Free Search in Multidimensional Space

Transcription

1 Free Search in Multidimensional Space Technology School, Maritime and Technology Faculty Southampton Solent University

2 Content Aims Test Problems Bump test function Michalewicz test function Norwegian test function Optimization Methods Free Search Differential Evolution Particle Swarm Optimisation Experimental methodology Experimental results Discussion Conclusion & Further Directions

3 Aims Evaluation on multidimensional tests of Free Search Differential Evolution Particle Swarm Optimisation Study abilities to avoid stagnation and trapping in local suboptimal solution Identify minimal number of iterations and time required to resolve multidimensional tasks with acceptable precision

4 Test Problems Criteria for tests selection: must be global optimisation test with many local suboptimal solutions; must not provide initial knowledge for optimal solution value and location; optimal solution must be dependent on dimensions number.

5 Test problems Norwegian test n i= 1 cos(πx 3 i 99 + x ) 100 i where x i [-1.1, 1.1] for i=1,,n

6 Test problems Mochalevicz test f 2 ( x1, x2 ) = sin( x i )(sin( ix i / π )) i= 1 2 2m where x i [0.0, 3.0] for i=1,,n

7 Test problems Bump test f(x ) i = n cos i= 1 4 (x i ) n 2 cos i= 1 2 (x i ) / n ix i= 1 2 i for: 0 < xi < 10 and i=1,,n subject to: n Π xi > 0.75, i=1 n Σ xi < 15*n/2, i= 1 i =1,,n starting from one location xi = 5, i =1,,n

8 Particle Swarm Optimization Modification strategy v id = w* v id + n 1 *random(0,1)*(p id - x id ) + n 2 *random(0,1)*(g d - x id ) x id = x id + v id w is inertia weight v - velocity vector n 1 is individual learning factor n 2 is social learning factor g - best achievement for all population X i - particle current position P i - the best particles achievement

9 Differential Evolution Modification strategy (1) X k ' = X k + F(X i - X j ), (2) X k ' = X* + F(X i - X j ), (3) X k ' = X k + F(X* - X k ) + F(X i - X j ), (4) X k ' = X* + F(X i - X j + X n - X m ), (5) X k ' = X k + F(X* - X k + X n - X m ), X k is a donor vector X k ' is mutated donor X* is the best vector for current population X i, X j, X n and X m are differential vectors F is differential factor

10 Free Search Modification strategy X = X + R*Rand*(Xmax Xmin) X is an initial location X is a new location R - neighbour space Rand random value between 0 and 1 Xmax & Xmin search space limits

11 Experimental methodology 320 experiments - start from different initial locations 3 series for each Norwegian and Michalewicz test limited to: iterations iterations iterations for Differential Evolution, Particle Swarm Optimisation and Free Search. Free Search exploration is 5 steps 10 iteration = 2 explorations * 5 steps Population size 10 for PSO, DE and FS Number of dimensions -100

12 Experimental results Maximal results on Norwegian and Michalewicz tests Iterations FS DE PSO F1 F

13 Experimental results Mean results on Norwegian and Michalewicz tests F1 Iterations FS DE PSO F

14 Experimental results Standard deviation on Norwegian and Michalewicz tests F1 Iterations FS DE PSO F

15 Experimental results Time for one experiment limited to iterations ( explorations * 5 steps for FS) for all methods this corresponds to objective function evaluations (OFEs) F1 - Norwegian test F2 - Michalewicz test OFEs FS DE PSO F min 15 min 37 min F min 31 min 55 min

16 Experimental results Results on Bump test (by 2013) Dimensions Free Search

17 Computer system CPU - Intel Core i7-3960x GHz RAM - G.Skill TridentX 16GB (4x4GB) DDR3 1866MHz Motherboard - ASUS Rampage IV Extreme 2011 SSD - SanDisk Extreme SSD SATA III

18 Further directions Higher dimensions tests evaluation Exploration of other tests Search methods improvement Application to real world problems

19 Thank you