Multiobjective Optimization Using Particle Swarm Optimization Introduction Evolutionary

Multi-objective Optimization Using Particle Swarm Optimization

Introduction • Evolutionary Algorithms (EAs) are not the only search techniques that have been used to deal with multiobjective optimization problems. • In fact, as other search techniques (e. g. , Tabu search and simulated annealing) have proved to have very good performance in many combinatorial (as well as other types of) optimization problems, it is only natural to think of extensions of such approaches to deal with multiple objectives.

• Search Techniques: – Simulated Annealing – Tabu Search and Scatter Search – Particle Swarm Optimization – Ant System – Artificial Immune Systems –. . .

Particle Swarm Optimization • Kennedy and Eberhart (1995) proposed an approach called “particle swarm optimization” (PSO) inspired by the choreography of a bird flock. • The idea of this approach is to simulate the movements of a group (or population) of birds which aim to find food. • The approach can be seen as a distributed behavioral algorithm that performs (in its more general version) multidimensional search. In the simulation, the behavior of each individual is affected by either the best local (i. e. , within a certain neighborhood) or the best global individual.

• As described by the inventers James Kennedy and Russell Eberhart, • “particle swarm algorithm imitates human (or insects) social behavior. Individuals interact with one another while learning from their own experience, and gradually the population members move into better regions of the problem space”.

• Both PSO and EC are population based. • PSO also uses the fitness concept, but, less-fit particles do not die. No “survival of the fittest”. • No evolutionary operators such as crossover and mutation. • Each particle (candidate solution) is varied according to its past experience and relationship with other particles in the population.

• In PSO, each single solution is a "bird" in the search space. • Call it "particle". • All of particles have fitness values – which are evaluated by the fitness function to be optimized, and • have velocities – which direct the flying of the particles. • The particles fly through the problem space by following the current optimum particles.

• Initialize with randomly generated particles. • Update through generations in search for optima • Each particle has a velocity and position • Update for each particle uses two “best” values. – Pbest(Personal Best): best solution (fitness) it has achieved so far. (The fitness value is also stored. ) – Gbest(Global Best): best value, obtained so far by any particle in the population.

• Population-based search procedure in which individuals called particles change their position (state) with time. • individual has position • & individual changes velocity Particle: Particle Presenting the Solution Set It contains: • coordinate of the position • velocity • fitness value

The particles are manipulated according to the following equation: (a) (b) Where and are two positive constants, are two random functions in the range [0, 1]. W is the inertia weight.

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=(pi 1, pi 2, …, pi. D) and the best position visited by the whole swarm (gbest) denoted as Pg=(pg 1, pg 2, …, pg. D).

Social (global) influence Personal influence

vn+1: Velocity of particle at n+1 th iteration Vn : Velocity of particle at nth iteration c 1 : acceleration factor related to lbest c 2 : acceleration factor related to gbest rand 1( ): random number between 0 and 1 rand 2( ): random number between 0 and 1 gbest: gbest position of swarm pbest: pbest position of particle

Particle Swarm Optimization (PSO) Process • • • Initialize population in hyperspace Evaluate fitness of individual particles Modify velocities based on previous best and global (or neighborhood) best positions Terminate on some condition Go to step 2

• • For each particle Initialize particle with feasible random number END Do For each particle Calculate the fitness value If the fitness value is better than the best fitness value (pbest) in history Set current value as the new pbest End Choose the particle with the best fitness value of all the particles as the gbest For each particle Calculate particle velocity according to velocity update equation Update particle position according to position update equation End While maximum iterations or minimum error criteria is not attained

Pseudo code • Initialize; while (not teminated) • { t = t +1 for i = 1: N // for each particle • { • Vi(t) = Vi(t-1) + c 1*rand()*(Pi –Xi(t-1)) +c 2*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 for MOP Three 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?

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.

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.