenses obtained using the AMPL modeling language and the nonlinear interior point solvers IPOPT, LOQO and KNITRO. Computational results are reported, as well as some examples of real-world progressive lenses calculated using this new model. Progressive lenses obtained are competitive in terms of quality with those resulting from previous models that are used in commercial glasses.
Glòria Casanellas, Jordi Castro, Using interior point solvers for optimizing progressive lens models with spherical coordinates, Research Report DR 2019/01, Dept. of Statistics and Operations Research, Universitat Politècnica de Catalunya, 2019