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

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

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

Net Input

Dynamics Basic idea:

Energy

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

Problem: Divergence

A Fix: Saturation

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

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

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

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

TSP 50 City Example

Random

Nearest-City

2-OPT

Centroid

Monotonic

Neural Network Approach neuron

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

Not Allowed

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

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

putting it all together:

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?

typical state of the network before convergence

“Fuzzy Readout”

Neural Activations Fuzzy Tour Initial Phase

Neural ActivationsFuzzy Tour Monotonic Phase

Neural ActivationsFuzzy Tour Nearest-City Phase

Fuzzy Tour Lengths tour length iteration

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

DEMO 2 Applet by Darrell Long

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.

EXTRA SLIDES

Brain Approximately 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

Fuzzy Tour Lengths iteration tour length

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

with external input e = 1/2

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