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Introducción a metaheurísticas y algoritmo SA

parent 604d166d
Showing with 12654 additions and 10 deletions
curso-search/IDAstar.tex.bak
......@@ -618,74 +618,81 @@ $h:S \to \mathbb{R}$ \tcp*{funcin heurstica}}
\only<71>{
\begin{center}
\includegraphics[scale=0.20]{IDAstar-ejemplo-71.pdf}
\includegraphics[scale=0.25]{IDAstar-ejemplo-70-5.pdf}
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}
\note<71>{}
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\note<72>{}
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}
\note<73>{}
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\includegraphics[scale=0.20]{IDAstar-ejemplo-73.pdf}
\end{center}
}
\note<74>{}
\only<75>{
\begin{center}
\includegraphics[scale=0.20]{IDAstar-ejemplo-75.pdf}
\includegraphics[scale=0.20]{IDAstar-ejemplo-74.pdf}
\end{center}
}
\note<75>{}
\only<76>{
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\includegraphics[scale=0.20]{IDAstar-ejemplo-76.pdf}
\includegraphics[scale=0.20]{IDAstar-ejemplo-75.pdf}
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}
\note<76>{}
\only<77>{
\begin{center}
\includegraphics[scale=0.20]{IDAstar-ejemplo-77.pdf}
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\end{center}
}
\note<77>{}
\only<78>{
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\includegraphics[scale=0.20]{IDAstar-ejemplo-77.pdf}
\end{center}
}
\note<78>{}
\only<79>{
\begin{center}
\includegraphics[scale=0.20]{IDAstar-ejemplo-79.pdf}
\includegraphics[scale=0.20]{IDAstar-ejemplo-78.pdf}
\end{center}
}
\note<79>{}
\only<80>{
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\includegraphics[scale=0.20]{IDAstar-ejemplo-79.pdf}
\end{center}
}
\note<80>{}
\only<81>{
\begin{center}
\includegraphics[scale=0.20]{IDAstar-ejemplo-80.pdf}
\end{center}
}
\note<81>{}
\end{frame}
......
curso-search/aceptance.svg 0 → 100644
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curso-search/espacio-soluciones-01.svg 0 → 100644
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curso-search/espacio-soluciones-02.svg 0 → 100644
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\documentclass[xcolor=dvipsnames]{beamer}
\usepackage[spanish]{babel}
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\title{Algoritmos metaheurísticos}
\author{Stalin Muñoz Gutiérrez}
\institute{
Centro de Ciencias de la Complejidad\\
Universidad Nacional Aut\'onoma de Mexico (UNAM)}
\date{}
\begin{document}
\parindent 2em
\frame
{
\titlepage
\note<1>{
Hoy trataremos de un tipo especial de algoritmos de búsqueda denominados algoritmos metaheurísticos.
}
}
\begin{frame}
\frametitle{Espacio de soluciones}
\begin{block}{Espacio de soluciones a un problema}
El conjunto donde existen todas las posibles soluciones a un problema.
\end{block}
\begin{center}
\includegraphics[scale=0.30]{espacio-soluciones-01.pdf}
\end{center}
\note{
Antes de abordar los algoritmos metaheurísticos debemos definir algunos conceptos importantes.
Entenderemos al espacio de soluciones a un problema como un conjunto donde existen todas las posibles soluciones a dicho problema.
Aquí los círculos representarían posibles soluciones a un problema.
Por ejemplo en el caso del problema de encontrar una ruta en un mapa, cada círculo representaría una posible secuencia de ciudades, independientemente de si es una ruta correcta o no.
Para la mayoría de los algoritmos metaheurísticos no es importante representar explicitamente el grafo de estados-acciones. Basta con describir el espacio de soluciones y definir operadores que permitan explorar este espacio.
}
\end{frame}
\begin{frame}
\frametitle{Espacio de soluciones}
\begin{block}{Soluciones candidatas}
Cualquier elemento del espacio de soluciones.
\end{block}
\begin{center}
\includegraphics[scale=0.30]{espacio-soluciones-02.pdf}
\end{center}
\note{
Todo elemento del espacio de soluciones lo denominamos una solución candidata.
}
\end{frame}
\begin{frame}
\frametitle{Espacio de soluciones}
\begin{block}{Soluciones factibles}
Satisfacen un conjunto de restricciones.
\end{block}
\begin{center}
\includegraphics[scale=0.30]{espacio-soluciones-03.pdf}
\end{center}
\note{
Aquellas soluciones candidatas que cumplen con las restricciones del problema a resolver se denominan \emph{soluciones factibles}.
Las soluciones factibles son las soluciones de interés.
En el ejemplo de encontrar una ruta en un mapa, las soluciones factibles son aquellas rutas que empiezan en la ciudad de origen y terminan en el destino deseado.
}
\end{frame}
\begin{frame}
\frametitle{Espacio de soluciones}
\begin{block}{Funciones heurísticas}
Guían al algoritmo hacia las regiones factibles.
\end{block}
\begin{center}
\includegraphics[scale=0.30]{espacio-soluciones-04.pdf}
\end{center}
\note{
Los algoritmos metaheurísticos también se valdrán de funciones heurísticas que explotan conocimiento del dominio del problema. Una buena heurística guiará al algoritmo hacia las regiones de interés, esto es, hacía las soluciones factibles.
}
\end{frame}
\begin{frame}
\frametitle{Espacio de soluciones}
\begin{block}{Soluciones óptimas}
Las mejores soluciones factibles posibles.
\end{block}
\begin{center}
\includegraphics[scale=0.30]{espacio-soluciones-05.pdf}
\end{center}
\note{
Típicamente, se desea un algoritmo que busque las mejores soluciones posibles.
En nuestro ejemplo de las rutas en un mapa, nos interesan las rutas más cortas en distancia o las más rápidas en tiempo.
Por esta razón los algoritmos metaheurísticos muchas veces se denominan algoritmos de optimización metaheurísticos.
Resolver un problema, en este caso, es encontrar el mínimo o el máximo de cierta función objetivo.
}
\end{frame}
\begin{frame}
\frametitle{Optimización de funciones}
\begin{block}{Problema de optimización}
Encontrar los valores extremos, máximos o mínimos, de una función objetivo.
\end{block}
\begin{center}
\includegraphics[scale=0.30]{optimizacion-01.pdf}
\end{center}
\note{
Una vez que planteamos el problema como un problema de optimización, el algoritmo metaheurístico tratará de encontrar los valores extremos de la función objetivo.
Esto es el valor mínimo o máximo de la función.
La función puede ser una función escalar, es decir un solo número o puede ser una función multidimensional, en cuyo caso se denomina \emph{problema de optimización multi-objetivo}.
}
\end{frame}
\begin{frame}[<+->]
\frametitle{Optimización de funciones}
\begin{block}{Algoritmos de búsqueda local}
Exploran el espacio de soluciones haciendo transformaciones locales a las soluciones candidatas.
\end{block}
\only<1>{
\begin{center}
\includegraphics[scale=0.30]{optimizacion-02.pdf}
\end{center}
}
\note<1>{
Un tipo de estrategia para buscar soluciones es el utilizado en los denominados algoritmos de búsqueda local.
Un algoritmo de búsqueda local comienza con una solución candidata inicial, típicamente generada de manera aleatoria.
}
\only<2>{
\begin{center}
\includegraphics[scale=0.30]{optimizacion-03.pdf}
\end{center}
}
\note<2>{
Este tipo de algoritmos es muy general en su aplicabilidad.
No asume propiedades matemáticas de la función objetivo, como en el caso de métodos clásicos basados en descenso de gradiente, donde se requiere que la función sea derivable.
La búsqueda local puede aplicarse aún sin existir una función matemática explícita de la función objetivo.
Podemos definir la transformación en términos muy generales, como una transformación que resulta en una solución que se parece mucho, pero no es igual, a la solución candidata original.
Como las soluciones se parecen mucho, decimos que son soluciones vecinas. Nuevamente, no se requiere de una definición matemática de vecindad para una solución.
Las transformaciones pueden estar definidas de manera algorítmica.
Generalmente las transformaciones se hacen con algún criterio aleatorio.
}
\only<3>{
\begin{center}
\includegraphics[scale=0.30]{optimizacion-04.pdf}
\end{center}
}
\note<3>{
El algoritmo de búsqueda local, transforma de manera iterada la solución inicial, decidiendo la dirección de búsqueda en función de la evaluación de soluciones vecinas.
Esto hace que el algoritmo vaya convergiendo a las soluciones que minimizan o maximizan el valor de la función objetivo.
Si la transformación resulta en una mejora. El algoritmo puede decidir aceptar la solución. Si siempre la acepta, entonces se denomina algoritmo de ascención de colinas (Hill climbing).
Si el algoritmo explora toda la vecindad antes de decidir sobre la dirección a tomar, el algoritmo se denomina de ascención de colinas de máxima pendiente.
Sin embargo, nada garantiza que convergerá al mínimo global.
}
\only<4>{
\begin{center}
\includegraphics[scale=0.30]{optimizacion-05.pdf}
\end{center}
}
\note<4>{
Aquí observamos que el algoritmo converge a un mínimo de la función objetivo.
Cualquier movimiento local, empeorará el valor de la función objetivo.
}
\only<5>{
\begin{center}
\includegraphics[scale=0.30]{optimizacion-06.pdf}
\end{center}
}
\note<5>{
Si el algoritmo hubiera empezado en otra solución candidata, es posible que encontrara una mejor solución.
Una estrategia posible para los algoritmos de búsqueda local es lanzar múltiples búsquedas con diferentes condiciones iniciales.
}
\end{frame}
\begin{frame}
\frametitle{El algoritmo de recocido simulado (SA)}
\begin{block}{Un algoritmo capaz de escapar óptimos locales}
El algoritmo de recocido (o templado) simulado (\emph{simulated annealing}) es un algoritmo metaheurístico capaz de escapar de óptimos locales. Esta inspirado en un proceso de calentamiento y enfriamiento controlado en el cual un sólido puede alcanzar un arreglo cristalino muy regular al ser enfriado de manera suficientemente lenta.
\end{block}
\note{
El primer algoritmo metaheurístico que presentaremos es el algoritmo de recocido o templado simulado, al que denominaremos \emph{SA} por las siglas de su nombre en inglés \emph{Simulated Annealing}.
Como muchos de los algoritmos de optimización metaheurísticos, esta inspirado en un proceso de optimización observado en la naturaleza. En este caso en un proceso físico.
En un proceso de templado, el material se calienta más allá de un punto de recristalización. Después se enfria de manera controlada.
Desde el punto de vista del algoritmo de búsqueda existirá una variable de temperatura que decidirá el régimen de exploración del algoritmo.
}
\end{frame}
\begin{frame}[<+->]
\frametitle{Soluciones peores pueden aceptarse}
\only<1>{
\begin{center}
\includegraphics[scale=0.6]{aceptance.pdf}
\end{center}
}
\note<1>{
La característica más interesante del algoritmo de templado simulado es que, a diferencia de un algoritmo ascensor de colinas estándar que puede fácilmente quedar atrapado en un óptimo local, SA no siempre se dirige en la dirección de máximo beneficio.
El algoritmo puede acepetar soluciones de mayor energía con cierta probabilidad.
La probabilidad de aceptar una solución que es peor que la solución candidata actual, esta dada por la distribución de Boltzman.
Aquí vemos como cambia la distribución de probabilidad para aceptar estados de mayor energía como una función de la temperatura.
La temperatura es alta al comenzar las iteraciones, $T_{\text{inicial}}$, y es baja al finalizar las iteraciones, $T_{\text{final}}$.
}
\only<2>{
\begin{center}
\includegraphics[scale=0.6]{exploracion-explotacion.pdf}
\end{center}
}
\note<2>{
En la gráfica que se ilustra que a temperaturas altas.
Esto es, en las primeras iteraciones,
la probabilidad cae lentamente, entonces la probabilidad de aceptar soluciones de mayor energía es alta.
Por lo cual, al inicio de las iteraciones el algoritmo esta en un régimen de exploración.
Conforme el sistema se enfria, la distribución es cada vez más estricta, indicando que se aceptará un número menor de soluciones que tengan mayor energía.
Al alcanzar la temperatura final, el sistema esta frio. La probabilidad de aceptar soluciones de mayor energía es casi nula. El algoritmo se encuentra en un régimen de explotación o aprovechamiento. En este régimen se comporta como un ascensor de colinas.
}
\end{frame}
\begin{frame}
\frametitle{Algoritmo SA}
\begin{center}
\includegraphics[scale=0.1]{pizarron.jpg}
\end{center}
\note{
Presentamos el algoritmo SA en el pizarrón.
}
\end{frame}
\begin{frame}
\frametitle{Ejemplo de aplicación de SA a un problema de permutaciones}
\begin{block}{El problema del agente viajero}
Un agente viajero debe visitar exactamente $n$ ciudades.
Se desea encontrar el orden en que dichas ciudades deben visitarse tal que el costo sea mínimo.
\end{block}
\begin{center}
\includegraphics[scale=0.18]{tsp.pdf}
\end{center}
\note{
Vamos a presentar un problema interesante y computacionalmente dificil de resolver.
El problema del agente viajero.
Un agente viajero, desea encontrar un circuito óptimo para visitar exactamente n ciudades, es decir, una secuencia ordenada de ciudades que minimice el costo de la ruta.
El espacio de posibles rutas es muy grande, es $n$ factorial.
En la parte superior izquierda vemos una ruta arbitraria para un problema con 30 ciudades, tiene un costo alto, pues hay muchos cruces con distancias muy grandes.
Una buena ruta no tiene cruces, como la mostrada en la grafica inferior.
El algoritmo SA, teoricamente puede encontrar la solución óptima, sin embargo en la práctica el tiempo que le tomaría encontrar la ruta óptima es muy grande, por lo que sólo podemos garantizar que la solución será subóptima.
}
\end{frame}
\begin{frame}
\frametitle{Soluciones candidatas y vecinas}
\begin{block}{Solución candidata}
Una solución candidata es una permutación de ciudades:
$x =
\begin{bmatrix}
B&C&G&H&J&D&A&E&G&F
\end{bmatrix}
$
\end{block}
\begin{block}{Solución vecina}
Una solución vecina puede construirse intercambiando dos ciudades elegidas aleatoriamente.
$x\phantom{'} =
\begin{bmatrix}
B&C&G&\color{blue}H&J&D&A&E&\color{blue}G&F
\end{bmatrix}
$
$x' =
\begin{bmatrix}
B&C&G&\color{blue}G&J&D&A&E&\color{blue}H&F
\end{bmatrix}
$
\end{block}
\begin{block}{Energía de una solución candidata}
La energía es el {\color{blue}costo} de la ruta.
\begin{center}
$\text{energía}(x) = \sum\limits_{i=1}^n{\text{costo}(x_i,x_{(i \mod n)+1})}$
\end{center}
\end{block}
\note{
En el problema del agente viajero una solución candidata es una permutación de ciudades.
En este ejemplo identificamos cada ciudad con una letra distinta.
Para encontrar soluciones vecinas, podemos definir un operador de intercambio de ciudades.
Aleatoriamente seleccionamos dos ciudades e intercambiamos su posición en la ruta.
Finalmente la energía de la solución es simplemente el costo sobre la trayectoria. En este caso la suma de los costos de cada transición.
}
\end{frame}
\end{document}
\ No newline at end of file
curso-search/metaheuristicas-intro.tex.bak 0 → 100644
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\usepackage{algpseudocode}
\usepackage{multicol}
% descomentar para generar handouts con notas
\setbeameroption{show notes on second screen=right}
\usepackage{tikz}
\tikzstyle{every picture}+=[remember picture]
\usefonttheme[onlymath]{serif}
\usepackage{verbatim}
\usepackage{pifont}
\usetheme{Antibes}
\usecolortheme{lily}
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\spanishdecimal{.}
\title{Algoritmos metaheurísticos}
\author{Stalin Muñoz Gutiérrez}
\institute{
Centro de Ciencias de la Complejidad\\
Universidad Nacional Aut\'onoma de Mexico (UNAM)}
\date{}
\begin{document}
\parindent 2em
\frame
{
\titlepage
\note<1>{
Hoy trataremos de un tipo especial de algoritmos de búsqueda denominados algoritmos metaheurísticos.
}
}
\begin{frame}
\frametitle{Espacio de soluciones}
\begin{block}{Espacio de soluciones a un problema}
El conjunto donde existen todas las posibles soluciones a un problema.
\end{block}
\begin{center}
\includegraphics[scale=0.30]{espacio-soluciones-01.pdf}
\end{center}
\note{
Antes de abordar los algoritmos metaheurísticos debemos definir algunos conceptos importantes.
Entenderemos al espacio de soluciones a un problema como un conjunto donde existen todas las posibles soluciones a dicho problema.
Aquí los círculos representarían posibles soluciones a un problema.
Por ejemplo en el caso del problema de encontrar una ruta en un mapa, cada círculo representaría una posible secuencia de ciudades, independientemente de si es una ruta correcta o no.
Para la mayoría de los algoritmos metaheurísticos no es importante representar explicitamente el grafo de estados-acciones. Basta con describir el espacio de soluciones y definir operadores que permitan explorar este espacio.
}
\end{frame}
\begin{frame}
\frametitle{Espacio de soluciones}
\begin{block}{Soluciones candidatas}
Cualquier elemento del espacio de soluciones.
\end{block}
\begin{center}
\includegraphics[scale=0.30]{espacio-soluciones-02.pdf}
\end{center}
\note{
Todo elemento del espacio de soluciones lo denominamos una solución candidata.
}
\end{frame}
\begin{frame}
\frametitle{Espacio de soluciones}
\begin{block}{Soluciones factibles}
Satisfacen un conjunto de restricciones.
\end{block}
\begin{center}
\includegraphics[scale=0.30]{espacio-soluciones-03.pdf}
\end{center}
\note{
Aquellas soluciones candidatas que cumplen con las restricciones del problema a resolver se denominan \emph{soluciones factibles}.
Las soluciones factibles son las soluciones de interés.
En el ejemplo de encontrar una ruta en un mapa, las soluciones factibles son aquellas rutas que empiezan en la ciudad de origen y terminan en el destino deseado.
}
\end{frame}
\begin{frame}
\frametitle{Espacio de soluciones}
\begin{block}{Funciones heurísticas}
Guían al algoritmo hacia las regiones factibles.
\end{block}
\begin{center}
\includegraphics[scale=0.30]{espacio-soluciones-04.pdf}
\end{center}
\note{
Los algoritmos metaheurísticos también se valdrán de funciones heurísticas que explotan conocimiento del dominio del problema. Una buena heurística guiará al algoritmo hacia las regiones de interés, esto es, hacía las soluciones factibles.
}
\end{frame}
\begin{frame}
\frametitle{Espacio de soluciones}
\begin{block}{Soluciones óptimas}
Las mejores soluciones factibles posibles.
\end{block}
\begin{center}
\includegraphics[scale=0.30]{espacio-soluciones-05.pdf}
\end{center}
\note{
Típicamente, se desea un algoritmo que busque las mejores soluciones posibles.
En nuestro ejemplo de las rutas en un mapa, nos interesan las rutas más cortas en distancia o las más rápidas en tiempo.
Por esta razón los algoritmos metaheurísticos muchas veces se denominan algoritmos de optimización metaheurísticos.
Resolver un problema, en este caso, es encontrar el mínimo o el máximo de cierta función objetivo.
}
\end{frame}
\begin{frame}
\frametitle{Optimización de funciones}
\begin{block}{Problema de optimización}
Encontrar los valores extremos, máximos o mínimos, de una función objetivo.
\end{block}
\begin{center}
\includegraphics[scale=0.30]{optimizacion-01.pdf}
\end{center}
\note{
Una vez que planteamos el problema como un problema de optimización, el algoritmo metaheurístico tratará de encontrar los valores extremos de la función objetivo.
Esto es el valor mínimo o máximo de la función.
La función puede ser una función escalar, es decir un solo número o puede ser una función multidimensional, en cuyo caso se denomina \emph{problema de optimización multi-objetivo}.
}
\end{frame}
\begin{frame}[<+->]
\frametitle{Optimización de funciones}
\begin{block}{Algoritmos de búsqueda local}
Exploran el espacio de soluciones haciendo transformaciones locales a las soluciones candidatas.
\end{block}
\only<1>{
\begin{center}
\includegraphics[scale=0.30]{optimizacion-02.pdf}
\end{center}
}
\note<1>{
Un tipo de estrategia para buscar soluciones es el utilizado en los denominados algoritmos de búsqueda local.
Un algoritmo de búsqueda local comienza con una solución candidata inicial, típicamente generada de manera aleatoria.
}
\only<2>{
\begin{center}
\includegraphics[scale=0.30]{optimizacion-03.pdf}
\end{center}
}
\note<2>{
Este tipo de algoritmos es muy general en su aplicabilidad.
No asume propiedades matemáticas de la función objetivo, como en el caso de métodos clásicos basados en descenso de gradiente, donde se requiere que la función sea derivable.
La búsqueda local puede aplicarse aún sin existir una función matemática explícita de la función objetivo.
Podemos definir la transformación en términos muy generales, como una transformación que resulta en una solución que se parece mucho, pero no es igual, a la solución candidata original.
Como las soluciones se parecen mucho, decimos que son soluciones vecinas. Nuevamente, no se requiere de una definición matemática de vecindad para una solución.
Las transformaciones pueden estar definidas de manera algorítmica.
Generalmente las transformaciones se hacen con algún criterio aleatorio.
}
\only<3>{
\begin{center}
\includegraphics[scale=0.30]{optimizacion-04.pdf}
\end{center}
}
\note<3>{
El algoritmo de búsqueda local, transforma de manera iterada la solución inicial, decidiendo la dirección de búsqueda en función de la evaluación de soluciones vecinas.
Esto hace que el algoritmo vaya convergiendo a las soluciones que minimizan o maximizan el valor de la función objetivo.
Si la transformación resulta en una mejora. El algoritmo puede decidir aceptar la solución. Si siempre la acepta, entonces se denomina algoritmo de ascención de colinas (Hill climbing).
Si el algoritmo explora toda la vecindad antes de decidir sobre la dirección a tomar, el algoritmo se denomina de ascención de colinas de máxima pendiente.
Sin embargo, nada garantiza que convergerá al mínimo global.
}
\only<4>{
\begin{center}
\includegraphics[scale=0.30]{optimizacion-05.pdf}
\end{center}
}
\note<4>{
Aquí observamos que el algoritmo converge a un mínimo de la función objetivo.
Cualquier movimiento local, empeorará el valor de la función objetivo.
}
\only<5>{
\begin{center}
\includegraphics[scale=0.30]{optimizacion-06.pdf}
\end{center}
}
\note<5>{
Si el algoritmo hubiera empezado en otra solución candidata, es posible que encontrara una mejor solución.
Una estrategia posible para los algoritmos de búsqueda local es lanzar múltiples búsquedas con diferentes condiciones iniciales.
}
\end{frame}
\begin{frame}
\frametitle{El algoritmo de recocido simulado (SA)}
\begin{block}{Un algoritmo capaz de escapar óptimos locales}
El algoritmo de recocido (o templado) simulado (\emph{simulated annealing}) es un algoritmo metaheurístico capaz de escapar de óptimos locales. Esta inspirado en un proceso de calentamiento y enfriamiento controlado en el cual un sólido puede alcanzar un arreglo cristalino muy regular al ser enfriado de manera suficientemente lenta.
\end{block}
\note{
El primer algoritmo metaheurístico que presentaremos es el algoritmo de recocido o templado simulado, al que denominaremos \emph{SA} por las siglas de su nombre en inglés \emph{Simulated Annealing}.
Como muchos de los algoritmos de optimización metaheurísticos, esta inspirado en un proceso de optimización observado en la naturaleza. En este caso en un proceso físico.
En un proceso de templado, el material se calienta más allá de un punto de recristalización. Después se enfria de manera controlada.
Desde el punto de vista del algoritmo de búsqueda existirá una variable de temperatura que decidirá el régimen de exploración del algoritmo.
}
\end{frame}
\begin{frame}[<+->]
\frametitle{Soluciones peores pueden aceptarse}
\only<1>{
\begin{center}
\includegraphics[scale=0.6]{aceptance.pdf}
\end{center}
}
\note<1>{
La característica más interesante del algoritmo de templado simulado es que, a diferencia de un algoritmo ascensor de colinas estándar que puede fácilmente quedar atrapado en un óptimo local, SA no siempre se dirige en la dirección de máximo beneficio.
El algoritmo puede acepetar soluciones de mayor energía con cierta probabilidad.
La probabilidad de aceptar una solución que es peor que la solución candidata actual, esta dada por la distribución de Boltzman.
Aquí vemos como cambia la distribución de probabilidad para aceptar estados de mayor energía como una función de la temperatura.
La temperatura es alta al comenzar las iteraciones, $T_{\text{inicial}}$, y es baja al finalizar las iteraciones, $T_{\text{final}}$.
}
\only<2>{
\begin{center}
\includegraphics[scale=0.6]{exploracion-explotacion.pdf}
\end{center}
}
\note<2>{
En la gráfica que se ilustra que a temperaturas altas.
Esto es, en las primeras iteraciones,
la probabilidad cae lentamente, entonces la probabilidad de aceptar soluciones de mayor energía es alta.
Por lo cual, al inicio de las iteraciones el algoritmo esta en un régimen de exploración.
Conforme el sistema se enfria, la distribución es cada vez más estricta, indicando que se aceptará un número menor de soluciones que tengan mayor energía.
Al alcanzar la temperatura final, el sistema esta frio. La probabilidad de aceptar soluciones de mayor energía es casi nula. El algoritmo se encuentra en un régimen de explotación o aprovechamiento. En este régimen se comporta como un ascensor de colinas.
}
\end{frame}
\begin{frame}
\frametitle{Algoritmo SA}
\begin{center}
\includegraphics[scale=0.1]{pizarron.jpg}
\end{center}
\note{
Presentamos el algoritmo SA en el pizarrón.
}
\end{frame}
\begin{frame}
\frametitle{Ejemplo de aplicación de SA a un problema de permutaciones}
\begin{block}{El problema del agente viajero}
Un agente viajero debe visitar exactamente $n$ ciudades.
Se desea encontrar el orden en que dichas ciudades deben visitarse tal que el costo sea mínimo.
\end{block}
\begin{center}
\includegraphics[scale=0.18]{tsp.pdf}
\end{center}
\note{
Vamos a presentar un problema interesante y computacionalmente dificil de resolver.
El problema del agente viajero.
Un agente viajero, desea encontrar un circuito óptimo para visitar exactamente n ciudades, es decir, una secuencia ordenada de ciudades que minimice el costo de la ruta.
El espacio de posibles rutas es muy grande, es $n$ factorial.
En la parte superior izquierda vemos una ruta arbitraria para un problema con 30 ciudades, tiene un costo alto, pues hay muchos cruces con distancias muy grandes.
Una buena ruta no tiene cruces, como la mostrada en la grafica inferior.
El algoritmo SA, teoricamente puede encontrar la solución óptima, sin embargo en la práctica el tiempo que le tomaría encontrar la ruta óptima es muy grande, por lo que sólo podemos garantizar que la solución será subóptima.
}
\end{frame}
\begin{frame}
\frametitle{Soluciones candidatas y vecinas}
\begin{block}{Solución candidata}
Una solución candidata es una permutación de ciudades. Si identificamos cada ciudad con una letra distinta entonces:
$x =
\begin{bmatrix}
B&C&G&H&J&D&A&E&G&F
\end{bmatrix}
$
\end{block}
\begin{block}{Solución vecina}
Una solución vecina puede construirse intercambiando dos ciudades elegidas aleatoriamente.
$x\phantom{'} =
\begin{bmatrix}
B&C&G&\color{blue}H&J&D&A&E&\color{blue}G&F
\end{bmatrix}
$
$x' =
\begin{bmatrix}
B&C&G&\color{blue}G&J&D&A&E&\color{blue}H&F
\end{bmatrix}
$
\end{block}
\begin{block}{Energía de una solución candidata}
La energía de una solución es el {\color{blue}costo} de la ruta.
$\text{energía}(x) = \sum\limits_{i=1}^n{\text{costo}(x_i,x_{(i \mod n)+1})}$
\end{block}
\end{frame}
\end{document}
\ No newline at end of file
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curso-search/optimizacion-06.svg 0 → 100644
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curso-search/tsp.svg 0 → 100644
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curso-search/tsp30-1.svg 0 → 100644
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