1. Neural Networks for Optimization William J. Wolfe California State University Channel Islands

2. Neural Models Simple processing units Lots of them Highly interconnected Exchange excitatory and inhibitory signals Variety of connection architectures/strengths “Learning”: changes in connection strengths “Knowledge”: connection architecture No central processor: distributed processing

3. Simple Neural Model a i Activation e i External input w ij Connection Strength Assume: w ij = w ji (“symmetric” network)  W = (w ij ) is a symmetric matrix

4. Net Input

5. Dynamics Basic idea:

6. Energy

8. Lower Energy da/dt = net = -grad(E)  seeks lower energy

9. Problem: Divergence

10. A Fix: Saturation

11. Keeps the activation vector inside the hypercube boundaries Encourages convergence to corners

12. Summary: The Neural Model a i Activation e i External Input w ij Connection Strength W (w ij = w ji ) Symmetric

13. Example: Inhibitory Networks Completely inhibitory –wij = -1 for all i,j –k-winner Inhibitory Grid –neighborhood inhibition

14. Traveling Salesman Problem Classic combinatorial optimization problem Find the shortest “tour” through n cities n!/2n distinct tours

15. TSP 50 City Example

16. Random

17. Nearest-City

18. 2-OPT

19. Centroid

20. Monotonic

21. Neural Network Approach neuron

22. Tours – Permutation Matrices tour: CDBA permutation matrices correspond to the “feasible” states.

23. Not Allowed

24. Only one city per time stop Only one time stop per city  Inhibitory rows and columns inhibitory

25. Distance Connections: Inhibit the neighboring cities in proportion to their distances.

26. putting it all together:

27. Research Questions Which architecture is best? Does the network produce: –feasible solutions? –high quality solutions? –optimal solutions? How do the initial activations affect network performance? Is the network similar to “nearest city” or any other traditional heuristic? How does the particular city configuration affect network performance? Is there any way to understand the nonlinear dynamics?

28. typical state of the network before convergence

29. “Fuzzy Readout”

30. Neural Activations Fuzzy Tour Initial Phase

32. Neural ActivationsFuzzy Tour Monotonic Phase

33. Neural ActivationsFuzzy Tour Nearest-City Phase

34. Fuzzy Tour Lengths tour length iteration

35. Average Results for n=10 to n=70 cities (50 random runs per n) # cities

36. DEMO 2 Applet by Darrell Long http://hawk.cs.csuci.edu/william.wolfe/TSP001/TSP1.html

37. Conclusions Neurons stimulate intriguing computational models. The models are complex, nonlinear, and difficult to analyze. The interaction of many simple processing units is difficult to visualize. The Neural Model for the TSP mimics some of the properties of the nearest-city heuristic. Much work to be done to understand these models.

38. EXTRA SLIDES

39. Brain Approximately 10 10 neurons Neurons are relatively simple Approximately 10 4 fan out No central processor Neurons communicate via excitatory and inhibitory signals Learning is associated with modifications of connection strengths between neurons

41. Fuzzy Tour Lengths iteration tour length

42. Average Results for n=10 to n=70 cities (50 random runs per n) # cities tour length

44. with external input e = 1/2

45. Perfect K-winner Performance: e = k-1/2