Probability Collectives

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Probability Collectives: A Distributed Optimization for Multi-Agent Systems

Anand J. Kulkarni, Tai KangOptimization and Agent Technology Research (OAT Research) Lab

www.oatresearch.org

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Outline

Introduction

Motivation and Objectives

Probability Collectives (PC)

Unconstrained PC Formulation

Validation of the Unconstrained PC

Constrained Handling TechniquesHeuristic ApproachPenalty Function ApproachFeasibility-based Rule IFeasibility-based Rule II

Conclusions

Future Recommendations

3

Introduction- What are Complex Systems?

Complex systems: a broad term encompassing a research approach to problems in the diverse areas such as Social Structures, earthquake prediction, climate change and weather forecasting, counter-terrorism, financial systems, project rescheduling, molecular biology, cybernetics, etc.

Complex systems generally have many (interconnected) components that not only interact but also compete with one another to deliver the best they can to reach the desired system objective.

Any move by a component affects the moves by other components and so on. So it is difficult to understand the behavior of the entire system simply by knowing the individual components and their behavior

Complex Systems in Engineering:

1) Internet Search2) Manufacturing and Scheduling3) Supply Chain4) Sensor Networks5) Aerospace Systems6) Telecommunication Infrastructure

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Introduction- Solving Complex Systems- Centralized System

Limitations:

1. Communication Overload

2. Computational Overload

3. Large Storage Space

4. Processing Bottleneck

5. Adds Latency (delay)

6. Limited Scalability

7. Reduced Robustness

A Single/Central Agent is supposed to have all the capabilities such as problem solving in order to alleviate user’s cognitive load.

The Agent is provided with general knowledge, storage space, etc. to deal with wide variety of tasks/computations.

Central Agent

Tasks/Sensors

Centralized System

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Introduction- Solving Complex Systems- Distributed System

Advantages

1. Reduced Risk of Bottleneck

2. Reduced Risk of Latency

3. Robustness

4. Highly Scalable

5. Easy to Maintain & Debug

In a Decentralized and Distributed System, the total work is decomposed into different expert modules. Each expert module is an autonomous agent, i.e. having local control, decision. All Agents achieve their individual goals contributing towards the system objective.

Local cooperation is to avoid the duplication of the work.

Challenges

1. Coordination

2. Handling Constraints

Probability Collectives (PC): Motivation and Objectives

• GA, PSO, ACO, Wasp Colony System, Swarm-bot, etc. have been used for solving complex problems

• As the complexity of the problem domain grew these problems became quite tedious to be solved using above algorithms.

• Probability Collectives is an emerging AI tool in the framework of COllective INtelligence (COIN) for modeling and controlling distributed MAS. Proposed by Dr. David Wolpert in 1999 in a technical report presented to NASA and further elaborated by S.R. Bieniawski in 2005.

• It is an obvious tool to deal with the increasing complexity as it decomposes the problem into sub-problems.

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State-of-the-Art - Probability Collectives (PC)

• Joint Routing and Resource Allocation in Wireless Sensor Networks --- Choosing the optimal number of nodes in a cluster and the cluster head

(Ryder et al. 2005, Mohammed et al. 2007)

• Solving the Benchmark Problems– Multimodality, non-separability, non-linearity, etc. (Huang et al. 2005)

– Robustness, rate of descent, trapping in false minima, etc.

• University Course Scheduling (Autry et al. 2008)

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State-of-the-Art - Probability Collectives (PC)

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Mechanical Design10 bar truss problem (Bieniawski et al. 2004)

Conflict ResolutionAirplanes Collision Avoidance (Sislak et al. 2011)

Airplane fleet assignment(Wolpert et al. 2004)

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Objectives: Probability Collectives (PC)

Develop a more generic and powerful approach of PC by incorporating constraint handling techniques necessary for solving constrained optimization problems and further test these techniques by solving a variety of challenging constrained problems

Solve the path planning of Multiple Unmanned Aerial Vehicles (MUAVs) by modeling it as a MTSP and solving by the PC approach

Modify the PC approach to make it more efficient and faster- inherent and desirable characteristics - key benefits of being a distributed, decentralized and

cooperative approach

Characteristics of PC

PC works through the COllective INtelligence (COIN) frameworkexploiting the advantages of Decentralized, Distributed & Cooperativeapproach.

• Deep connections to Game Theory, Statistical Physics & Optimization

• Successfully exploits the important concept of “Nash Equilibrium”

• PC can be applied to continuous, discrete or mixed variables, etc.,

• Works on Probability Distribution directly incorporating Uncertainty

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Characteristics of PC

• The Homotopy function for each agent (variable) helps the algorithm to jump out of the local minima and further reach the global minima.

• It can successfully avoid the tragedy of commons, skipping the local minima and further reach the true global minimum.

• It can efficiently handle problems with a large number of variables i.e. scalable.

• It is robust and can accommodate the agent failure case.

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Formulation of Unconstrained PC

• Consider a general unconstrained problem (in minimization sense) comprising variables

• Variables Agents/Players of a game being played iteratively. • Initially, every agents is given a sampling interval/space

• Every agent randomly samples strategies from within the corresponding sampling interval .

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1 2 1, ,..., ,..., ,i N NG f X X X X XX

,lower upperi i i

N

[ ][1] [2] [ ]{ , ,..., ,..., } , 1,2,...,imri i i i iX X X X i N and X

1 2 1... ...i N Nm m m m m

i imi

13


[1] [?] [?] [1] [?] [?]1 2 1, ,..., ,..., ,i i N NX X X X XY

Agent selects its first strategy and samples randomly from other agents’ strategies as well.

[1]iG Y

1 [ ] [ ][ ][1] [2] [1] [2] [1] [2]1 1 1 1{ , ,..., } ,..., { , ,..., } ,..., { , ,..., }i Nm mm

i i i i N N N NX X X X X X X X X X X X

[2] [?] [?] [2] [?] [2]1 2

[3] [?] [?] [3] [?] [3]1 2

[ ] [?] [?] [ ] [?] [ ]1 2

[ ] [ ] [ ][?] [?] [?]1 2

, ,..., ,...,

, ,..., ,...,

, ,..., ,...,

, ,..., ,...,i i i

i i N i

i i N i

r r ri i N i

m m mi i N i

X X X X G

X X X X G

X X X X G

X X X X G

Y Y

Y Y

Y Y

Y Y

[ ]

1

imr

ir

G

Y

i


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• The ultimate goal of every agent is to identify its strategy value

which contributes the most towards the minimization of the sum

(collection) of these system objectives i.e. .

• Possibly many local minima

• Directly minimizing may require excessive computational efforts

• Homotopy Method: modify the function by converting it into

another topological space by constructing a related and easier

function . This forms the Homotopy function:

[ ]

1

imr

ir

G Y

i

[ ]

1

, ( ) , 0,im

ri i i i

r

J q T G T f T

X Y X

if X


• Analogy to Helmholtz free energy

One of the ways to achieve the thermal equilibrium and hence minimize

the energy to do work is actually minimizing the internal energy through

an annealing schedule, i.e. stepwise drop the temperature of the

system from to achieving the equilibrium in

every step.

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[ ]

1

, ( ) , 0,im

ri i i i

r

J q T G T f T

X Y X

L D T S

Energy available to do work Internal energy Spontaneous (Random) energy

initialT T 0 finalT or T T


Deterministic Annealing• It suggests conversion of the variables into random real valued

probabilities which converts the into .

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[ ] [ ] [ ]2

1 1

, ( ) ( ) log ( ) , 0,i im m

r r ri i i i i

r r

J q T E G T q X q X T

X Y

[ ]

1

( )im

ri

r

G Y [ ]

1

( )im

ri

r

E G Y

[ ]

1

, ( ) , 0,im

ri i i i

r

J q T G T f T

X Y X

[ ]

1

, ( ) , 0,im

ri i i i

r

J q T E G T S T

X Y


17

0

0.05

0.1

0.15

1 2 3 4 5 6 7 8 9 100

0.05

0.1

0.15

1 2 3 4 5 6 7 8 9 10

11 1 1... 1 /imq X q X m 1 ... 1 /im

N N Nq X q X m

0

0.05

0.1

0.15

1 2 3 4 5 6 7 8 9 10

1 ... 1 /imi i iq X q X m

Agent 1 Agent i Agent N

[1] [?] [1] [?]1

[ ] [?] [ ] [?]1

[ ] [ ][?] [?]1

,..., ,...,

,..., ,...,

,..., ,...,i i

i i N

r ri i N

m mi i N

X X X

X X X

X X X

Y

Y

Y

[ ]

1

imr

ir

E G Y

1 1 ?[?] [1] [?] [1]1

?[?] [ ] [?] [ ]1

?[ ] [ ][?] [?]1

,..., ,..., Y

,..., ,..., Y

,..., ,..., Y i ii i

i N i i iii

r rr ri N i i ii

i

m mm mi N i i ii

i

q X q X q X G q X q X E G



Y

Y

Y

Strategies Strategies Strategies


• The minimization of the Homotopy function can be carried out using a suitable second order optimization approach such Nearest Newton Descent Scheme as well as Broyden-Fletcher-Goldfarb-Shanno (BFGS) scheme.

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0

0.05

0.1

0.15

1 2 3 4 5 6 7 8 9 100

0.05

0.1

0.15

1 2 3 4 5 6 7 8 9 10

0

0.05

0.1

0.15

1 2 3 4 5 6 7 8 9 10

00.20.40.60.81

1 2 3 4 5 6 7 8 9 10

00.20.40.60.81

1 2 3 4 5 6 7 8 9 10

0

0.2

0.4

0.6

0.8

1 2 3 4 5 6 7 8 9 10

1 2 1, ,..., ,..., ,fav fav fav fav fav fav favi N NX X X X X G Y Y

Favorable Strategy Favorable Strategy Favorable Strategy



• Updating of the Sampling Interval (Neighboring Method)

• Convergence and Final Solution

If

If there is no significant change in the system objectives for

successive considerable number of iterations

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, , 0 1fav favupper lower upper loweri i down i i i down i i downX X

, , , , , ,1 2 1, ,..., , ( )fav final fav final fav final fav final fav final fav final

N NX X X X G Y Y

0finalT T or T

, , 1( ) ( )fav n fav nG G Y Y

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Nash Equilibrium (Necessary Properties):

Rationality: Select the best possible strategy by guessing other agents’

strategies

Convergence: Same class policy of selecting the best possible strategy and

guessing other agents’ strategies (guaranteed: policy does not change)

Nash Equilibrium in PC

: by guessing other agents’ strategies

and : is communicated to every other agent


faviX

faviX ( )favG Y

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Solution to Rosenbrock Function using PC

1 2 22

11

100 1N

i i ii

f x x x

X

where 1 2 3....... Nx x x xX

lower limit upper limit1,2,...,

ixi N

Agents/(Variables)

Strategy Values Selected with maximum Probability

Trial-1 Trial-2 Trial-3 Trial-4 Trial-5 Range of Values

Agent-1 1.0000 0.9999 1.0002 1.0001 0.9997 -1.0 to 1.0

Agent-2 1.0000 0.9998 1.0001 1.0001 0.9994 -5.0 to 5.0

Agent-3 1.0001 0.9998 1.0000 0.9999 0.9986 -3.0 to 3.0

Agent-4 0.9998 0.9998 0.9998 0.9995 0.9967 -3.0 to 8.0

Agent-5 0.9998 0.9999 0.9998 0.9992 0.9937 1.0 to 10.0

Fun. Value 2 x 10-5 1 x 10-5 2 x 10-5 2 x 10-5 5 x 10-5

Fun. Evals. 288100 223600 359050 204750 242950

Results

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Solution to Rosenbrock using PC (Comparison)Method No. of Var./

AgentsFunction

ValueFunction

EvaluationsVariable Range(s)/

Strategy Sets

CGA 2 0.000145 250 -2.048 to 2.048PAL 2

5≈ 0.01≈ 2.5

5250100000

-2.048 to 2.048-2.048 to 2.048

Modified DE 25

1 × 10-6

1 × 10-61089

11413-5 to 10-5 to 10

LCGA 2 ≈ 0.00003 -- -2.12 to 2.12

PC 5 0.00001 223600 -1.0 to 1.0-5.0 to 5.0-3.0 to 3.0-3.0 to 8.01.0 to 10.0

Unconstrained Test Problems

1. Ackley Function2. Beale Function3. Bohachevsky Function4. Booth Function5. Branin Function6. Colville Function7. Dixon & Price Function8. Easom Function9. Goldstein & Price Function10. Griewank Function11. Hartmann Functions12. Hump Function13. Levy Function14. Matyas Function15. Michalewicz Function

16. Perm Functions17. Powell Function18. Power Sum Function19. Rastrigin Function20. Rosenbrock Function21. Schwefel Function22. Shekel Function23. Shubert Function24. Sphere Function25. Sum Squares Function26. Trid Function27. Zakharov Function

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Constrained PC

• Approach 1: Heuristic Approach Two variations of the MDMTSP and several cases of

the SDMTSP

• Approach 2: Penalty Function Approach Three Test Problems

• Approach 3: Feasibility-based Rule I Two cases of the Circle Packing Problem Feasibility-based Rule II Two cases and associated cases of the Sensor

Network Coverage Problem

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Constrained PC Approach 1: Heuristic Approach

• Explicitly uses the problem specific information and combines them with the unconstrained optimization technique to push the objective function into the feasible region.

• Validated by solving two cases of the Multiple Depot Multiple Traveling Salesmen Problem (MDMTSP) and several cases of the Single Depot Multiple Traveling Salesmen Problem (SDMTSP)

– Solve the path planning of Multiple Unmanned Aerial Vehicles (MUAVs) by modeling it as a MTSP

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Multiple Traveling Salesmen Problem (MTSP)

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1 3 5 7 9 11 13 15 17 19 21 23 25 27

Iterations

Tota

l Tra

velin

g Co

st

(a) Test Case 1 (b) Solution to Test Case 1

Convergence Plot for Test Case 1

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Multiple Traveling Salesmen Problem (MTSP)

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(a) Test Case 2 (b) Solution to Test Case 2

60

110

160

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260

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37

Iterations

Tota

l Tra

velin

g C

ost

Convergence Plot for Test Case 2

-10 -5 0 5 10-10

-8

-6

-4

-2

0

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6

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14

1

3

7

1213

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68

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MTSP (Randomly Located Nodes)

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D

Randomly Located Nodes (Sample Case 1) Randomly Located Nodes (Sample Case 2)

20 30 40 50 60 70 80 90 1000

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D

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(MTSP) ComparisonMethod Nodes Vehicles Avg. CPU Time

(Minutes)PDP*

MTSP to std. TSP 202020

234

2.052.472.95

--

Cutting Plane 202020

234

1.711.501.44

----

Elastic Net+ 222222

234

12.0313.1012.73

28.7174.1833.33

Branch on an Arc with LB0

15 3 0.56 --

Branch on an Arc with LB2

1520

34

1.012.32

--

Branch on an Route withLB0

15 3 0.44 --

PC (MDMTSP)

PC (SDMTSP)

15 (Case 1)15 (Case 2)

15

3

3

2.091.27

3.34

0.000.00

2.94

PDP*: percent deviation from the average solution+: unable to reach the optimum

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MTSP (Heuristic Techniques)

Insertion Heuristic

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Vehicle 1: D1-5-6-7-8-9-10-D1 Vehicle 2: D2-5-1-2-3-4-10-D2Vehicle 1: D1-5-6-7-8-6-9-10-D1Vehicle 2: D2-5-1-2-3-4-11-D2

Vehicle 1: D1-5-6-7-8-6-9-10-D1Vehicle 2: D2-5-1-2-3-4-10-D2

Inter-vehicle Repetition

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Vehicle 1: D1-5-6-7-8-6-9-10-D1 Vehicle 2: D2-5-1-2-3-4-11-D2

Elimination Heuristic

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MTSP (Heuristic Techniques)Elimination Heuristic

Swapping Heuristic

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Vehicle 2: D2-1-2-3-4-11-D2Vehicle 1: D1-5-6-7-8-9-10-D1-20 -15 -10 -5 0 5 10 15 20 25

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Vehicle 2: D2-5-6-7-8-9-10-D2Vehicle 1: D1-1-2-3-4-11-D1 Vehicle 1: D1-1-2-3-4-11-D1Vehicle 2: D2-5-6-7-8-9-10-D2

Vehicle 1: D1-5-6-7-8-9-10-D1Vehicle 2: D2-1-2-3-4-11-D2

Vehicle 1: D1-5-6-7-8-6-9-10-D1 Vehicle 2: D2-1-2-3-4-11-D2

Vehicle 1: D1-5-6-7-8-9-10-D1 Vehicle 2: D2-1-2-3-4-11-D2

Intra-vehicle Repetition

Constrained PC (Approach 2): Penalty Function Approach

32

• Penalty based methods are the most generalized constraint handling

methods: simplicity, ability to handle non linear constraints and

compatibility with most of the unconstrained optimization methods

• Converts constrained optimization problem into unconstrained one.

2 2

[ ] [ ] [ ]

1 1

s tr r r r

i i j i j ij j

G g h

Y Y Y Y

[ ] [ ]max 0,r rj i j iwhere g g and is scalar parameter Y Y

Every agent obtains the probability distribution identifying its favorable strategy

START

Every agent sets up a strategy set. Initialize ‘n’, ‘T’

Every agent forms a combined strategy set for its every strategy and computes system objectives and

constraints, and corresponding collection of pseudo system objectives

Every agent assigns uniform probabilities to its strategies and computes expected collection of system

objectives

Every agent forms a modified Homotopy function

Every agent minimizes the Homotopy function using Nearest Newton Method/BFGS Method

Compute the global objective function and associated constraints

1

2

33

Accept current objective function and related favorable strategies

N

Discard current and retain previous objective function with related favorable strategies

STOP

Accept final values

Convergence ?

Y

YN

Maximum constraint value ≤

1

2

PC…

34

Every agent updates its sampling interval and forms corresponding updated strategy set, and

Update the Penalty Parameter

A

B

Spring Design

35

23 2 1

32 3

1 41

22 1 2

2 23 412 1 1

13 2

2 3

1 24

1 2 3

Minimize 2

Subject to 1 071785

4 1 1 0510812566

140.451 0

1 01.5

where 0.05 2, 0.25 1.3, 2 15

f x x x

x xgx

x x xgxx x x

xgx x

x xg

x x x

X

X

X

X

X

2x

1x

PP

Spring Design

No. of runs Avg. CPU time Best Sol. Mean Sol. Worst Sol. % with Best Sol.

10 24.5 Sec 0013500 0.02607 0.05270 6.6336

Designvariables &Constrains

Best Solutions FoundCulturalalgorithm

Constraintcorrectionalgorithm

Self-adaptive penalty app.

Multi-obj. app. GA

HPSO Proposed PC

0.050000 0.053390 0.051480 0.051980 0.051700 0.0506000.317390 0.399180 0.351660 0.363960 0.357120 0.327810

14.031790 9.185400 11.632200 10.890520 11.265080 14.0567000.000000 0.000010 -0.003300 -0.001900 -0.000000 -0.052900-0.000070 -0.000010 -0.000100 0.000400 0.000000 -0.007400-3.967960 -4.123830 -4.026300 -4.060600 -4.054600 -3.704400-0.755070 -0.698280 -0.731200 -0.722700 -0.727400 -0.7476900.012720 0.012730 0.012700 0.012680 0.012660 0.013500

Fun. Evals 80000 5214

2x

3x1g

2g3g

4gf

0 100 200 300 400 5000

10

20

30

40

50

60

70

Iterations

f(X)

1x

Himmelblau Function

No of runs 10

Avg. CPU time 11 Mins

Best Sol. -30641

Mean Sol. -30635

Worst Sol. -30626

% with Best Sol 0.078

37

23 1 5 1

1 2 5 1 4 3 5

2 2 5 1 4 3 5

3

Minimize 5.3578547 0.8356891 37.293239 40792.141

Subject to 85.334407 0.0056858 0.0006262 0.0022053 92 0

85.334407 0.0056858 0.0006262 0.0022053 0

80.51249 0

f x x x x

g x x x x x x

g x x x x x x

g

X

X

X

X

22 5 1 2 3

24 2 5 1 2 3

5 3 5 1 3 3 4

6 3 5

.0071317 0.0029955 0.0021813 110 0

80.51249 0.0071317 0.0029955 0.0021813 90 0

9.300961 0.0047026 0.0012547 0.0019085 25 0

9.300961 0.0047026 0.0012547

x x x x x

g x x x x x

g x x x x x x

g x x

X

X

X

1 3 3 4

1 2

0.0019085 20 0

where 78 102, 33 45, 27 45, 3, 4,5i

x x x x

x x x i

0 500 1000 1500 2000 2500 3000 3500-3.1

-3

-2.9

-2.8

-2.7

-2.6

-2.5

-2.4

-2.3

-2.2 x 104

Iterations

f(X)

Algorithm Best Mean Worst Std Dev Average FE

Cultural Algorithm -30665.5000 -30662.5000 -30636.2000 9.3

Cultural Differential Evolution

-30665.5386 -30665.5386 -30665.5386 0.00000

Homomorphous Mapping

-30664.0000 -30655.0000 -30645.0000 - 1400000

Filter SA -30665.5380 -30665.4665 -30664.6880 0.173218 86154

Self Adaptive Penalty Approach

-31020.8590 -30984.2400 -30792.4070 73.633

Gradient Repair Method

-30665.5386 -30665.3538 -30665.5386 0.000000 26981

Multiobjective Approach

-30665.5386 -30665.3539 -30665.5386 0.000000 66400

Bubble Sort Algorithm

-30665.5390 -30665.5390 -30665.5390 1.1E-11 350000

PSO (Zahara et al. 2008)

-30665.5386 -30665.35386 -30665.5386 0.000000 19658

Feasibility-based Rule (Deb (2000))

-30665.5370 -- -29846.6540 --

Dynamic Penalty Scheme

-30665.5000 -30665.2000 -30663.3000 4.85E-01 --

PSO (Hu et al. 2002) -30665.5000 -30665.5000 -30665.5000 -- --

PSO (Dong et al. (2005))

-30664.7000 -30662.8000 -30656.1000 -- --

Multi-criteria Approach

-30651.6620 -30647.1050 -30619.0470 35408

HPSO -30665.5390 -30665.5390 -30665.5390 1.7E-06

Stratum Approach -30373.9500 -- -30175.8040 -- --

PC -30641.5702 -30635.4157 -30626.7492 7.5455 278044938

Chemical Equilibrium Problem

39

10

1 1 2 10

1 1 2 3 6 10

2 4 5 6 7

3 3 7 8 9 10

1 2 3 4 5

Minimize ln...

Subject to 2 2 2 0

2 1 0

2 1 0

0.000001, 1,2,...,10

where 6.089 17.164 34.054 5.914 24.721

jj j

j

i

xf x c

x x x

h x x x x x

h x x x x

h x x x x x

x i

c c c c c

X

X

X

X

6 7 8 9 1014.986 24.100 10.708 26.662 22.179c c c c c

Chemical Equilibrium Problem

40

Best Solutions Found

DesignVariables

Hock et al. (1981)

GENOCOP PC

0.01773548 0.04034785 0.0308207485

0.08200180 0.15386976 0.2084261218

0.88256460 0.77497089 0.6708869580

0.0007233256 0.00167479 0.0371668767

0.4907851 0.48468539 0.3510055351

0.0004335469 0.00068965 0.1302810195

0.01727298 0.02826479 0.1214712339

0.007765639 0.01849179 0.0343070642

0.01984929 0.03849563 0.0486302636

0.05269826 0.10128126 0.0486302636

8.6900E-08 6.0000E-08 -0.0089160590

0.0141 1.0000E-08 -0.0090697995

5.9000E-08 -1.0000E-08 -0.0047181958

-47.707579 -47.760765 -46.7080572120

Average FE -- -- 389546

1x

2x

3x

4x

5x

6x

7x

8x

9x

10x

1h X

2h X

3h X

f X

0 200 400 600 800 1000-7

-6

-5

-4

-3

-2

-1

0 x 104

Iterations

f(X)

8000 8200 8400 8600 8800 9000-7000

-6000

-5000

-4000

-3000

-2000

-1000

Iterations

f(X)

No of runs 10

Avg. CPU time 21.60 Mins

Best Sol. -46.7080572120

Mean Sol. -45.6522267370

Worst Sol. -44.4459333503

% with Best Sol 2.20

Constrained PC (Approach 3): Feasibility-based Rule I

• Feasibility-based rule allows the objective and constraint information to be considered separately.

• The constraint violation tolerance is tightened iteratively to obtain the fitter solution and further drive the solution towards the feasibility.

• Convert the equality constraint into inequality constraints

41

MinimizeSubject to 0 , 1,2,...,

0, 1, 2,...,j

j

Gg j s

h j t

0 1,2,...,0

0

MinimizeSubject to 0 , 1, 2,...,

s j jj

s w j j

j

g h j wh

g h

Gg j t


Feasibility-based Rule I:

• Any feasible solution is preferred over any infeasible solution.

• Between two feasible solutions, the one with better objective is preferred.

• Between two infeasible solutions, the one with fewer violated constraints

is preferred.

42

Constrained PC (Approach 3): Feasibility-based Rule I• Updating of the Sampling Space and Perturbation Approach

In order to jump out of this possible local minimum, every agent perturbs its current feasible strategy

The value of and +/- sign are selected based on preliminary trials.

Every agent expands the sampling space as follows:

43

i

1 1

2 2

1,

1,

fav fav favi i i i

lower upperfav

ii

lower upperfav

i

X X X fact

randomvalue ifX

where factrandomvalue if

X

1 1 2 20 1lower upper lower upper

, , 0 1lower upper lower upper upper loweri i up i i i up i i up

44

2 2

1

2 2

Minimize

Subject to

0.001 2, 1,2,...,

z

ii

i j i j i j

i i l

i i u

i i l

i i u

i

f L r

x x y y r r

x r xx r xy r yy r y

Lr

i j z i j

Circle Packing Problem FormulationTragedy of Commons

4 5 6 7 8 9 10 11

5

5.5

6

6.5

7

7.5

8

8.5

9

9.5

10

Shipping, Apparel, Automobile, Aerospace, Food Industry, etc.

Circle Packing Problem (Case 1): Solution History

45

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Randomly Generated Initial Solution Iteration 401 Iteration 901

Iteration 1001 Stable Solution at iteration 1055

No. of Circles Avg CPU time Avg F.E. No. of runs

5 14.05 Mins 17515 30

Circle Packing Problem (Case 2): Solution History

46

3 4 5 6 7 8 9 10 11 123

4

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Randomly Generated Initial Solution Iteration 301 Iteration 401

Stable Solution at iteration 955Iteration 801Iteration 601

No. of Circles Avg CPU time Avg F.E. No. of runs

5 15.70 Min 68406 30

4 5 6 7 8 9 10 11

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Circle Packing Problem with Agent Failure: Solution History

47

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Randomly Generated Initial Solution Iteration 124 Iteration 231 Iteration 377

Iteration 561 Iteration 723 Stable Solution at iteration 901

Stable Solution at iteration 1051

4 5 6 7 8 9 10 11

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1

No. of Circles Avg CPU time Avg F.E. No. of Trials

5 -- 17365 30

Constrained PC (Approach 3): Feasibility-based Rule II

• Feasibility-based rule II allows the objective and constraint information to be considered separately.

• In addition to the iterative tightening of the constraint violation obtaining the fitter solution and further drive the solution towards the feasibility, the rule helps the solution jump out of possible local minima.

• Procedure starts with initializing the number of constraints improved

initialized to , i.e. . The value of is updated iteratively.

48

0 0

Constrained PC (Approach 3): Feasibility-based Rule IIFeasibility-based Rule II:

• Any feasible solution is preferred over any infeasible solution.

• Between two feasible solutions, the one with better objective is preferred.

• Between two infeasible solutions, the one with more number of improved

constraint violations is preferred.

• If the solution remains feasible and unchanged for successive number of

iterations, and current feasible system objective is worse than the

previous feasible solution, accept the current solution.

49

50

Sensor Network Coverage Problem

• Strategic Applications of Sensor NetworkNatural disaster relief, Hostile and Hazardous environment monitoring, criticalinfrastructure monitoring and protection, Habitat exploration and surveillance,Situational awareness in battlefield and target detection, Industrial sensing anddiagnosis, Biomedical health monitoring, Seismic sensing, etc.

• How to best deploy/position the sensors over a FoI to achieve best possible Coverage and Detection capability, connectivity, etc.

• Coverage directly affects the quality and effectiveness of the surveillance/ monitoring provided by the sensor network

51

(Sweep) Barrier coverageBlanket Coverage

Point Set CoverageComplete coverage

Coverage Classification

Deterministic

Static and systematic deployment of the sensors over certain (or weighted) FoI.

Sensor Network Coverage Problem

Stochastic

Sensor positions are selected based on some distributions such as uniform, Gaussian, Poission, etc.

52

1 2 1 2

1 2 1 2

Minimize

max , ,..., ,..., min , ,..., ,...,

max , ,..., ,..., min , ,..., ,...,

Subject to, 2 , 1, 2,..., ,

1,2,...,, ~

i z s i z s

i z s i z s

s

i s l

i s u

i s l

i s u

A x x x x r x x x x r

y y y y r y y y y r

d i j r i j z i j

x r xx r xy r yy r yi zd i j i j

, , , 1,2,...,i j i j z

Sensor Network Coverage Problem Formulation

32

, ,1 1

3z

collective c i c ii i

A A A r

Deploy a set ofHomogeneous sensors over a certain FoI to achieve the maximum possible Deterministic,Connected Blanket Coverage

Sensor Network Coverage Problem (Variation 1) Solution History and Convergence

53

5 6 7 8 9 10 11

4

5

6

7

8

9

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11

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0 500 1000 1500 2000 2500 30005

10

15

20

25

30

35

40

45

Iterations

Are

a of

the

Enc

losi

ng R

ecta

ngle

0 500 1000 1500 2000 2500 30000

0.5

1

1.5

2

2.5

3

3.5

4

Iterations

Col

lect

ive

Cov

erag

e

Sensor Network Coverage Problem (Variation 2- Case 1) Solution History and Convergence

54

2 4 6 8 10 12 14

2

4

6

8

10

12

14

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0 1000 2000 3000 4000 5000 6000 7000 800020

40

60

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100

120

140

160

180

200

220

Iterations

Are

a of

the

Encl

osin

g R

ecta

ngle

0 1000 2000 3000 4000 5000 6000 7000 80000

2

4

6

8

10

12

14

16

18

Iterations

Col

lect

ive

Cov

erag

e


55

2 4 6 8 10 12 140

2

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2

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Summary of Sensor Network Coverage Problem ResultsSN Particulars Variation 1 Variation 21 Cases -- Case 1 Case 2 Case 32 Number of Sensors ( ) 5 5 10 203 The Sensing Range ( ) 0.5 1.2 1 0.64 Average Collective Coverage 3.927 18.5237 19.4856 16.36315 Minimum and Maximum

Collective Coverage3.9270, 3.9270 18.0920, 18.7552 17.5427, 20.8797 15.5347, 17.3377

6 Standard Deviation associated with Collective Coverage

0.0000 0.1687 1.1837 1.2217

7 Average area of the Enclosing Rectangle

5.8311 34.3014 49.0938 39.3480

8 Minimum and Maximum area of the Enclosing Rectangle

5.7046, 5.9750 33.0448, 39.7099 44.7135, 52.6277 34.1334, 43.8683

9 Standard Deviation associated with the area of Enclosing Rectangle

0.1040 1.9899 2.6995 2.8829

10 Average CPU time (Approx.) 20 Mins 1 Hr 2Hrs 3.5 Hrs11 Average number of Function

Evaluations90417 315063 1172759 3555493

56

zsr

57

• Discrete Problems

Variables GA [31]

HS [34]

PSO [36]

PSOPC [36]

HPSO [36]

DHPSACO [37]

Proposed PC

1x 0.4 0.01 0.01 0.01 0.01 0.01 0.01

2 5~x x 2.0 2.0 2.6 2.0 2.0 1.6 0.4

6 9~x x 3.6 3.6 3.6 3.6 3.6 3.2 3.6

10 11~x x 0.01 0.01 00.01 0.01 0.01 0.01 0.01

12 13~x x 0.01 0.01 0.4 0.01 0.01 0.01 2

14 17~x x 0.8 0.8 0.8 0.8 0.8 0.8 0.8

18 21~x x 2.0 1.6 1.6 1.6 1.6 2.0 0.01

22 25~x x 2.4 2.4 2.4 2.4 2.4 2.4 4

( )f lb 563.52 560.59 566.44 560.59 560.59 551.61 477.16684

Variables GA [31]

PSO [36]

PSOPC [36]

HPSO [36]

DHPSACO [37]

Proposed PC

1x 0.307 1.000 0.111 0.111 0.111 0.111

2 5~ x x 1.990 2.620 1.563 2.130 2.130 0.563

6 9~ x x 3.130 2.620 3.380 3.380 3.380 3.13

10 11~ x x 0.111 0.250 0.111 0.111 0.111 0.141

12 13~ x x 0.141 0.307 0.111 0.111 0.111 1.8

14 17~ x x 0.766 0.602 0.766 0.766 0.766 0.766

18 21~x x 1.620 1.457 1.990 1.620 1.620 0.111

22 25~ x x 2.620 2.880 2.380 2.620 2.620 3.88 ( )f lb 556.49 567.49 567.49 551.14 551.14 464.14708

Case 1: The discrete variables are selected from the set

{0.01, 0.4, 0.8, 1.2, 1.6, 2.0, 2.4, 2.8, 3.2, 3.6, 4.0, 4.4, 4.8, 5.2, 5.6, 6.0}

.

58

Discrete Problems

59

Mixed Problems

Conclusions and Original Contributions

60

Improvements to the original PC approach:• The original PC approach was improved with a reduction in the

computational complexity.

- A neighboring scheme developed for updating the solution space was developed which contributed to faster convergence and improved efficiency of the overall algorithm.

- the modified PC was successfully validated optimizing Rosenbrock Function

- Nash Equilibrium successfully formalized and demonstrated


Constraint Handling Techniques• A number of constraint handling techniques were developed . This

allowed PC to solve practical problems which inevitably are constrained problems.

• Problem specific heuristics were developed and incorporated into the PC algorithm for solving the NP-hard problem such as MTSP.

• True optimum solution was achieved for two specially developed cases of the MDMTSP, several cases of the SDMTSP were also solved.

• For the first time, the MTSP was solved using a distributed, decentralized and cooperative approach such as PC.

61


• Penalty function approach was successfully incorporated and tested by solving variety of test problems with in/equality constraints.

• Feasibility-based rule I was successfully formalized and demonstrated solving two specially developed cases of the Circle Packing Problem (CPP).

• In order to make the solution jump out of possible local minima, a perturbation approach and voting heuristic were developed.

• Demonstrate the desirable and key characteristic of a distributed approach to avoid the tragedy of commons.

• Important ability of PC to deal with the practically significant agent failure problem was demonstrated solving the CPP.

62


• Feasibility-based rule II was successfully formalized and demonstrated solving two variations and associated cases of the Sensor Network Coverage Problem (SNCP).

• Two variations and associated cases produced sufficiently robust results.

• BFGS method was successfully used as an alternative to the Nearest Newton Descent Scheme.

• CPP and SNCP were first time solved using a distributed, decentralized approach such as PC.

63

Recommendations for Future Work

• Make the approach more generalized and increase the efficiency of the PC algorithm by developing a self adaptive scheme for the parameters, improving diversification of sampling, etc.

• More realistic path planning problems of the Multiple Unmanned Vehicles (MUVs) can then be solved with the MTSP and VRP approaches.

• Multi-Objective Probability Collectives (MOPC)

64

Recommendations for Future Work

Solve the Traffic Control Problem using PC

• Distributed, decentralized approach• Every intersection represents an independent agent dynamically optimizing the signal durations, cycle time, phase sequence, etc.• Local traffic optimization → Network traffic optimization• Traffic simulator will be used to set up the traffic scenario• Flow rate will be measured at intersections (agents)• PC will optimize the variables such as signal durations, cycle time, phase sequence, etc.• Optimized variables will be fed back to evaluate the performance.

65

66

Thanks for your attendance

Q & A

68

Solution to Rosenbrock using PC

69

Nash Equilibrium

The basic concept states that when a social game is being played iteratively

by number of agents, if a state comes when any agent changes its

strategy/state unilaterally without taking into consideration the other agents’

strategy/state, it does not benefit that agent and also does not benefit the

entire game output. If the game is in such state then the agents are assumed

to be in Nash Equilibrium.

It is worth to mention that Nash Equilibrium does not necessarily gives

best payoffs to agents but as a social system best collective / global /

system objective can be achieved.


n i

70

Probability Collectives (PC) ComparisonSampling, Convergence criterion and Neighboring makes the PC presented here different than the originally proposed by Dr. David Wolpert.

Proposed PC Original PC

Sampling Pseudorandom scalar values drawn from uniform distribution Fewer number of samples

Monte Carlo sampling

Computationally expensive and slower

Convergence criterion Predefined number of iterations and/or there is no change in the final goal value for considerable number of iterations.

No change in the probability values for considerable number of iterations

71

Probability Collectives (PC) Comparison

Proposed PC Original PC [1, 3]

Neighboring Sample around the ‘favorable strategy values’ and continue from the beginning.

Narrows down the sampling options of Agents forcing

them to sample only from the neighbored range.

Increases convergence speed.

Computationally cheaper

Regression

Data-aging

Computationally expensive/Large memory


• Procedure starts with initializing the constraint violation tolerance

where is the cardinality of .

Feasibility-based Rule I• Any feasible solution is preferred over any infeasible solution

If the current system objective as well as the previous

solution are infeasible, accept the current system objective

and corresponding as the current solution if the number of

constraints violated is less than or equal to , i.e. ,

and then the value of is updated to , i.e. .

73

C 1 2 ... tg g gCC

favG Y

favY

favG Y

violatedC violatedC

violatedC violatedC

Constrained PC (Approach 3): Feasibility-based Rule I• Between two feasible solutions, the one with better objective is

preferred

If the current system objective is feasible, and the previous

solution is infeasible, accept the current system objective

and corresponding as the current solution and then the value of

is updated to , i.e. .

74

favG Y

favY

favG Y

0

0violatedC


• Between two infeasible solutions, the one with fewer violated constraints is preferred.

If the current system objective is feasible, i.e. and

is not worse than the previous feasible solution, accept the current

system objective and corresponding as the current

solution.

• If all the above conditions are not met, then discard current system

objective and corresponding , and retain the previous

iteration solution.

75

favG Y

favY favG Y

0violatedC

favG Y favY


Updating of the Sampling Space and Perturbation Approach• On completion of pre-specified iterations,

• If then shrink the sampling intervals:

• If and are feasible and

the system objective is referred to as .

76

, ,( ) ( )testfav n fav n nG G Y Y

testn

, , 0 1fav favupper lower upper loweri i down i i i down i i downX X

,( )fav nG Y ,( )testfav n nG Y , ,( ) ( )testfav n fav n nG G Y Y ,( )fav nG Y ,( )fav sG Y

Constrained PC (Approach 3): Feasibility-based Rule I• Updating of the Sampling Space and Perturbation Approach

In order to jump out of this possible local minimum, every agent perturbs its current feasible strategy

The value of and +/- sign are selected based on preliminary trials.

Every agent expands the sampling space as follows:

77

i

1 1

2 2

1,

1,

fav fav favi i i i

lower upperfav

ii

lower upperfav

i

X X X fact

randomvalue ifX

where factrandomvalue if

X

1 1 2 20 1lower upper lower upper

, , 0 1lower upper lower upper upper loweri i up i i i up i i up


• How about the convergence or the stable solution acceptance

78

Constrained PC (Approach 3): Feasibility-based Rule IIFeasibility-based Rule II:• Any feasible solution is preferred over any infeasible solution

If the current system objective as well as the previous

solution are infeasible, accept the current system objective

and corresponding as the current solution if the number of

improved constraints is greater than or equal to , i.e. ,

and then the value of is updated to , i.e. .

80

favG Y

favY

favG Y

improvedC

improvedC improvedC

Constrained PC (Approach 3): Feasibility-based Rule II• Between two feasible solutions, the one with better objective is

preferred

If the current system objective is feasible, and the previous

solution is infeasible, accept the current system objective

and corresponding as the current solution and then the value of

is updated to , i.e. .

• Between two infeasible solutions, the one with more number of improved constraint violations is preferred.

If the current system objective is feasible, and is not worse

than the previous feasible solution, accept the current system

objective and corresponding as the current solution.

81

favG Y

favY

favG Y

0

0improvedC

favG Y

favY favG Y

Constrained PC (Approach 3): Feasibility-based Rule II

• If the solution remains feasible and unchanged for successive predefined number of iterations, and current feasible system objective is worse than the previous iteration feasible solution, accept the current solution.

If the solution remains feasible and unchanged for successive pre-

specified iterations i.e. and are feasible and ,

and the current feasible system objective is worse than the previous

iteration feasible solution, accept the current system objective

and corresponding as the current solution.

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[1] [?] [?] [1] [?] [?]1 2 1, ,..., ,..., ,i i N NX X X X XY

Agent selects its first strategy and samples randomly from other agents’ strategies as well.

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[3] [?] [?] [3] [?] [3]1 2

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Yr r r r

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Multiple Unmanned Aerial Vehicles (MUAVs) Path Planning

Related Work

Probabilistic Map Approach- real-time and local updating of the map

Flock formation- collision avoidance, obstacle avoidance, formation keeping- single objective function Vs individual objective function

Gyroscope force- real-time change in the path avoiding the collision

Magnetic forces - Attraction and Repulsion

Concept of Auto-pilot – the airplanes with conflicting trajectories change their ways with local communication avoiding latency in decision making


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Probability Collectives

Technology

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