Simultaneous Recurrent Neural Networks for Static Optimization Problems

By: Amol Patwardhan

Adviser: Dr. Gursel Serpen

August, 1999

The University of Toledo

Drawbacks of conventional computing systems:-• Perform poorly on complex problems• Lack the computational power• Do not utilize the inherent parallelism of problems

Advantages of Artificial Neural Networks:-• Perform well even on complex problems• Very fast computational cycles if implemented in hardware• Can take advantage of the inherent parallelism of problems

Driving Force for the Research

Earlier Efforts to solve Optimization Problems

• Many ANN algorithms with feedforward and recurrent architectures have been used to solve unconstrained and

combinatorial optimization problems

• The Hopfield network and its derivatives including Boltzmann machine and MFA seem to be most prominent and extensively applied ANN algorithms to solve these static optimization problems.

• However HN and their derivatives do not scale well with the increase in the size of the optimization problem.

Statement of Thesis

Simultaneous Recurrent Neural Network, a trainable and recurrent ANN, to address the scaling problem Artificial Neural Network algorithms currently experience for static optimization problems

Can we use ….

Research Approach

• A neural network simulator is developed for simulation of Simultaneous Recurrent Neural Network

• An extensive simulation study is conducted on two well known static optimization problems

- Traveling Salesman Problem- Graph Path Search Problem

• Simulation results are analyzed

Significance of Research

• A powerful and efficient optimization tool

• Optimizer can solve real-life size and complex static optimization problems

• Will require a fraction of time if implemented in hardware

• Applications in many fields like- Routing in computer networks- VLSI circuit design- Planning in operational and logistic systems- Power distribution systems- Wireless and satellite communication systems

Hopfield Network and Static Optimization Problems

• Most widely used ANN algorithms

• Offer a computationally simple way for a class of optimization problems

• HN dynamics minimizes a quadratic Lyapunov function

• Employed as fixed-point attractors

• Performance greatly depends on constraint weight parameters

Shortcoming of the Hopfield Network

• Constraint weight parameters are set empirically• All weights and connections are specified in advance

• Difficult to guess weights for large-scale problems

• Lack mechanism to incorporate the experience gained

• Quality of solution not good for large scale TSP

• Do not scale well with increase in the problem size

• Acceptable solution for Graph Path Search Problem can not be found

Why Simultaneous Recurrent Neural Network

• Hopfield Network do not employ any learning that can benefit from prior relaxations

• A relaxation based neural search algorithm, which can learn from its own experience is needed Simultaneous Recurrent Neural Network is a …..

- Recurrent algorithm- has relaxation search capability- has ability to learn

Simultaneous Recurrent Neural Network

Feedforward

Mapping

(.,W)

Outputs Y

Feedback Path

Inputs X

Simultaneous Recurrent Neural Network is a feedforward network with simultaneous feedback from outputs of the network to its inputs without any time delay

Simultaneous Recurrent Neural Network

• Follows a trajectory in the state space to relax to a fixed point

• The network is provided with the external inputs and initial outputs are typically assumed randomly

• The output of previous iteration is fedback to the network along with the external inputs to compute the output of next iteration

• The network iterates until it reaches a stable equilibrium point

Training of SRN

Methods available for training of SRN in literature

Backpropagation Through Time (BTT) which requires the knowledge of desired outputs throughout the trajectory path

Error critics (EC) has no quarantee of yielding exact results in equilibrium

Truncation did not provide satisfactory results and needs to be further tested

Recurrent Backpropagation requires only knowledge of desired outputs at the end of trajectory path and hence chosen to train SRN

Traveling Salesman Problem

Network Topology for Traveling Salesman Problem

Hidden Layers

Cost Matrix

Output Array

N Nnodes

N Nnodes

N Nnodes

Input Layer Hidden Layer(s) Output Layer

Path

Specification

Error Computation for TSP

Constraints used for training TSP

Asymmetry of the path traveled Column inhibition Row inhibition Cost of the path traveled Values of the solution matrix

SourceCities

Destination Cities

Output Matrix

Graph Path Search Problem

Source

Destination

Network Topology for Graph Path Search Problem

Hidden LayersCost

Matrix

Output Array

N Nnodes

N Nnodes

N Nnodes

Input Layer Hidden Layer(s) Output Layer

Path

SpecificationN Nnodes

Adjacency Matrix

Error Computation for GPSP

Constraints used for training GPSP Asymmetry of the sub-graph Column inhibition Row inhibition Source and target vertex inhibition Column/row excitation Row/column excitation Cost of the solution path Number of vertices in the path

SourcesVertices

Destinations Vertices

Output Matrix

Simulation:-

Software Environment

Language: C, MATLAB 5.2GUI: Xwindows11Plotting of Graphs: C program calling MATLAB functions for plotting of graphs

Hardware Environment

Sun OS 5.7Sun Ultra machine 300MHzPhysical Memory (RAM) 1280 MBVirtual Memory (Swap) 1590 MB

Simulation:- GUI for Simulator

Simulation:- Initialization

• Randomly initialize weights and initial outputs (Range: 0.0 - 1.0)

• Randomly initialize cost matrix for TSP (Range: 0.0 - 1.0)

• Randomly initialize adjacency matrix ( 0.0 or 1.0) depending on the connectivity level parameter for GPSP

• For TSP, values along the diagonal of the cost matrix are clamped to 1.0 to avoid self looping.

• For GPSP, values along the diagonal of the adjacency matrix and cost matrix are clamped to 1.0 to avoid self looping.

• Values of constraint weight parameters are set depending on the size of the problem

Simulation:- Initialization for TSP

Cities Values Asymmetry Row/Column Output Value Cost

Initial 0.008 0.0050 0.0010 0.015040

Inc (30) ------- 0.0080 0.0030 0.0020

Initial 0.008 0.0020 0.0010 0.015050

Inc (30) ------ 0.0060 0.0030 0.0020

Initial 0.005 0.0010 0.0010 0.015075

Inc (30) ------- 0.0050 0.0015 0.0020

Initial 0.005 0.0010 0.0010 0.0150100

Inc (30) ------- 0.0040 0.0015 0.0015

Initial 0.005 0.0010 0.0010 0.0150200

Inc (30) ------ 0.0030 0.0015 0.0015

Initial 0.005 0.0005 0.0010 0.0150300

Inc (30) ------ 0.0022 0.0015 0.0015

Initial 0.004 0.0004 0.0010 0.0150400

Inc (30) ------ 0.0016 0.0010 0.0010

Initial 0.003 0.0004 0.0010 0.0120500

Inc (30) ------ 0.0012 0.0010 0.0010

Initial values and Increments per 30 relaxation of constraint weight parameters for the TSP

Simulation:- Training

Error function vs. Simulation Time for TSP

Simulation:- Results

Convergence criteria of network is checked after every 100 relaxations Criteria: 95% of active nodes have value greater than 0.9

Cities NormalizedDistance

between Cities

ComputationalTime inmin/100

Relaxations

Average Numberof Relaxations for

Solution

TotalComputational

Time

40 0.26 0.13 1700 2.21 min.

50 0.27 0.19 2400 4.56 min.

75 0.32 0.42 3200 13.44 min.

100 0.31 0.75 3600 27.00 min.

200 0.25 3.20 4200 134.40 min.

300 0.28 7.82 5100 398.82 min.

400 0.30 14.65 5700 835.05 min.

500 0.27 22.50 6400 1440.00 min.

Simulation:- Results

Normalized Distance vs. Problem Size

0

0.1

0.2

0.3

0.4

0.5

0.6

0 40 50 75 100 200 300 400 500

Problem Size

No

rma

lize

d D

ista

nc

e

Plot of Normalized Distance between cities after the convergence of networkto an acceptable solution vs. Problem Size

Simulation:- Results

Plot of Number of Relaxations required for a solution and values ofConstraint Weight Parameters gc and gr after 300 Relaxations vs. Problem Size

Problem Size vs Number of Relaxations and

Problem Size vs Constraint Weight Parameter gc or gr

0

1000

2000

3000

4000

5000

6000

7000

40 50 75 100 200 300 400 500

Problem Size

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Relaxations Constraint Weight Parameter

Simulation:- Initialization for GPSP

Vertices Values Asymmetry Row/Column

Cost Row/Column andColumn/Row

excitation

Total PathLength

Source andTarget vertex

Initial 0.0001 0.000050 0.005 0.001 0.000500 0.001040

Inc (30) 0.0001 0.000100 0.005 0.080 0.001000 0.0200

Initial 0.0001 0.000050 0.005 0.001 0.000600 0.001050Inc (30) 0.0001 0.000100 0.005 0.080 0.000800 0.0150

Initial 0.0001 0.000030 0.005 0.001 0.000050 0.000575

Inc (30) 0.0001 0.000100 0.005 0.080 0.000350 0.0050

Initial 0.0001 0.000030 0.005 0.001 0.000020 0.0001100

Inc (30) 0.0001 0.000100 0.005 0.080 0.000200 0.0005Initial 0.0001 0.000020 0.005 0.001 0.000020 0.0001200

Inc (30) 0.0001 0.000030 0.005 0.080 0.000050 0.0005

Initial 0.0001 0.000020 0.005 0.001 0.000010 0.0001300

Inc (30) 0.0001 0.000030 0.005 0.080 0.000020 0.0005

Initial 0.0001 0.000020 0.005 0.001 0.000005 0.0001400Inc (30) 0.0001 0.000015 0.005 0.080 0.000010 0.0005

Initial 0.0001 0.000010 0.005 0.001 0.000002 0.0001500

Inc (30) 0.0001 0.000010 0.005 0.080 0.000005 0.0002

Initial values and Increments per 30 relaxation of constraint weight parameters for the GPSP

Simulation:- Results for GPSP

Convergence criteria of network is checked after every 100 relaxations Criteria: Active nodes have value greater than 0.8

Vertices Computational Timein min/100 Relaxations

ConnectivityLevel Parameter

Path LengthAchieved

Average Numberof Relaxations

Total ComputationTime

0.300 4 1100 1.540.200 6 1500 2.10

40 0.14

0.080 8 3200 4.480.200 4 1500 3.000.100 6 2300 4.60

50 0.2

0.050 8 3800 7.600.100 4 2200 9.460.050 6 3700 15.91

75 0.43

0.030 8 4500 19.350.050 4 2500 19.750.030 6 4200 33.18

100 0.79

0.020 8 5100 40.290.050 4 2600 89.700.020 6 3200 110.40

200 3.45

0.020 8 5500 189.750.020 4 2700 219.240.010 6 4700 381.64

300 8.12

0.010 8 5900 479.080.010 4 2700 409.590.010 6 4800 728.16

400 15.17

0.008 8 6500 986.050.010 4 2900 685.850.010 6 5100 1206.15

500 23.65

0.006 8 6600 1560.90

Simulation:- Results for GPSP

Problem Size vs Number of Relaxations and

Problem Size vs Constraint Weght Parameter g i

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2000

3000

4000

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6000

40 50 75 100 200 300 400 500

Problem Size

Ave

rag

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Average Number of Relaxations Constraint Weight Parameter

Plot of Number of Relaxations required for a solution and values ofConstraint Weight Parameters gi after 300 Relaxations vs. Problem Size

Conclusions

• The SRN with the RBP was able to find “good quality” solutions, in the range of 0.25-0.35, for large-scale (40 to 500 city) Traveling Salesman Problem

• Solutions were obtained within acceptable computation efforts

• Normalized Distance between cities remained almost consistent as the problem size was varied from 40 to 500 cities

• The simulator developed does not require weights to be predetermined before simulation as is the case with the HN and its derivatives

• The initial and incremental values of constraint weight parameters play very important role in the training of the network

Conclusions (continued)

• Computational effort and memory requirement increased proportional to the square of the problem size

• The SRN with the RBP was able to find a solution for large-scale Graph Path Search Problem in the range of 40 to 500 vertices

• The solutions were obtained within acceptable computation efforts and time

• The computation effort required for the GPSP is 1.1 to 1.2 times more than that of the TSP

• The number of relaxations required increased with the increase in the problem size

• The GPSP was very sensitive to the constraint weight parameters

Conclusions (continued)

Thus we can say that ….

Simultaneous Recurrent Neural Network with Recurrent Backpropagation training algorithm scaled wellfor large-scale static optimization problems like the TravelingSalesman Problem and the Graph Path Search Problem within acceptable computation effort bounds.

Recommendations for Future Study

• The feasibility of the hardware implementation of the network and algorithm for the TSP should be thought over

• More number of simulations should be done for the GPSP to find the effect of change in each constraint weight parameter on the solution

• The effect of incorporating a stochastic or probabilistic component into the learning for network dynamics can also be studied to find the better approach

• Simulation study on weighted GPSP should be done for more practical use

Questions ?