Multi-objective Optimization Using Particle Swarm Optimization

Satchidananda Dehuri, Ph.D.(CS),

Department of Information and Communication Technology, Fakir Mohan University, vyasa Vihar, Balasore-756019,

ORISSA, INDIA.

Single Vs Multi-objectiveSingle Objective Optimization: When an optimization problem involves only one objective function, the task of finding the optimal solution is called single-objective optimization.

Example: Find out a CAR for me with Minimum cost.

Multi-objective Optimization: When an optimization problem involves more than one objective function, the task of finding one or more optimal solutions is known as multi-objective optimization.

Example: Find out a CAR for me with minimum cost and maximum comfort.

Single Vs Multi-objective: A Simple Visualization

Price

Luxury

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A

B2

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60

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60-0.5

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Multi-objective Problem (ctd.)

Mapping: Rd to Fn

Reference: S. Dehuri, A. Ghosh, and S.-B. Cho, “Particle Swarm Optimized Polynomial Neural Network for Classification: A Multi-objective View”, International Journal of Intelligent Defence Support Systems, vol. 1, no. 3, pp.225-253, 2008.

Concept of Domination

A solution x1 dominate other solution x2, if both conditions 1 and 2 are true:

1. The solution x1 is no worse than x2 in all objectives, or fi(x1) ♣ fj(x2) for all j.

2. The solution x1 is strictly better than x2 in at least one objective or fi(x1) ♠ fj(x2) for at least one j.

A Simple Visualization

Maximize f1

Minimize f2

1

2

3 4 5

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Time Complexity of non-dominated set: O(MN2)

Properties of Dominance

Reflexive: The dominance relation is not reflexive.

Symmetric: The dominance relation is also not symmetric.

Transitive: The dominance relation is transitive.

In order for a binary relation to qualify as an ordering relation, it must be at least transitive [3].

Thus dominance relation is only a strict partial order relation.

Pareto Optimality

Non-dominated Set: Among a set of solutions P, the non-dominated set of solutions P’ are those that are not dominated by any member of the set P.

Global Pareto Optimality Set: The non-dominated set of the entire feasible search space S is the globally Pareto-optimal set.

Locally Pareto Optimality Set: If for every member x in a set P” there exists no solution ‘y’ (in the neighborhood of x such that ||y-x||∞ <= eps, where eps is a small pos. number) dominating any member of the set P”, then solutions belonging to the set P” constitute a locally Pareto-optimal set.

Multi-objective Problem (ctd.)

Why PSO for MOP• It would not be surprising to apprehend that the

development of preference-based approaches was motivated by the fact that available optimization methods could find only a single optimized solution in a single simulation run.

• How to get multiple trade off solutions?

• Probably the non-classical, unorthodox and stochastic search such as PSO can help us to find multiple trade –off solutions in a single run of the algorithm.

HOW?

Examples of MOPMinimization Problem:

1. Minimize f1(x)=x1

f2(x)=(1+x2)/(x1)

Domain: {0.1 <= x1<=1, 0 <=x2 <=5}

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 12

4

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Examples (ctd.)Maximize f1(x)=x1

f2(x)=1+x2-(x1*x1)

Domain:{0<=x1<=1, 0<=x2<=3}

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2-3

-2

-1

0

1

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MOPS Approaches

1) Weighted Sum Approaches

2) Lexicography Approaches

3) Pareto Approaches

Weighted Sum Approach

Optimize

F(x)=

M

mmm xfw

1

)(.

M

mmm ww

1

1],1,0[

As we converted it into single objective we can now proceed using PSO with its associated operators.

Hopefully we will get an optimal solutions.

Problem: How to fix these weights? (Static/Dynamic)

ExampleMinimize f1(x)=x1

Minimize f2(x)=1+x2*x2-x1-a*sin(b*PI*x1)

Domain: x1=[0,1], x2=[-2,2], a=0.2, b=1.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

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2.5

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3.5

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Lexicography Approach

• In the lexicographic approach, different priorities are assigned to different objectives, and then the objectives are optimized in order of their priority.

Review of the Classical Methods

1. Only one Pareto optimal solution can be expected to be found in one simulation run of a classical algorithm.

2. Not all Pareto optimal solution can be found by some algorithms in non-convex MOOPs.

3. All algorithms require some problem knowledge, such as suitable weights, epsilon, or target values, etc.

Pareto Approach from EA DomainVEGA (Vector Evaluated Genetic Algorithms)

(Contributed by David Schaffer in 1984).

VOES(Vector Optimized Evolution Strategy) contributed by Frank Kursawe in 1990.

MOGA (Multi-objective GA) introduced by Fonseca and Fleming in 1993.

NSGA (Non-dominated Sorting GA) introduced by Srinivas and Deb in 1994.

NPGA (Niched-Pareto Genetic Algorithm) introduced by Horn et al. in 1994.

PPES (Predator-Prey Evolution Strategy) introduced by Laumanns et al. in 1998.

DSGA (Distributed Sharing GA) introduced by Hiroyasgu et al. in 1999.

DRLA (Distributed Reinforcement Learning Approach) introduced by Mariano and Morales in 2000.

Nash GA introduced by Sefrioui and Periaux in 2000, motivated by a game theoretic approach.

REMOEA (Rudolph’s Elitist MOEA) introduced by Rudolph in 2001.

NSGA-II by Deb et a. in 2000. and so on……………………..

Potential Research Directions

• MOEA in Data Mining [-1]

• MOEA in real time task scheduling [0]

• MOEA for Ground-water Contamination [1]

• MOEA for Land-use Management [2]

More about MOGA

Please Visit:

KANGAL-Kanpur Genetic Algorithm Laboratory

(Prof. Kalyanmoy Deb)

CINVESTA-Mexico (Prof. Carlos A. Coello Coello)

Particle Swarm Optimization

• A new Paradigm of Swarm Intelligence

– What is a Swarm Intelligence (SI)?

– Examples from nature

– Origins and Inspirations of SI

What is a Swarm?• Collection of interacting agents (Soft/Hardware).

– Agents (Soft/Hardware):• Individuals that belong to a group (but are not necessarily

identical). • They contribute to and benefit from the group.• They can recognize, communicate, and/or interact with each

other.

• The instinctive perception of swarms is a group of agents in motion – but that does not always have to be the case.

• A swarm is better understood if thought of as agents exhibiting a collective behavior.

Example of Swarms in Nature• Classic Example: Swarm of Wasps/Bees• Can be extended to other similar systems:

– Ant colony• Agents: ants

– Flock of birds• Agents: birds

– Traffic• Agents: cars

– Crowd• Agents: humans

– Immune system• Agents: cells and molecules

Beginnings of Swarm Intelligence

• First introduced by Beni and Wang in 1989 with their study of cellular robotic systems

• Extended by Theraulaz, Bonabeau, Dorigo, Kennedy,…..

Definition of Swarm Intelligence

• SI is also treated as an artificial intelligence (AI) technique based on the collective behavior in decentralized, self-organized systems.

• Generally made up of agents who interact with each other and the environment.

• No centralized control structures.• Based on group behavior found in nature.

Swarm Intelligence Techniques

• A few popular and recent SI techniques:

• Particle Swarm Optimization,

• Ant Colony Optimization,

• Bee colony Optimization,

• Wasp Colony Optimization,

• Intelligent Water Drops

Success and On-going Research on SI• Tim Burton's “Batman Returns” was the first movie to

make use of swarm technology for rendering, realistically depicting the movements of a group of bats using the Boids system. Entertainment industry is applying for battle and crowd scenes.

• U.S. military is investigating swarm techniques for controlling unmanned vehicles.

• NASA is investigating the use of swarm technology for planetary mapping.

• Swarm intelligence to control nanobots within the body for the purpose of killing cancer tumors.

• Load balancing in telecommunication networks

Swarm Robotics

• The application of SI principles to collective robotics.

• A group of simple robots that can only communicate locally and operate in a biologically inspired manner.

• A currently developing area of research.

Advantage of SI Techniques• The systems are scalable because the same

control architecture can be applied to a couple of agents or thousands of agents.

• The systems are flexible because agents can be easily added or removed without influencing the structure.

• The systems are robust because agents are simple in design, the reliance on individual agents is small, and failure of a single agents has little impact on the system’s performance.

• The systems are able to adapt to new situations easily.

Particle Swarm Optimization

• A population based stochastic optimization technique.

• Searches for an optimal solution in the computable search space.

• Developed in 1995 by Eberhart and Kennedy.

• Inspiration: Flocks of Birds, Schools of Fish.

Particle Swarm Optimization (ctd.)

• In PSO individuals strive to improve themselves and often achieve this by observing and imitating their neighbors

• Each PSO individual has the ability to remember

• PSO has simple algorithms and low overhead– Making it more popular in some circumstances

than Genetic/Evolutionary Algorithms– Has only one operation calculation:

• Velocity: a vector of numbers that are added to the position coordinates to move an individual

In General: How PSO Work• Individuals in a population learn from previous

experiences and the experiences of those around them.– The direction of movement is a function of:

• Current position• Velocity • Location of individuals “best” success• Location of neighbors “best” successes

• Therefore, each individual in a population will gradually move towards the “better” areas of the problem space.

• Hence, the overall population moves towards “better” areas of the problem space.

Particle Swarm Optimization (ctd.)

• A swarm consists of N particles in a D-dimensional search space. Each particle holds a position (which is a candidate solution to the problem) and a velocity (which means the flying direction and step of the particle).

• Each particle successively adjust its position toward the global optimum based on two factors: the best position visited by itself (pbest) denoted as Pi=(pi1,pi2,…,piD) and the best position visited by the whole swarm (gbest) denoted as Pg=(pg1,pg2,…,pgD) .

Particle Swarm Optimization (ctd.)

Here I am! The best

perf. of team

My best perf.

xpg

pi

v

PBest

gBest

1 2( 1) ( ) ( ) ( )() ( ) () ( )gi i i i it t t tv w v c rand p x c rand p x

( 1) ( ) ( )i i it t tx x v

Pseudo code

• Initialize;while (not teminated)

• { t = t +1for i = 1:N // for each particle

• {• Vi(t) = Vi(t-1) + c1*rand()*(Pi –Xi(t-1))

+c2*rand()*(Pg –Xi(t-1)) • Xi(t) = Xi(t-1) + Vi(t) • Fitness i(t) = f(Xi(t));

if needed, update Pi and Pg;• }// end for i• } // end for while

PSO Vs.GA• Similarity

– Both algorithms start with a group of a randomly generated population– Both have fitness values to evaluate the population. – Both update the population and search for the optimum with random

techniques. – Both systems do not guarantee success.

• Dissimilarity– However, unlike GA, PSO has no evolution operators such as crossover

and mutation.– In PSO, the potential solutions, called particles, fly through the problem

space by following the current optimum particles.– Particles update themselves with the internal velocity. – They also have memory, which is important to the algorithm.

• advantages– PSO is easy to implement and there are few parameters to adjust. – Compared with GA, all the particles tend to converge to the best solution

quickly even in the local version in most cases

Our Contribution towards PSO[1]Mishra, B.B., and Dehuri, S., “A Novel Stranger Sociometry

Particle Swarm Optimization (S2PSO)”, ICFAI Journal of Computer Science, vol. 1, no. 1, 2007.

[2]Dehuri, S., “An Empirical Study of Particle Swarm Optimization for Cluster Analysis”, ICFAI Journal of Information Technology, 2007.

[3]Dehuri, S.,3]Dehuri, S., Ghosh A., and Mall, R, “Particles’ with Age for Data Clustering”, Proceedings of International Conference on Information Technology, Dec. 18-21, Bhubaneswar, 2006.

[4]Dehuri, S, and Rath, B. K., “gbest Multi-swarm for Multi-objective Rule Mining”, Proceedings of National Conference on Advance Computing, March 22-23, Tezpur University, 2007.

PSO for MOPThree main issues to be considered when extending PSO to multi-

objective optimization are:

• How to select particles (to be used as leaders) in order to give preference to non-dominated solutions over those that are dominated?

• How to retain the non-dominated solutions found during the search process in order to report solutions that are non-dominated with respect to all the past populations and not only with respect to the current one? Also, it is desirable that these solutions are well spread along the Pareto front.

• How to maintain diversity in the swarm in order to avoid convergence to a single solution?

Statistics of MOPSO Development

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1999 2000 2001 2002 2003 2004 2005 2006 2007 2008

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Publications

Growth of GA, PSO and ACO for MOP

Algorithm: MOPSO• INITIALIZATION of the Swarm• EVALUATE the fitness of each particle of the swarm.• EX_ARCHIVE = SELECT the non-dominated solutions from the

Swarm.• t = 0. • REPEAT• FOR each particle• SELECT the gbest• UPDATE the velocity• UPDATE the Position• MUTATION /* Optional */• EVALUATE the Particle• UPDATE the pbest• END FOR• UPDATE the EX_ARCHIVE with gbests.• t = t+1• UNTIL (t <= MAXIMUM_ITERATIONS)• Report Results in the EX_ARCHIVE.

Dehuri, S., Cho, S.-B., "Multi-criterion Pareto based particle swarm optimized polynomial neural network for classification: A Review and State-of-the-Art. Computer Science Review, Elsevier Science, vol. 3, no. 1, pp. 19-40, 2009.

A Few Contributions…• Parsopoulos and Vrahatis [a]

• Baumgartner et al. [b]

• Hu and Eberhart [c]

• Parsopoulos et al. [d]

• Chow and Tsui [e]

• Moore and Chapman [f]

• Ray and Liew [g]

• Fieldsend and Singh [h]

• Coello et al. [i] and so on…

References [-1] A. Ghosh, S. Dehuri, and S. Ghosh, Multi-objective Evolutionary

Algorithms for KDD, Springer-Verlag, 2008.

[0] J. Oh and C. Wu, “Genetic Algorithms based real time task scheduling with multiple goals”, The journal fo systems and Software, vol. 71, pp. 245-258, 2004.

[1]R. Farmani, et al., “An Evolutionary Bayesian Belief Network Methodology for Optimum Management of Groundwater Contamination”, environmental Modeling and Software, vol.24, pp.303-310, 2009.

[2]D. Dutta, et al., “Multi-objective Evolutionary Algorithms for Land-

Use Management Problem”, International Journal of Computational Intelligence Research, vol. 3, no. 4, pp/ 371-384, 2007.

[3] V. Chankong and Y. Y. Haimes, Multi-objective Decision Making Theory and Methodology, New York: North-Holland, 1983.

References[a]Konstantinos E. Parsopoulos and Michael N. Vrahatis. Particle swarm

optimization method in multiobjective problems. In Proceedings of the 2002 ACM Symposium on Applied Computing (SAC’2002), pages 603–607, Madrid, Spain, 2002. ACM Press.

[b]U. Baumgartner, Ch. Magele, and W. Renhart. Pareto optimality and particle swarm optimization. IEEE Transactions on Magnetics, 40(2):1172–1175, March 2004.

[c]Xiaohui Hu and Russell Eberhart. Multiobjective optimization using dynamic neighborhood particle swarm optimization. In Congress on Evolutionary Computation (CEC’2002), volume 2, pages 1677–1681, Piscataway, New Jersey, May 2002. IEEE Service Center.

[d]Konstantinos E. Parsopoulos, Dimitris K. Tasoulis, and Michael N. Vrahatis. Multiobjective optimization using parallel vector evaluated particle swarm optimization. In Proceedings of the IASTED International Conference on Artificial Intelligence and Applications (AIA 2004), volume 2, pages 823–828, Innsbruck, Austria, February 2004. ACTA Press.

References[e]Chi-kin Chow and Hung-tat Tsui. Autonomous agent response learning by a

multi-species particle swarm optimization. In Congress on Evolutionary Computation (CEC’2004), volume 1, pages 778–785, Portland, Oregon, USA, June 2004. IEEE Service Center.

[f]Jacqueline Moore and Richard Chapman. Application of particle swarm to multiobjective optimization. Department of Computer Science and Software Engineering, Auburn University, 1999.

[g]Tapabrata Ray and K.M. Liew. A swarm metaphor for multiobjective design optimization. Engineering Optimization, 34(2):141–153, March 2002.

[h]Jonathan E. Fieldsend and Sameer Singh. A multiobjective algorithm based upon particle swarm optimisation, an efficient data structure and turbulence. In Proceedings of the 2002 U.K. Workshop on Computational Intelligence, pages 37–44, Birmingham, UK, September 2002.

[i]Carlos A. Coello Coello and Maximino Salazar Lechuga. MOPSO: A proposal for multiple objective particle swarm optimization. In Congress on Evolutionary Computation (CEC’2002), volume 2, pages 1051–1056, Piscataway, New Jersey, May 2002. IEEE Service Center.