Stochasticity in optimization and game theory is a very important aspect to model more accurately real-world problems in many different areas (see for instance [6]). In optimization problems as well as in one-level games, namely, Nash equilibrium problems, stochasticity aspects have received quite a lot of attention for a while and have also been well studied [22]. However, for the branch of bilevel games quite few studies have included in their analysis stochastic aspects in their models.
A bilevel game is basically to split a finite set of players into two levels: the leaders or upper-level players, and the followers or lower-level players. In the model, the followers react in a passive way to the leaders’ actions, while the
leaders compete in the upper level trying to actively anticipate the followers’ reaction. Moreover, in each level, the interaction is non-cooperative as in Nash equilibrium problems. Bilevel games have been recognized as one of the most complex and at the same time very useful models in the literature [17].
Bilevel games, and more precisely the problem of the leaders in a bilevel game, face an ambiguity/uncertainty whenever the followers’ reaction is not necessarily uniquely determined for each leaders’ decision. To deal with this ambiguity two main approaches are well-known the optimistic and the pessimistic. The weakness of these two approaches is that both are quite extreme and the optimistic one lacks of real modeling foundations, putting the leaders in a quite naive position.
Recently, in [9] a general stochastic approach has been proposed to solve this ambiguity, which is seen as an uncertainty of the problem, providing also a specific approach that seems to be more reasonable than the optimistic one, from a modeling point of view. The stochasticity in the stochastic approach is an endogenous one since it corresponds to a decision-dependent uncertainty [1, 23].
But, of course, stochasticity might also come from an exogenous side, that is, when some of the parameters defining the game, such as future demand and prices, 1 forecasts of winds and clouds, are uncertain and possibly follow some probability distribution. This has been considered in [34, 12, 13, 16].
In the second part of the project, which is the applied part, we are interested in using the developed theoretical framework of bilevel games with stochastic aspects to a problem of contaminated water resource management, which has high levels of stochasticity.
The scarcity of water resource and its efficient use has been recognized as an extremely important problem in Chile and the whole world, for agricultural, industrial, and human use. Moreover, after any use, there is an outflow of water
which has generally more contaminants than the inflow. Depending on the type and quantity of contaminants the outflow of water could be reused, but sometimes giving less profit to the entity. The general situation is full of uncertainties, since the entities do not share their information. Moreover, the main source of information for us will be
measurements on the quality of water at different strategic points and punctual events of contamination registered by inspection, which is simply a qualitative data.
Therefore, we propose first to apply predictive models and machine learning to do inverse engineering in order to understand the game played by the different actors. Then we want to study the underlying one-level game and study the design of mechanisms (bilevel game) so that we can move the equilibrium to a desired goal. In a somehow similar spirit, in [30, 31, 10] game theory techniques have been used to analyze the behavior of companies sharing contaminated water in the context of eco-industrial parks, while in [36] also a bilevel model is used for a water resource optimal allocation problem.

La contaminación ambiental extradomiciliaria es un importante problema de salud ambiental ya que es responsable del 7,6% de la mortalidad anual total y de la pérdida de 103,1 millones de años de vida saludable. Entre algunos de los efectos adversos que se han reportado para el Material Particulado, uno de los principales contaminantes, se encuentran aumento en la mortalidad, morbilidad, muerte prematura, enfermedades cardiovasculares y respiratorias, cáncer de pulmón, impacto adverso en la actividad del sistema nervioso central que resulta en deterioro cognitivo, y efectos nocivos sobre el desarrollo fetal y el embarazo, afectando mayormente a grupos más vulnerables. Si bien, a nivel global, se cuentan con estrategias y criterios de monitoreo de calidad del aire para identificar la concentración de los contaminantes relacionados con la salud, es posible que estos no permitan analizar con mayor profundidad las diferencias en la distribución geográfica de los contaminantes y su impacto en poblaciones específicas.
En ese sentido, el presente proyecto tiene el objetivo de establecer una metodología para la generación de un mapa confiable de la contaminación del aire a partir de una red de monitoreo, el cual permita correlacionar de manera más precisa a través de modelos de interpolación, la contaminación del aire con los problemas de salud respiratorio de las personas del sector considerado, hacer una validación de la metodología propuesta con un programa piloto en un área a determinar de la comuna de Rancagua y realizar divulgación de resultados en colegios y jardines de la región con la finalidad de generar un debate más amplio y democrático sobre tanto de prácticas personales así como de las nuevas políticas relacionadas con la contaminación del aire.
Para esto se propone emplear diversos métodos propios de cada disciplina. Primeramente, se estimará un radio máximo de cobertura de un sensor dentro del cual se garantice cierto grado de confianza (Variograma empírico) respecto de los valores entregados por el sensor como promedios representativos, para así determinar el tamaño/forma del área geográfica total donde instalar la red de sensores (Obj 1). Con esta información se determinará una configuración inicial de los sensores en un área geográfica determinada considerando la capacidad de la red, usando como ubicación potencial establecimientos educativos. Luego se hará un análisis estadístico con los datos proporcionados por la red y por sensores móviles para determinar una nueva configuración que maximice la confianza de la nueva interpolación. Una vez instalados los sensores y a partir de los datos entregados por ellos se escogerá una nueva configuración a través de un análisis estadístico de covarianzas (Obj 2). Además, se realizará un estudio ecológico para correlacionar las mediciones de contaminación de aire y algunos indicadores de salud respiratoria y atención de salud considerando distintos puntos geográficos de la sexta región (Obj 3). Finalmente, el proyecto desarrollará un plan de educación ambiental para concientizar sobre el tema y favorecer el cambio de comportamientos a nivel individual y comunitario (Obj 4).
Con este proyecto se espera abordar con una visión interdisciplinaria un tema que es de especial preocupación en la región, como es la contaminación ambiental, fortaleciendo esta línea de investigación tanto en el instituto de ingeniería como en el de Salud, involucrando de forma activa también a estudiantes de pregrado.

Estudio de problemas de optimización y juegos con incertidumbre dependiente de decisiones. Estudio a nivel teórico y algorítmico. Estudio de aplicaciones.

Nonsmooth dynamical system involving regular structures

Stochastic Optimization and Chance Constraints with Applications to Energy (SOCCAE)

Stochasticity in optimization and game theory is a very important aspect to model more accurately real-world problems in many different areas (see for instance [6]). In optimization problems as well as in one-level games, namely, Nash equilibrium problems, stochasticity aspects have received quite a lot of attention for a while and have also been well studied [22]. However, for the branch of bilevel games quite few studies have included in their analysis stochastic aspects in their models.
A bilevel game is basically to split a finite set of players into two levels: the leaders or upper-level players, and the followers or lower-level players. In the model, the followers react in a passive way to the leaders’ actions, while the
leaders compete in the upper level trying to actively anticipate the followers’ reaction. Moreover, in each level, the interaction is non-cooperative as in Nash equilibrium problems. Bilevel games have been recognized as one of the most complex and at the same time very useful models in the literature [17].
Bilevel games, and more precisely the problem of the leaders in a bilevel game, face an ambiguity/uncertainty whenever the followers’ reaction is not necessarily uniquely determined for each leaders’ decision. To deal with this ambiguity two main approaches are well-known the optimistic and the pessimistic. The weakness of these two approaches is that both are quite extreme and the optimistic one lacks of real modeling foundations, putting the leaders in a quite naive position.
Recently, in [9] a general stochastic approach has been proposed to solve this ambiguity, which is seen as an uncertainty of the problem, providing also a specific approach that seems to be more reasonable than the optimistic one, from a modeling point of view. The stochasticity in the stochastic approach is an endogenous one since it corresponds to a decision-dependent uncertainty [1, 23].
But, of course, stochasticity might also come from an exogenous side, that is, when some of the parameters defining the game, such as future demand and prices, 1 forecasts of winds and clouds, are uncertain and possibly follow some probability distribution. This has been considered in [34, 12, 13, 16].
In the second part of the project, which is the applied part, we are interested in using the developed theoretical framework of bilevel games with stochastic aspects to a problem of contaminated water resource management, which has high levels of stochasticity.
The scarcity of water resource and its efficient use has been recognized as an extremely important problem in Chile and the whole world, for agricultural, industrial, and human use. Moreover, after any use, there is an outflow of water
which has generally more contaminants than the inflow. Depending on the type and quantity of contaminants the outflow of water could be reused, but sometimes giving less profit to the entity. The general situation is full of uncertainties, since the entities do not share their information. Moreover, the main source of information for us will be
measurements on the quality of water at different strategic points and punctual events of contamination registered by inspection, which is simply a qualitative data.
Therefore, we propose first to apply predictive models and machine learning to do inverse engineering in order to understand the game played by the different actors. Then we want to study the underlying one-level game and study the design of mechanisms (bilevel game) so that we can move the equilibrium to a desired goal. In a somehow similar spirit, in [30, 31, 10] game theory techniques have been used to analyze the behavior of companies sharing contaminated water in the context of eco-industrial parks, while in [36] also a bilevel model is used for a water resource optimal allocation problem.

En este proyecto, se busca general nuevas tecnologías que permitan mejorar el manejo de recursos hídricos en la sexta región.

Director de línea “Gestión de Riego Intrapredial con Inteligencia Artificial”

El Centro de Modelamiento Matemático (CMM) es un centro científico líder en Chile para la investigación y aplicaciones de las matemáticas. Fue inaugurado en abril del 2000 y forma parte de la Facultad de Ciencias Físicas y Matemáticas (FCFM) de la Universidad de Chile, en la que se encuentra la principal y más antigua escuela de ingeniería del país. Su objetivo es crear nuevas matemáticas y utilizarlas para resolver problemas provenientes de otras ciencias, la industria y las políticas públicas.

Resumen

Implementación de una microrred de energías renovables (solar, eólica y geotérmica) en el distrito salinero artesanal Barranca-La Villa de Cáhuil. Implementación de una planta piloto geotérmica de producción de sal y electrificación de bombas y planta de yodación comunitaria mediante energías renovables no convencionales.