Publication: Algunas cuestiones notables sobre el modelo de Hopfield en optimización
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2018-11-07
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Universidad Complutense de Madrid
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Abstract
El modelo de Hopfield continuo puede utilizarse como heurística para resolver problemas de optimización combinatoria, como es el caso del problema del viajante (TSP). Consiste en una red neuronal recurrente con una ecuación diferencial asociada, cuyos estados evolucionan de un punto inicial a un punto de equilibrio mediante la minimización de una función de Lyapunov. Asociando dicha función con la función objetivo y restricciones del problema de optimización, se consigue que los puntos de equilibrio se correspondan con soluciones factibles del problema de optimización. Aplicando la formulación original de Hopfield al TSP, este proceso deja un parámetro libre. Históricamente, los investigadores en el campo han utilizado valores pequeños de dicho parámetro, sin explicar por qué se obtienen mejores soluciones.
Esta memoria analiza la relación entre el parámetro libre y el punto de silla del modelo de Hopfield. Mientras que en problemas pequeños este resultado garantiza poder obtener siempre la solución óptima, en problemas más complejos como el TSP resulta más complicado. No obstante, las cuencas de atracción para las mejores soluciones se hacen más amplias en las proximidades del punto de silla cuando el parámetro libre decrece. Así, los puntos situados en las proximidades del punto de silla serán excelentes candidatos a punto inicial de la simulación.
Con el objetivo de continuar mejorando la calidad de las soluciones del modelo de Hopfield, se propone el modelo Divide-y-Vencerás, un modelo basado en dos modelos de Hopfield consecutivos. Este modelo conecta en la primera fase ciudades vecinas y en una segunda fase los grupos de ciudades conectados en la fase anterior. En ambos casos, se deducen las correspondientes parametrizaciones del problema y se resuelven con el modelo de Hopfield.
Finalmente, se estudia cómo un caso particular del modelo de Hopfield utilizado en la segunda fase del modelo Divide-y-Vencerás, se comporta en realidad como un algoritmo 2-opt, siempre y cuando se elija adecuadamente el punto inicial. Todos estos resultados mejoran el rendimiento del modelo de Hopfield como heurística, equiparándolo al algoritmo 2-opt en calidad de solución.
The continuous Hopfield model can be used as a heuristic to solve combinatorial optimization problems, such as the Traveling Salesman Problem (TSP). It consists of a recurrent neural network with an associated differential equation, whose states evolve from an initial point to an equilibrium point by minimizing a Lyapunov function. By associating this function with the objective function and constraints of the optimization problem, the equilibrium points correspond to feasible solutions of the optimization problem. Applying Hopfield's original formulation to solve the TSP, this process leaves a free parameter. Historically, researchers in the field have used small values of this free parameter, without explaining why better solutions are obtained. This dissertation analyzes the relation between the free parameter and the saddle point of the Hopfield model. Whereas in small problems this result guarantees that the global optimum is always obtained, in more complex instances, such as the TSP, this is far more complicated. However, in the surroundings of the saddle point, the attractor basins for the best solutions grow as the free parameter decreases. Thus, saddle point neighbors become excellent starting point candidates for the simulation. In order to continue improving the quality of Hopfield model solutions, the Divide-and-Conquer model is proposed, a model based in two consecutive Hopfield models. This model connects neighboring cities in the first phase and, in a second phase, the connected groups of cities from the previous phase. In both cases, the corresponding parametrizations of the problem are deduced and solved with the Hopfield model. Finally, it is studied how a particular case of the Hopfield model used in the second phase of the Divide-and-Conquer model actually behaves as a 2-opt algorithm, provided that the starting point is chosen appropriately. All these results improve the performance of the Hopfield model as a heuristic, matching it to the 2-opt algorithm in terms of quality of the solution.
The continuous Hopfield model can be used as a heuristic to solve combinatorial optimization problems, such as the Traveling Salesman Problem (TSP). It consists of a recurrent neural network with an associated differential equation, whose states evolve from an initial point to an equilibrium point by minimizing a Lyapunov function. By associating this function with the objective function and constraints of the optimization problem, the equilibrium points correspond to feasible solutions of the optimization problem. Applying Hopfield's original formulation to solve the TSP, this process leaves a free parameter. Historically, researchers in the field have used small values of this free parameter, without explaining why better solutions are obtained. This dissertation analyzes the relation between the free parameter and the saddle point of the Hopfield model. Whereas in small problems this result guarantees that the global optimum is always obtained, in more complex instances, such as the TSP, this is far more complicated. However, in the surroundings of the saddle point, the attractor basins for the best solutions grow as the free parameter decreases. Thus, saddle point neighbors become excellent starting point candidates for the simulation. In order to continue improving the quality of Hopfield model solutions, the Divide-and-Conquer model is proposed, a model based in two consecutive Hopfield models. This model connects neighboring cities in the first phase and, in a second phase, the connected groups of cities from the previous phase. In both cases, the corresponding parametrizations of the problem are deduced and solved with the Hopfield model. Finally, it is studied how a particular case of the Hopfield model used in the second phase of the Divide-and-Conquer model actually behaves as a 2-opt algorithm, provided that the starting point is chosen appropriately. All these results improve the performance of the Hopfield model as a heuristic, matching it to the 2-opt algorithm in terms of quality of the solution.
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Tesis de la Universidad Complutense de Madrid, Facultad de Ciencias Matemáticas, Departamento de Estadística e Investigación Operativa, leída el 15-12-2017