Multi-objective Optimization

Transcription

1 Some introductory figures from : Deb Kalyanmoy, Multi-Objective Optimization using Evolutionary Algorithms, Wiley 2001 Multi-objective Optimization Implementation of Constrained GA Based on NSGA-II

2 Optimization Optimization refers to finding one or more feasible solutions which correspond to extreme values of one or more objectives Finding out design variable : x Minimize f(x) - Single objective Subjected to g j (x) 0, j=1,,n j h k (x) = 0, k=1,,n k x (L) i x i x (U) i

3 Optimization Model Classification Basic classifications are: Constrained or unconstrained Linear or non-linear Single objective or multi-objective Another classification can be made by variables: continuous/discrete/mixed-integer

4 Single and Multi-objective Optimization Single Objective : Only one objective function Multi-Objective : Two or more and often conflicting objective functions e.g. Buying a car : minimize cost and maximize comfort

5 Pareto Optimal Front Mapping between feasible decision space and objective space Dominated solutions : Set of design points performing worse than some other better points Domination criterion : A feasible solution x 1 dominates an other feasible solution x 2 (denoted as x 1 < x 2 ), if both of the following conditions are true: 1) The solution x 1 is no worse than x 2 in all objectives, i.e. f i (x 1 ) f i (x 2 ) 2) The solution x 1 is strictly better than x 2 in at least one objective, i.e. f i (x 1 ) < f i (x 2 ) Non-dominated solutions : If two solutions are compared, then the solutions are said to be non-dominated with respect to each other IF neither solution dominates the other Pareto optimal front : The function space representation of all the nondominated solutions

6 Pareto Optimal Front.. contd Options : Min Min Min Max Max Min Max Max Which one is which?

7 Solution Methods Methods that try to avoid generating the Pareto front Generate utopia point Define optimum based on some measure of distance from utopia point Generating entire Pareto front Weighted sum of objectives with variable coefficients Optimize one objective for a range of constraints on the others Niching methods with population based algorithms

8 Implementation of Multiobjective Constrained GA, Based on NSGA-II

9 Genetic Algorithms Genetic algorithms imitate natural optimization process, natural selection in evolution Coding: replace design variables with a continuous string of digits or genes Binary Integer Real Population: Create population of design points Selection: Select parents based on fitness Crossover: Create child designs Mutation: Mutate child designs

10 Problem Formulation Inequalities defined 0 Current program is written for 2 objectives (M=2), it is possible to change it

11 NSGA-II Non-dominated Sorting Genetic Algorithm (NSGA)-II performs better than other constrained multi-objective optimizers* (PAEA, SPEA) Better and faster convergence to true optimal front Better spread on Pareto optimal front NSGA-II ranks designs based on non-domination For example : min-max problem Design Cost Comfort A 25K 65% B 45K 80% 3 55K 50% Design 3 is dominated by both design A and B (and thus undesirable), but design A and B are non-dominated with respect to one another (and thus Pareto optimal). 3 * Deb, K, et al, A Fast and Elitist Multiobjective Genetic Algorithm: NSGA-II, IEEE Transactions on Evolutionary Computations, Vol. 6, No. 2, pp , 2002

12 * From presentation of Tushar Goel Flow Chart *

13 Initialize population Implementation Fixed number of population size (N_pop) Fixed number of variables (N_var) Discrete variables Variable upper (UB) and lower bounds (LB) Number of increments (N_increments) Randomly distributed throughout the design space

14 Ranking Ranks designs based on nondomination The Pareto front is all rank 1 designs If the rank 1 designs are removed, the next Pareto front will be all rank 2 designs, etc. Sorting method is different than what NSGA-II* details Constraints : handled with constraint-domination ideas If two designs are both feasible, the standard non-domination techniques are used If one design is feasible and the other is not, the former is obviously favored (ranked lower) If both designs are infeasible, the design with a smaller overall constraint violation is favored (ranked lower) Rank 1 Rank 2 * Deb, K, et al, A Fast and Elitist Multiobjective Genetic Algorithm: NSGA-II, IEEE Transactions on Evolutionary Computations, Vol. 6, No. 2, pp , 2002

15 Selection and Fitness More fit designs have higher chance of passing their genes to the next generation Fitness is based on rank, low rank designs have higher fitness Selection : Using fitness based roulette wheel Create roulette wheel with ns segments Create random number between 0 and 1 Find segment on roulette wheel that contains the random number Segment number corresponds to design number Build parent database * * From presentation of Gerhard Venter

16 Child Population Creation Select two parents for each reproduction randomly from parent database Crossover : Probability close to 1 One point crossover randomly select crossover point Child = [parent1(1:cross_pt),parent2(cross_pt+1:n_var)] Mutation : Exploration parameter Probability of mutation is typically small (e.g. 0.2) Randomly select gene to mutate Randomly modify gene

17 Keeps best individuals Elitism Combine the child and parent population Select best individuals from the combined population * Figure from presentation of Tushar Goel

18 Nitching Guides the selection process toward a uniformly spread-out Pareto front Uses a parameter based upon crowding distance (c = a + b), where designs which provide the greatest spread along the Pareto front are favored Between two solutions with differing nondomination ranks, we prefer the solution with the lower (better) rank If both solutions belong to the same front, then we prefer the solution that is located in a lesser crowded region * Figure from presentation of Tushar Goel

19 Example Laminate Design

20 Problem Formulation Objectives : Design a symmetric laminate Maximize D11, maximize D22 Design Variables : 8 to 16 layers Layup orientations, 0 θi 90 (15 step) Constraints : D12 0.5*D11 D12 0.5*D22

21 Optimization Settings N_pop = 10; % size of the population N_gen = 30; % # of generations cross = 1.0; % crossover probability mut = 0.2; % mutation probability LB = [ ]; UB = [ ]; N_increments = [ ]; Use higher values

22 Layup Orientations For last 4 layers If variable is 0,the ply does not exist for i = 1:4 ply_angles(i) = (X(i)-1)*15; end count = 5; for i=5:8 if X(i) > 0 ply_angles(count) = (X(i)-1)*15; count = count+1; end end

23 Pareto Front 30 Generations A B C D Layup orientation ( ) Design D11 D22 con1 con2 θ1 θ2 θ3 θ4 θ5 θ6 θ7 θ8 A B C D θ1 is outermost layer