Using interior point solvers for optimizing progressive lens models with spherical coordinates
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Using interior point solvers for optimizing progressive lens models with spherical coordinates Glòria Casanellas1 · Jordi Castro2 Received: 17 May 2019 / Revised: 10 December 2019 / Accepted: 11 December 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract 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. Keywords Nonlinear optimization · Interior point methods · Optical lens design · Progressive lenses · Optimization industry applications
* Jordi Castro [email protected] Glòria Casanellas [email protected] 1
Horizons Optical, Av. Alcalde Barnils 72, 08174 Sant Cugat del Vallès, Spain
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Department of Statistics and Operations Research, Universitat Politècnica de Catalunya, Edifici C5, Planta 2, Campus Nord, C. Jordi Girona 1–3, 08034 Barcelona, Catalonia
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1 Introduction Presbyopia is the gradual inability of the eyes to focus on near objects. It appears among people in their forties, who thus require lenses to correct their near vision. Progressive lenses correct presbyopia and their design is complex: an upper region for far vision (far region); the corridor for middle vision; and the low region for near vision (near region). The different parts of a progressive lens surface are shown in Fig. 1 (left). The two main properties of a progressive lens are the power and the astigmatism (whose formulas will be provided in the next section), which are defined at each point on the lens. From now on, by power we will be referring to surface power, or, more precisely, to mean surface power. Similarly, by astigmatism we will be referring to surface astigmatism. In geometrical terms, the optical power is the product of the me
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