Optimization methods for the design of progressive lenses

This work is the result of an Industrial Doctorate developed through a partnership agreement between the Universitat Politècnica de Catalunya and the company Horizons Optical. 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 curvatures 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 modeling 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 constraints. 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 converged 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 decreased 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 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.Aquest document és el resultat d'un Doctorat Industrial desenvolupat a través d'un acord entre la Universitat Politècnica de Catalunya i l'empresa Horizons Optical. Aquesta tesi resol el disseny complex de les lents progressives per ulleres, que és un problema real de la indústria. El cristal·lí és la part transparent de l'ull, situada darrera la pupil·la, que ens permet veure-hi nítidament enfocant la llum a la retina. Amb el pas del temps, el cristal·lí perd la seva elasticitat i disminueix la seva capacitat d'acomodació a la visió de prop. Aquest fenomen s'anomena presbícia i explica per què necessitem ulleres quan ens fem grans. Les lents progressives corregeixen la presbícia i tenen un disseny complex: la zona superior s'utilitza per a la visió de lluny, la zona inferior per a la visió de prop (lectura) i el corredor que connecta aquestes dues zones permet una visió nítida per a distàncies intermèdies, per exemple per a mirar la pantalla d'un ordinador. La superfície de la lent progressiva dissenyada en aquesta tesi és la superfície més distant de l'ull, és a dir, la potència de la zona de prop és més gran que la potència de la zona de lluny. En termes geomètrics, la potència i l'astigmatisme es calculen utilitzant les curvatures principals de la superfície de la lent. En augmentar la potència verticalment, apareixen aberracions laterals en forma d'astigmatisme com a conseqüència del teorema de Minkwitz. L'objectiu d'aquesta tesi és utilitzar mètodes d'optimització per a dissenyar lents progressives minimitzant les aberracions laterals (astigmatisme) i proporcionant la potència demanada per a cada zona de la lent. Aquesta tesi presenta dos models diferents per a calcular lents progressives. Tots dos models són altament no lineals, no convexos i continus i han estat resolts utilitzant el llenguatge de modelització AMPL i els solvers de punt interior IPOPT, LOQO i KNITRO. Tots dos models tenen aproximadament 900 variables (les variables són els coeficients d'una base de B-splines de grau tres). El primer model té unes 7000 restriccions, mentre que el segon model té unes 15000 restriccions. Cada restricció correspon a una propietat de potència o astigmatisme d'un punt de la malla que defineix la superfície de la lent. El primer model utilitza coordenades cartesianes i és una versió millorada d'un model previ de la mateixa autora, publicat en un treball final de màster. El temps de CPU en el treball final de màster era entre 10 i 33 minuts, i en aquesta tesi s'ha reduït a menys de 3 minuts utilitzant el mateix ordinador i el solver LOQO. En aquesta tesi, totes les instàncies han convergit utilitzant el solver LOQO i el model amb coordenades cartesianes, la qual cosa no passava en el treball final de màster. Tanmateix, en aquesta tesi algunes de les instàncies no han convergit utilitzant el model amb coordenades cartesianes i altres solvers. El segon model utilitza coordenades esfèriques i presenta millor convexitat que l'anterior model de coordenades cartesianes. Totes les instàncies han convergit utilitzant qualsevol dels solvers, i la qualitat de la solució ha millorat. El temps de CPU utilitzant coordenades esfèriques ha estat superior que el temps del model de coordenades cartesianes, a causa dels llargs càlculs, tot i que el nombre d'iteracions ha disminuït considerablement (per exemple, d'un màxim de 192 iteracions utilitzant coordenades cartesianes a un màxim de 84 iteracions utilitzant coordenades esfèriques i el mateix solver LOQO). Aquests dos models han tingut com a resultat dues publicacions. La primera és la patent d'invenció que utilitza coordenades cartesianes i orienta el gradient d'astigmatisme, la qual cosa és útil a l'hora de personalitzar les lents progressives per als usuaris finals. La segona publicació és un article científic publicat a la revista Optimization and Engineering, que presenta el model amb coordenades esfèriques.Postprint (published version

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

Research Report UPC-DEIO DR 2019Designing progressive lenses is a complex problem that has beenpreviously solved by formulating an optimization model based on Cartesiancoordinates. In this work a new progressive lens model using spherical co-ordinates is presented, and interior point solvers are used to solve this newoptimization model. Although this results in a highly nonlinear, nonconvex,continuous optimization problem, the new spherical coordinates model exhibitsbetter convexity properties compared to previous ones based on Cartesian co-ordinates. The real-world instances considered gave rise to nonlinear optimiza-tion problems of about 900 variables and 15000 constraints. Each constraintcorresponds to a point of the grid used to define the lens surface. The numberof variables depends on the precision of a B-spline basis used for the repre-sentation of the surface, and the number of constraints depends on the shapeand quality of the design. We present results of progressive lenses obtainedusing the AMPL modeling language and the nonlinear interior point solversIPOPT, LOQO and KNITRO. Computational results are reported, as wellas some examples of real-world progressive lenses calculated using this newmodel. Progressive lenses obtained are competitive in terms of quality withthose resulting from previous models that are used in commercial glasses.Peer ReviewedPreprin

Generació de la corba d'oferta a partir de les dades públiques del MIBEL

Research Report, Dept. of Statistics and Operations Research, UPC.Preprin

Disseny de lents progressives

Presbyopia starts to show up at the age of 40. The lens of the eye loses its flexibility to the point where we can no longer accommodate clearly in near vision. Progression-addition lenses correct the lack of accommodation and have three different zones: distance view, near view and intermediate zone, with a gradual change in power from the distance zone to the near zone. In geometrical terms, the optical power of a lens can be expressed as the difference between the principal curvatures of the surface lens, and the power can be expressed as the mean of the principal curvatures. The Minkwitz theorem states the apparition of unwanted astigmatism in the lateral zones of the lens, due to the change of power. This master thesis proposes a model for the design of progressive-addition lenses, taking into account the power lens and minimizing the unwanted astigmatism, as well as the gradient of astigmatism and the gradient of power.El treball de fi de màster proposat consistirà en modelitzar i optimitzar el disseny de les superfícies de lents progressives utilitzant rutines d'optimització tenint en compte la potència de la superfície desitjada i la minimització dels astigmatismes i gradient d'astigmatisme i de potènci

Optimization methods for the design of progressive lenses

This work is the result of an Industrial Doctorate developed through a partnership agreement between the Universitat Politècnica de Catalunya and the company Horizons Optical. 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 curvatures 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 modeling 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 constraints. 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 converged 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 decreased 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 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.Aquest document és el resultat d'un Doctorat Industrial desenvolupat a través d'un acord entre la Universitat Politècnica de Catalunya i l'empresa Horizons Optical. Aquesta tesi resol el disseny complex de les lents progressives per ulleres, que és un problema real de la indústria. El cristal·lí és la part transparent de l'ull, situada darrera la pupil·la, que ens permet veure-hi nítidament enfocant la llum a la retina. Amb el pas del temps, el cristal·lí perd la seva elasticitat i disminueix la seva capacitat d'acomodació a la visió de prop. Aquest fenomen s'anomena presbícia i explica per què necessitem ulleres quan ens fem grans. Les lents progressives corregeixen la presbícia i tenen un disseny complex: la zona superior s'utilitza per a la visió de lluny, la zona inferior per a la visió de prop (lectura) i el corredor que connecta aquestes dues zones permet una visió nítida per a distàncies intermèdies, per exemple per a mirar la pantalla d'un ordinador. La superfície de la lent progressiva dissenyada en aquesta tesi és la superfície més distant de l'ull, és a dir, la potència de la zona de prop és més gran que la potència de la zona de lluny. En termes geomètrics, la potència i l'astigmatisme es calculen utilitzant les curvatures principals de la superfície de la lent. En augmentar la potència verticalment, apareixen aberracions laterals en forma d'astigmatisme com a conseqüència del teorema de Minkwitz. L'objectiu d'aquesta tesi és utilitzar mètodes d'optimització per a dissenyar lents progressives minimitzant les aberracions laterals (astigmatisme) i proporcionant la potència demanada per a cada zona de la lent. Aquesta tesi presenta dos models diferents per a calcular lents progressives. Tots dos models són altament no lineals, no convexos i continus i han estat resolts utilitzant el llenguatge de modelització AMPL i els solvers de punt interior IPOPT, LOQO i KNITRO. Tots dos models tenen aproximadament 900 variables (les variables són els coeficients d'una base de B-splines de grau tres). El primer model té unes 7000 restriccions, mentre que el segon model té unes 15000 restriccions. Cada restricció correspon a una propietat de potència o astigmatisme d'un punt de la malla que defineix la superfície de la lent. El primer model utilitza coordenades cartesianes i és una versió millorada d'un model previ de la mateixa autora, publicat en un treball final de màster. El temps de CPU en el treball final de màster era entre 10 i 33 minuts, i en aquesta tesi s'ha reduït a menys de 3 minuts utilitzant el mateix ordinador i el solver LOQO. En aquesta tesi, totes les instàncies han convergit utilitzant el solver LOQO i el model amb coordenades cartesianes, la qual cosa no passava en el treball final de màster. Tanmateix, en aquesta tesi algunes de les instàncies no han convergit utilitzant el model amb coordenades cartesianes i altres solvers. El segon model utilitza coordenades esfèriques i presenta millor convexitat que l'anterior model de coordenades cartesianes. Totes les instàncies han convergit utilitzant qualsevol dels solvers, i la qualitat de la solució ha millorat. El temps de CPU utilitzant coordenades esfèriques ha estat superior que el temps del model de coordenades cartesianes, a causa dels llargs càlculs, tot i que el nombre d'iteracions ha disminuït considerablement (per exemple, d'un màxim de 192 iteracions utilitzant coordenades cartesianes a un màxim de 84 iteracions utilitzant coordenades esfèriques i el mateix solver LOQO). Aquests dos models han tingut com a resultat dues publicacions. La primera és la patent d'invenció que utilitza coordenades cartesianes i orienta el gradient d'astigmatisme, la qual cosa és útil a l'hora de personalitzar les lents progressives per als usuaris finals. La segona publicació és un article científic publicat a la revista Optimization and Engineering, que presenta el model amb coordenades esfèriques

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

This is a post-peer-review, pre-copyedit version of an article published in Optimization and engineering. The final authenticated version is available online at: http://dx.doi.org/10.1007/s11081-019-09480-zDesigning 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.Peer ReviewedPostprint (author's final draft

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

Research Report UPC-DEIO DR 2019Designing progressive lenses is a complex problem that has beenpreviously solved by formulating an optimization model based on Cartesiancoordinates. In this work a new progressive lens model using spherical co-ordinates is presented, and interior point solvers are used to solve this newoptimization model. Although this results in a highly nonlinear, nonconvex,continuous optimization problem, the new spherical coordinates model exhibitsbetter convexity properties compared to previous ones based on Cartesian co-ordinates. The real-world instances considered gave rise to nonlinear optimiza-tion problems of about 900 variables and 15000 constraints. Each constraintcorresponds to a point of the grid used to define the lens surface. The numberof variables depends on the precision of a B-spline basis used for the repre-sentation of the surface, and the number of constraints depends on the shapeand quality of the design. We present results of progressive lenses obtainedusing the AMPL modeling language and the nonlinear interior point solversIPOPT, LOQO and KNITRO. Computational results are reported, as wellas some examples of real-world progressive lenses calculated using this newmodel. Progressive lenses obtained are competitive in terms of quality withthose resulting from previous models that are used in commercial glasses.Peer Reviewe

Generació de la corba d'oferta a partir de les dades públiques del MIBEL

Research Report, Dept. of Statistics and Operations Research, UPC

Generació de figures eps a partir de resultats d'AMPL

Research Report, Dept. of Statistics and Operations Research, UPC.Preprin

Generació de figures eps a partir de resultats d'AMPL

Research Report, Dept. of Statistics and Operations Research, UPC