Title: Hopfield Neural Networks for Optimization
1Hopfield Neural Networksfor Optimization
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2Content
- Introduction
- A Simple Example ? Race Traffic Problem
- Example ? A/D Converter
- Example ? Traveling Salesperson Problem
3Hopfield Neural Networksfor Optimization
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4Energy Function of a Hopfield NN
Interaction btw neurons
Interaction to the external
constant
Running a Hopfield NN asynchronously, its energy
is monotonically non-increasing.
5Solving Optimization Problems Using Hopfield NNs
- Reformulating the cost of a problem in the form
of energy function of a Hopfield NN. - Build a Hopfield NN based on such an energy
function. - Running the NN asynchronously until the NN
settles down. - Read the answer reported by the NN.
6Hopfield Neural Networksfor Optimization
- A Simple Example
- Race Traffic Problem
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7A Simple Hopfield NN
8The Race Traffic Problem
9The Race Traffic Problem
10The Race Traffic Problem
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1
Stable State
11The Race Traffic Problem
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1
Stable State
12The Race Traffic Problem
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How about if to run synchronously?
13Hopfield Neural Networksfor Optimization
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14Reference
Tank, D.W., and Hopfield, J.J., Simple "neural"
optimization networks An A/D converter, signal
decision circuit and a linear programming
circuit, IEEE Transactions on Circuits and
Systems, Vol. CAS-33 (1986) 533-541.
15A/D Converter
A/D
I
Analog
Using Unipolar Neurons
16A/D Converter
Using Unipolar Neurons
17A/D Converter
18Hopfield Neural Networksfor Optimization
- Example
- Traveling Salesperson Problem
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19Reference
J. J. Hopfield and D. W. Tank, Neural
computation of decisions in optimization
problems, Biological Cybernetics, Vol. 52,
pp.141-152, 1985.
20Traveling Salesperson Problem
21Traveling Salesperson Problem
Given n cities with distances dij, what is the
shortest tour?
22Traveling Salesperson Problem
2
3
4
1
5
6
9
11
8
10
7
23Traveling Salesperson Problem
Distance Matrix
Find a minimum cost Hamiltonian Cycle.
24Search Space
Assume we are given a fully connection graph with
n vertices and symmetric costs (dijdji).
The size of search space is
Find a minimum cost Hamiltonian Cycle.
25Problem Representation Using NNs
Time
2
1
4
City
3
5
26Problem Representation Using NNs
The salesperson reaches city 5 at time 3.
Time
2
1
4
City
3
5
27Problem Representation Using NNs
Goal Find a minimum cost Hamiltonian
Cycle.
Time
2
1
4
City
3
5
28The Hamiltonian Constraint
Goal Find a minimum cost Hamiltonian
Cycle.
Time
- Each row and column can have only one neuron
on. - For a n-city problem, n neurons will be on.
2
1
4
City
3
5
29Cost Minimization
Goal Find a minimum cost Hamiltonian
Cycle.
Time
- The total distance of the valid tour have to be
very low.
2
1
4
City
The summation of these dijs is very low.
3
5
30Indices of Neurons
i
Time
vxi
City
x
31Energy Function
32Energy Function
n neurons on
Each row one or zero neuron on
Each column one or zero neuron on
33Energy Function
Total distance of the tour
34Energy Function
35Build NN for TSP
Mapping
Energy function of a 2-D neural network
36Analog Hopfield NN for 10-City TSP
37Analog Hopfield NN for 10-City TSP
The shortest path
38Analog Hopfield NN for 10-City TSP
The shortest path
39Analog Hopfield NN for 30-City TSP
40Analog Hopfield NN for 30-City TSP