1. Neural Networks Chapter 4
Joost N. Kok
Universiteit Leiden

2. 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

3. Hopfield Networks Three Problems Weighted Matching Traveling Salesman
Graph Bipartitioning

4. 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

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

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

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

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

10. Hopfield Networks
Constraints:

11. 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

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

13. 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

15. Hopfield Networks

16. Hopfield Networks