A Multiobjective Memetic Algorithm Based on Particle Swarm Optimization
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1 A Multiobjective Memetic Algorithm Based on Particle Swarm Optimization Dr. Liu Dasheng James Cook University, Singapore / 48
2 Outline of Talk. Particle Swam Optimization 2. Multiobjective Particle Swarm Optimization 3. A Multiobjective Memetic Algorithm Based on Particle Swarm Optimization 4. Research Ideas 2 / 48
3 Particle Swarm Optimization Particle swarm optimization (PSO) was first introduced by James Kennedy (a social psychologist) and Russell Eberhart (an electrical engineer) in 995, which originates from the simulation of behavior of bird flocks. 3 / 48
4 Particle Swarm Optimization There are a number of algorithms to simulate the movement of a bird flock or fish school. Kennedy and Eberhart became particularly interested in the models developed by Heppner (a zoologist) [62]. 4 / 48
5 Heppner s Model In Heppner s model, birds would begin by flying around with no particular destination and in spontaneously formed flocks until one of the birds flew over the roosting area. 5 / 48
6 Particle Swarm Optimization To Eberhart and Kennedy, finding a roost is analogous to finding a good solution in the field of possible solutions. They revised Heppner s methodology so that particles will fly over a solution space and try to find the best solution depending on their own discoveries and past experiences of their neighbors. 6 / 48
7 Working Principle of PSO 7 / 48
8 Original Version In the original version of PSO, each individual is treated as a volume-less particle in the D dimensional solution space. The equations for calculating velocity and position of particles are shown below: 8 / 48
9 Adjustable Step Size Further research shows that to adjust velocity not by a fixed step size but according to the distance between current position and best position can improve performance. 9 / 48
10 V max One parameter V max is introduced, and the particle s velocity on each dimension cannot exceed V max. If V max is too large, particle may fly past good solutions. If V max is too small, particle may not explore sufficiently beyond locally good regions. V max is usually set at -2% of the dynamic range of each dimension. / 48
11 Inertial Weight To better control the exploration and exploitation in particle swarm optimization, the concept of inertial weight (w) was developed. v w v c r ( p x ) k k k k k i, d i, d i, d i, d c r ( p x ) k k k 2 2 g, d i, d / 48
12 Particle Swarm Optimization Formula Each individual in PSO is assigned a random velocity and flies across the solution space with a memory of its own best position called pbest and a knowledge of the whole swarm s global best position called gbest. v w v c r ( p x ) k k k k k i, d i, d i, d i, d c r ( p x ) x x v k k k 2 2 g, d i, d k k k i, d i, d i, d w is the inertia weight; c is the cognition weight and c 2 is the social weight; r and r 2 are two random values uniformly distributed in the range of [, ]. 2 / 48
13 Terminology 3 / 48
14 Terminology 4 / 48
15 Multiobjective Optimization Real world problems usually involve simultaneous optimization of several competing objectives. Solution exists in the form of alternative tradeoffs. Non-inferior solutions are known as nondominated solutions The set of nondominated solutions form the Pareto solution set A minimization problem Trade-off Curve f 2 Unfeasible Region f 5 / 48
16 MOPSO Conventional optimization search techniques Hardly handle multiple objectives The gradients need to be well-defined and differentiable May trap in local optima Multi-objective particle swarm optimization (MOPSO) is a powerful tool for solving MO optimization problems. Capable of searching for the global trade-off. Maintain a diverse set of solutions Robust and applicable to a wide variety of problems. f 2 f 6 / 48
17 MOPSO 7 / 48
18 Minimization Performance Assessments For MOO, performance metrics must be able to measure quality in terms of: Diversity. Proximity between the generated and true Pareto f 2 Non-dominated solution Pareto Frontier Non-dominated set front. Minimization f 8 / 48
19 Performance Measures Generational Distance (GD) (Veldhuizen, 999) Represents how far the evolved solution set is from the true Pareto front. Spacing (S) (Schott, 995) Measures how evenly evolved solutions distribute itself. Maximum Spread (MS) (Zitzler, 999) Measures how well the true Pareto front is covered by the evolved solution set. 9 / 48
20 Problem Test Suite ZDT ZDT2 Test Problem Features ZDT Pareto front is convex. 2 ZDT2 Pareto front is non-convex. 3 ZDT3 Pareto front consists of several noncontiguous convex parts. 4 ZDT4 Pareto front is highly multi-modal where there are 2 9 local Pareto fronts. 5 ZDT6 The Pareto optimal solutions are non-uniformly distributed along the global Pareto front. The density of the solutions is low near the Pareto front and high away from the front ZDT3 ZDT4 ZDT6 2 / 48
21 Problem Test Suite Test Problem Features 6 FON Pareto front is non-convex. 7 KUR Pareto front consists of several noncontiguous convex parts. 8 POL Pareto front and Pareto optimal solutions consist of several noncontiguous convex parts FON KUR POL 2 / 48
22 Two modification introduced to improve performance Fuzzy global best Synchronous particle local search Memetic algorithm : evolution algorithm with local improvement technique 22 / 48
23 Two Modifications Fuzzy Global Best (f-gbest) A new particle updating strategy is proposed based upon the concept of fuzzy global-best to deal with the problem of premature convergence and diversity maintenance within the swarm. Synchronous Particle Local Search (SPLS) Hybridized with a directed local search operator for local fine tuning, which helps to discover a well-distributed Pareto front. 23 / 48
24 Fuzzy gbest Fuzzy Global Best (f-gbest) accounts for the uncertainty of global best knowledge to prevent premature convergence Incorporates fuzzy number x to represent global best. F-gbest is characterized by normal distribution. Degree of uncertainty reduces with generations synonymous with information gain. Search trajectory using conventional gbest gbest Possible location of gbest Search Region incorporating f-gbest Particle x2 x3 24 / 48
25 Formula for Fuzzy gbest The calculation of particle velocity can be rewritten as k k p N( p, ) c, d g, d f( k) k k k k k k k k v w v c r ( p x ) c r ( p x ) i, d i, d i, d i, d 2 2 c, d i, d f-gbest is characterized by a normal distribution, N( p k, ), where g, d representing the degree of uncertainty about the optimality of the global-best. 25 / 48
26 Synchronous Particle Local Search SPLS of assimilated particles along x and x3 x Assimilated Particle A' Trajectory along assigned search direction, x A B Possible location of assigned gbest Trajectory along assigned search direction, x3 Assimilated Particle B' x2 x3 26 / 48
27 Synchronous Particle Local Search SPLS is performed in the vicinity of the particles. SPLS: Select S LS particles randomly from particle swarm Select N LS nondominated particles from the archive with the best niche count into a selection pool Assign an arbitrary nondominated solution from the selection pool to each of the S particles as gbest Assign an arbitrary dimension to each of the LS SLS particles Assimilation: With the exception of the assigned dimension, update the position of S LS particles in the desion space with the selected gbest position Update the position of all SLS assimilated particles using fuzzy gbest along the pre-assigned dimension 27 / 48
28 Implementation Initialize Particle Swarm Evaluate Particles Return Archive Archiving Update Particle Position using f-gbest SPLS Yes cycle = max_cycles? Select S LS particles for SPLS No Select Personal Best Select Global Best 28 / 48
29 Simulation Results (a) (b) (c) (d) (e) (f) Evolved tradeoffs by a) FMOPSO, b) CMOPSO, c) SMOPSO, d) IMOEA, e) NSGA II, and f) SPEA2 for ZDT 29 / 48
30 Simulation Results (a) (b) (c) (d) (e) (f) (g) (i) (h) Statistical performance of the different algorithms: a) GD, b) MS, c) S for ZDT4; d) GD, e) MS, f) S for ZDT6; and g) GD, i) MS, h) S for FON 3 / 48
31 Research Ideas Researchers are facing the challenge of increasing dimensionality and computational cost of today s applications. Handling of high dimensional problems. Use of dimensional reduction techniques. Use of learning techniques to gain information on the shape and position of Pareto front/pareto set. Apply surrogates to reduce evaluation time. Surrogate models are cheap and approximate evaluation models. Solving real world problems of your interest 3 / 48
32 The Papers Zhinzhong Ding, Fuqiang Lu, and Hualing Bi, A TWO-STAGE PARTICLE SWARM OPTIMIZATION FOR VIRTUAL ENTERPRISE RISK MANAGEMENT, International Journal of Innovative Computing, Information and Control, vol., no. 4, pp , 24. Marco Corazza, Giovanni Fasano, S. Y., and Riccardo Gusso, Particle Swarm Optimization with non-smooth penalty reformulation, for a complex portfolio selection problem, Applied Mathematics and Computation 244, pp , 23. Kuo, R. J. and Hong, C. K. Integration of Genetic Algorithm and Particle Swarm Optimization for Investment Portfolio Optimization, Applied Mathematics & Information Sciences, vol. 7, no. 6, pp , 23. Jui-Fang Chang, Peng Shi, Using investment satisfaction capability index based particle swarm optimization to construct a stock portfolio, Information Sciences 8, pp , 2 Liu, D. S., Tan, K. C., Goh, C. K. and Ho, W. K., A Multiobjective Memetic Algorithm Based on Particle Swarm Optimization, IEEE Transactions on Systems, Man and Cybernetics: Part B (Cybernetics), vol. 37, no., pp. 42-5, / 48
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