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Neural Networks for Optimization

This research examines the application of simple neural network models to optimize the Traveling Salesman Problem (TSP), a classic combinatorial challenge of finding the shortest route through a set of cities. Utilizing a neural framework with excitatory and inhibitory connections, the study investigates various neural architectures and learning dynamics that affect convergence and solution quality. Key questions focus on network performance, initial activations, and the similarities to traditional heuristic methods. Results are presented for multiple city configurations, contributing insights into computational models inspired by neural dynamics.

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Neural Networks for Optimization

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  1. Neural Networks for Optimization Bill 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 • aiActivation • ei External input • wij Connection Strength Assume: wij = wji (“symmetric” network) W = (wij) is a symmetric matrix

  4. Net Input

  5. Dynamics • Basic idea:

  6. Energy

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

  8. Problem: Divergence

  9. A Fix: Saturation

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

  11. Summary: The Neural Model aiActivation eiExternal Input wijConnection Strength W (wij = wji) Symmetric

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

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

  14. TSP solution for 15,000 cities in Germany

  15. TSP50 City Example

  16. Random

  17. Nearest-City

  18. 2-OPT

  19. An Effective Heuristic for the Traveling Salesman Problem S. Lin and B. W. Kernighan Operations Research, 1973 http://www.jstor.org/view/0030364x/ap010105/01a00060/0

  20. Centroid

  21. Monotonic

  22. Neural Network Approach neuron

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

  24. Not Allowed

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

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

  27. putting it all together:

  28. 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?

  29. typical state of the network before convergence

  30. “Fuzzy Readout”

  31. Initial Phase Fuzzy Tour Neural Activations

  32. Monotonic Phase Fuzzy Tour Neural Activations

  33. Nearest-City Phase Fuzzy Tour Neural Activations

  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 1010 neurons • Neurons are relatively simple • Approximately 104 fan out • No central processor • Neurons communicate via excitatory and inhibitory signals • Learning is associated with modifications of connection strengths between neurons

  40. Fuzzy Tour Lengths tour length iteration

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

  42. with external input e = 1/2

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

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