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Optimization Methods for the Design of Progressive Lenses Optimization methods for the design of progressive lenses Glòria Casanellas Peñalver ADVERTIMENT La consulta d’aquesta tesi queda condicionada a l’acceptació de les següents condicions d'ús: La difusió d’aquesta tesi per mitjà del r e p o s i t o r i i n s t i t u c i o n a l UPCommons (http://upcommons.upc.edu/tesis) i el repositori cooperatiu TDX ( h t t p : / / w w w . t d x . c a t / ) ha estat autoritzada pels titulars dels drets de propietat intel·lectual únicament per a usos privats emmarcats en activitats d’investigació i docència. No s’autoritza la seva reproducció amb finalitats de lucre ni la seva difusió i posada a disposició des d’un lloc aliè al servei UPCommons o TDX. No s’autoritza la presentació del seu contingut en una finestra o marc aliè a UPCommons (framing). Aquesta reserva de drets afecta tant al resum de presentació de la tesi com als seus continguts. En la utilització o cita de parts de la tesi és obligat indicar el nom de la persona autora. ADVERTENCIA La consulta de esta tesis queda condicionada a la aceptación de las siguientes condiciones de uso: La difusión de esta tesis por medio del repositorio institucional UPCommons (http://upcommons.upc.edu/tesis) y el repositorio cooperativo TDR (http://www.tdx.cat/?locale- attribute=es) ha sido autorizada por los titulares de los derechos de propiedad intelectual únicamente para usos privados enmarcados en actividades de investigación y docencia. No se autoriza su reproducción con finalidades de lucro ni su difusión y puesta a disposición desde un sitio ajeno al servicio UPCommons No se autoriza la presentación de su contenido en una ventana o marco ajeno a UPCommons (framing). Esta reserva de derechos afecta tanto al resumen de presentación de la tesis como a sus contenidos. En la utilización o cita de partes de la tesis es obligado indicar el nombre de la persona autora. WARNING On having consulted this thesis you’re accepting the following use conditions: Spreading this thesis by the i n s t i t u t i o n a l r e p o s i t o r y UPCommons (http://upcommons.upc.edu/tesis) and the cooperative repository TDX (http://www.tdx.cat/?locale- attribute=en) has been authorized by the titular of the intellectual property rights only for private uses placed in investigation and teaching activities. Reproduction with lucrative aims is not authorized neither its spreading nor availability from a site foreign to the UPCommons service. Introducing its content in a window or frame foreign to the UPCommons service is not authorized (framing). These rights affect to the presentation summary of the thesis as well as to its contents. In the using or citation of parts of the thesis it’s obliged to indicate the name of the author. Optimization methods for the design of progressive lenses Author: Gl`oriaCasanellas Pe~nalver A thesis submitted for the degree of Doctor of Philosophy Supervisor: Jordi Castro P´erez Department of Statistics and Operations Research Universitat Polit`ecnicade Catalunya - BarcelonaTech Barcelona 2020 Amb el suport de la Secretaria d'Universitats i Recerca del Departament d'Economia i Coneixement de la Generalitat de Catalunya. Abstract This work is the result of an Industrial Doctorate developed through a partnership agree- ment between the Universitat Polit`ecnicade Catalunya and the company Horizons Op- tical. This thesis solves the complex design of progressive lenses for eyeglasses, which is a real problem in the industry. The lens is the transparent part of the eye behind the pupil that helps humans to see clearly by focusing light onto the retina. Over time, the lens loses some of its elasticity and therefore can no longer accommodate clearly for near vision. This phenomenon is called presbyopia and explains why people need reading glasses as they become older. Progressive lenses correct presbyopia and have a complex design: they have an upper region for far vision, a low region for near vision (reading), and a corridor that connects these areas which allows clearly vision at an intermediate distance, for example, when looking at a computer screen. The surface of the progressive lens designed in this thesis is the surface that is farthest from the eye, thus the power in the near region is bigger than the power in the far region. In geometrical terms, power and astigmatism are calculated using the principal curva- tures of the lens surface. When the power changes vertically, unwanted lateral astigmatism (aberrations) appear as a result of the Minkwitz theorem. The focus of this thesis is the use of optimization methods in order to design progressive lenses minimizing the lateral aberrations (astigmatism) and providing the power required in each zone. This thesis presents two different models for computing progressive lens. Both models are highly nonlinear, nonconvex and continuous and were solved using the AMPL mod- eling language and the interior point solvers IPOPT, LOQO and KNITRO. Both models have approximately 900 variables (the coefficients of a third-degree B-spline basis). The first model has about 7000 constraints, while the second model has about 15000 con- straints. Each constraint corresponds to a property of power or astigmatism at a point on the grid that defines the lens surface. The first model uses Cartesian coordinates and is an improved version of a previous model by the same author, published in a master's thesis. The CPU time in the master thesis was between 10 and 33 minutes, and in this thesis it has been reduced to less than 3 minutes using the same machine and the LOQO solver. In this thesis, all of the proposed instances converged using the LOQO solver and the Cartesian coordinate model, which was not the case in the master's thesis. However, with other solvers some of the instances did not converge using the Cartesian coordinate model of this thesis. The second model uses spherical coordinates and exhibits better convexity properties than the previous one based on Cartesian coordinates. All of the problem instances con- verged using all the proposed solvers, and the quality of the solution was improved. CPU time for spherical coordinates increased in relation to the Cartesian coordinate model, due to large calculations involved, but the number of iterations needed to converge de- creased considerably (for example, from a maximum of 192 iterations using the Cartesian coordinate model to a maximum of 84 iterations using the spherical coordinate model and i the same LOQO solver). These models resulted in two publications. The first one is a patent for an invention that uses the Cartesian coordinate model and orients the astigmatism gradient, which is useful when personalizing progressive lenses for real users. The second publication is a scientific article published in Optimization and Engineering that proposes the spherical coordinate model. ii Resum Aquest document ´esel resultat d'un Doctorat Industrial desenvolupat a trav´esd'un acord entre la Universitat Polit`ecnicade Catalunya i l'empresa Horizons Optical. Aquesta tesi resol el disseny complex de les lents progressives per ulleres, que ´esun problema real de la ind´ustria.El cristal.l´ı´esla part transparent de l'ull, situada darrera la pupil.la, que ens permet veure-hi n´ıtidament enfocant la llum a la retina. Amb el pas del temps, el cristal.l´ı perd la seva elasticitat i disminueix la seva capacitat d'acomodaci´oa la visi´ode prop. Aquest fenomen s'anomena presb´ıciai explica per qu`enecessitem ulleres quan ens fem grans. Les lents progressives corregeixen la presb´ıciai tenen un disseny complex: la zona superior s'utilitza per a la visi´ode lluny, la zona inferior per a la visi´ode prop (lectura) i el corredor que connecta aquestes dues zones permet una visi´on´ıtidaper a dist`ancies interm`edies,per exemple per a mirar la pantalla d'un ordinador. La superf´ıciede la lent progressiva dissenyada en aquesta tesi ´esla superf´ıciem´esdistant de l'ull, ´esa dir, la pot`encia de la zona de prop ´esm´esgran que la pot`enciade la zona de lluny. En termes geom`etrics,la pot`enciai l'astigmatisme es calculen utilitzant les curvatures principals de la superf´ıcie de la lent. En augmentar la pot`enciaverticalment, aparei- xen aberracions laterals en forma d'astigmatisme com a conseq¨u`encia del teorema de Minkwitz. L'objectiu d'aquesta tesi ´esutilitzar m`etodes d’optimitzaci´oper a dissenyar lents progressives minimitzant les aberracions laterals (astigmatisme) i proporcionant la pot`encia demanada per a cada zona de la lent. Aquesta tesi presenta dos models diferents per a calcular lents progressives. Tots dos models s´onaltament no lineals, no convexos i continus i han estat resolts utilitzant el llenguatge de modelitzaci´oAMPL i els solvers de punt interior IPOPT, LOQO i KNITRO. Tots dos models tenen aproximadament 900 variables (les variables s´onels coeficients d'una base de B-splines de grau tres). El primer model t´eunes 7000 restriccions, mentre que el segon model t´eunes 15000 restriccions. Cada restricci´ocorrespon a una propietat de pot`enciao astigmatisme d'un punt de la malla que defineix la superf´ıciede la lent. El primer model utilitza coordenades cartesianes i ´esuna versi´omillorada d'un model previ de la mateixa autora, publicat en un treball final de m`aster. El temps de CPU en el treball final de m`asterera entre 10 i 33 minuts, i en aquesta tesi s'ha redu¨ıta menys de 3 minuts utilitzant el mateix ordinador i el solver LOQO. En aquesta tesi, totes les inst`ancieshan convergit utilitzant el solver LOQO i el model amb coordenades cartesianes, la qual cosa no passava en el treball final de m`aster.Tanmateix, en aquesta tesi algunes de les inst`anciesno han convergit utilitzant el model amb coordenades cartesianes i altres solvers.
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