Neural Networks Chapter 4 Joost N. Kok Universiteit Leiden

Hopfield Networks Optimization Problems (like Traveling Salesman) can be encoded into Hopfield Networks Fitness corresponds to energy of network Good solutions are stable points of the network

Hopfield Networks Three Problems Weighted Matching Traveling Salesman Graph Bipartitioning

Hopfield Networks Weighted matching Problem: Let be given N points with distances dij Connect points together in pairs such that the total sum of distances is as small as possible

Hopfield Networks Variables: nij (i<j) with values 0/1 Constraint: Sj nij = 1 for all i Optimize: Si<j dij nij

Hopfield Networks Penalty Term approach: put constraints in optimization criterion Weights and thresholds of Hopfield Network can be derived from

Hopfield Networks Travelling Salesman Problem (TSP): Given N cities with distances dij . What is the shortest tour?

Hopfield Networks Construct a Hopfield network with N2 nodes Semantics: nia = 1 iff town i on position a in tour

Hopfield Networks Constraints:

Hopfield Networks 0/1 Nodes Nodes within each row connected with weight –g Nodes within each column connected with weight –g Each node is connected to nodes in columns left and right with weight –dij (Often) continuous activation

Hopfield Networks postion city 1 2 3 4 A B C D

Hopfield Networks Graph bipartitioning: divide nodes in two sets of equal size in such a way as to minimize the number of edges going between the sets +1/-1 Nodes 0/1 Connection matrix Cij

Hopfield Networks

Hopfield Networks