Multi-Objective Evolutionary Algorithms

Transcription

1 Multi-Objective Evolutionary Algorithms Kalyanmoy Deb a Kanpur Genetic Algorithm Laboratory (KanGAL) Indian Institute o Technology Kanpur Kanpur, Pin 0806 INDIA deb@iitk.ac.in a Currently visiting TIK, ETH Zürich

2 Overview o the Tutorial Multi-objective optimization Classical methods History o multi-objective evolutionary algorithms (MOEAs) Non-elitst MOEAs Elitist MOEAs Constrained MOEAs Applications o MOEAs Salient research issues

3 Multi-Objective Optimization We oten ace them 90% B C Comort A 40% 0k Cost 00k 3

4 More Examples A cheaper but inconvenient light A convenient but expensive light 4

5 Which Solutions are Optimal? Domination: x () dominates x () i (minimize) 5 6. x () is no worse than x () in all objectives x () is strictly better than x () in at least one objective (maximize) 5

6 Pareto-Optimal Solutions Non-dominated solutions: Among a set o solutions P, the nondominated set o solutions P (minimize) are those that are not dominated 5 4 by any member o the set P. 3 O(MN 3 ) algorithms exist. Pareto-Optimal solutions: When 6 0 P = S, the resulting P is Paretooptimal set A number o solutions are optimal (maximize) Non dominated ront 6

7 Pareto-Optimal Fronts Min Min Min Max Max Min Max Max 7

8 Preerence-Based Approach Multi objective optimization problem Estimate a Minimize Higher level relative Minimize importance inormation... vector Minimize M (w.. w w ) M subject to constraints Single objective optimization problem F = w + w w M M or a composite unction Single objective optimizer One optimum solution Classical approaches ollow it 8

9 Classical Approaches No Preerence methods (heuristic-based) Posteriori methods (generating solutions) A priori methods (one preerred solution) Interactive methods (involving a decision-maker) 9

10 Weighted Sum Method Construct a weighted sum o objectives and optimize F (x) = M m= w m m (x). User supplies weight vector w A Feasible objective space a w w b c d Pareto optimal ront 0

11 Diiculties with Weighted Sum Method Feasible objective space Need to know w Non-uniormity in Paretooptimal solutions Inability to ind some Pareto-optimal solutions A B C a b D Pareto optimal ront

12 ɛ-constraint Method Optimize one objective, constrain all other Minimize µ (x), subject to m (x) ɛ m, m µ; User supplies a ɛ vector B C a b c d ε ε ε ε D Need to know relevant ɛ vectors Non-uniormity in Pareto-optimal solutions

13 Diiculties with Most Classical Methods Need to run a singleobjective optimizer many times 0.9 A Expect a lot o problem knowledge Even then, good distribution is not guaranteed Multi-objective optimization as an application o single-objective optimization B Pareto optimal ront C D 3

14 Ideal Multi-Objective Optimization Multi objective optimization problem Minimize Minimize... Minimize M subject to constraints Step IDEAL Multi objective optimizer Multiple trade o solutions ound Higher level inormation Choose one solution Step Step Find a set o Pareto-optimal solutions Step Choose one rom the set 4

15 Advantages o Ideal Multi-Objective Optimization Decision-making becomes easier and less subjective Single-objective optimization is a degenerate case o multi-objective optimization Step inds a single solution No need or Step Multi-modal optimization is a special case o multi-objective optimization 5

16 Two Goals in Ideal Multi-Objective Optimization. Converge on the Paretooptimal ront. Maintain as diverse a distribution as possible 6

17 Why Evolutionary? Population approach suits well to ind multiple solutions Niche-preservation methods can be exploited to ind diverse solutions

18 History o Multi-Objective Evolutionary Algorithms (MOEAs) Early penalty-based approaches VEGA (984) Goldberg s suggestion (989) MOGA, NSGA, NPGA (993-95) Elitist MOEAs (SPEA, NSGA-II, PAES, MOMGA etc.) (998 Present) and beore Year Number o Studies

19 9

20 What to Change in a Simple GA? Modiy the itness computation Begin Initialize Population t = 0 Cond? No Yes Stop Evaluation Assign Fitness t = t + Reproduction Crossover Mutation 0

21 Identiying the Non-dominated Set Step Set i = and create an empty set P. Step For a solution j P (but j i), check i solution j dominates solution i. I yes, go to Step 4. Step 3 I more solutions are let in P, increment j by one and go to Step ; otherwise, set P = P {i}. Step 4 Increment i by one. I i N, go to Step ; otherwise stop and declare P as the non-dominated set. O(MN ) computational complexity

22 An Eicient Approach Kung et al. s algorithm (975) Step Sort the population in descending order o importance o Step, Front(P ) I P =, T return P as the output o Front(P ). Otherwise, T = Front(P () P ( P /) ) and T B = Front(P ( P /+) P ( P ) ). I the i-th solution o B is not dominated by any solution o T, create a merged set M = T {i}. Return B B M as the output o Front(P ). O ( N(log N) M ) or M 4 and O(N log N) or M = and 3 Merging

23 A Simple Non-dominated Sorting Algorithm Identiy the best non-dominated set Discard them rom population Identiy the next-best non-dominated set Continue till all solutions are classiied We discuss a O(MN ) algorithm later 3

24 Non-Elitist MOEAs Vector evaluated GA (VEGA) (Schaer, 984) Vector optimized EA (VOES) (Kursawe, 990) Weight based GA (WBGA) (Hajela and Lin, 993) Multiple objective GA (MOGA) (Fonseca and Fleming, 993) Non-dominated sorting GA (NSGA) (Srinivas and Deb, 994) Niched Pareto GA (NPGA) (Horn et al., 994) Predator-prey ES (Laumanns et al., 998) Other methods: Distributed sharing GA, neighborhood constrained GA, Nash GA etc. 4

25 Non-Dominated Sorting GA (NSGA) A non-dominated sorting o the population First ront: Fitness F = N to all Niching among all solutions in irst ront Note worst itness (say Fw) Second ront: Fitness Fw ɛ to all Niching among all solutions in second ront Continue till all ronts are assigned a itness 5

26 Non-Dominated Sorting GA (NSGA) 30 Fitness 5 6 x Front beore ater Niching in parameter space Non-dominated solutions are emphasized Diversity among them is maintained 6

27 Vector-Evaluated GA (VEGA) Divide population into M equal blocks Each block is reproduced with one objective unction Complete population participates in crossover and mutation Bias towards to individual best objective solutions A non-dominated selection: Non-dominated solutions are assigned more copies Mate selection: Two distant (in parameter space) solutions are mated Both necessary aspects missing in one algorithm 7

28 Multi-Objective GA (MOGA) Count the number o dominated solutions (say n) Fitness: F = n + A itness ranking adjustment Niching in itness space Rest all are similar to NSGA F Asgn. Fit

29 Niched Pareto GA (NPGA) Solutions in a tournament are checked or domination with respect to a small subpopulation (t dom ) I one dominated and other non-dominated, select second I both non-dominated or both dominated, choose the one with smaller niche count in the subpopulation Algorithm depends on t dom Nevertheless, it has both necessary components 9

30 NPGA (cont.) X Y Check or domination t_dom Population Parameter Space X Y 30

31 Shortcoming o Non-Elitist MOEAs Elite-preservation is missing Elite-preservation is important or proper convergence in SOEAs Same is true in MOEAs Three tasks Elite preservation Progress towards the Pareto-optimal ront Maintain diversity among solutions 3

32 Elitist MOEAs EA Elite Elite-preservation: Maintain an archive o non-dominated solutions Progress towards Pareto-optimal ront: Preerring non-dominated solutions Maintaining spread o solutions: Clustering, niching, or grid-based competition or a place in the archive (minimize) (maximize) 5 Non dominated ront clusters 3

33 Elitist MOEAs (cont.) Distance-based Pareto GA (DPGA) (Osyczka and Kundu, 995) Thermodynamical GA (TDGA) (Kita et al., 996) Strength Pareto EA (SPEA) (Zitzler and Thiele, 998) Non-dominated sorting GA-II (NSGA-II) (Deb et al., 999) Pareto-archived ES (PAES) (Knowles and Corne, 999) Multi-objective Messy GA (MOMGA) (Veldhuizen and Lamont, 999) Other methods: Pareto-converging GA, multi-objective micro-ga, elitist MOGA with coevolutionary sharing 33

34 Elitist Non-dominated Sorting Genetic Algorithm (NSGA-II) Non-dominated sorting: O(MN ) Calculate (n i, S i ) or each solution i n i : Number o solutions dominating i S i : Set o solutions dominated by i (5, Nil) 9 5 (0, {9,}) 34

35 NSGA-II (cont.) Elites are preserved Non dominated sorting Crowding distance sorting P t+ P t F F F 3 Q t Rejected R t 35

36 NSGA-II (cont.) Diversity is maintained: O(MN log N) 0 i- Cuboid i i+ l Overall Complexity: O(MN ) 36

37 NSGA-II Simulation Results NSGA II NSGA II (binary coded)

38 Strength Pareto EA (SPEA) Stores non-dominated solutions externally Pareto-dominance to assign itness External members: Assign number o dominated solutions in population (smaller, better) Population members: Assign sum o itness o external dominating members (smaller, better) Tournament selection and recombination applied to combined current and elite populations A clustering technique to maintain diversity in updated external population, when size increases a limit 38

39 SPEA (cont.) Fitness assignment and clustering methods Population Ext_pop x x x x x Fitness Assignment x x x x Function Space Clustering (d and p_max) Function Space 39

40 Pareto Archived ES (PAES) An (+)-ES Parent p t and child c t are compared with an external archive A t I c t is dominated by A t, p t+ = p t I c t dominates a member o A t, delete it rom A t and include c t in A t and p t+ = c t I A t < N, include c t and p t+ = winner(p t, c t ) I A t = N and c t does not lie in highest count hypercube H, replace c t with a random solution rom H and p t+ = winner(p t, c t ). The winner is based on least number o solutions in the hypercube 40

41 Niching in PAES-(+) Ospring 3 Parent Pareto optimal ront Ospring Parent Pareto optimal ront 4

42 Constrained Handling Penalty unction approach F m = m + R m Ω( g). Explicit procedures to handle ineasible solutions Jimenez s approach Ray-Tang-Seow s approach Modiied deinition o domination Fonseca and Fleming s approach Deb et al. s approach 4

43 Constrain-Domination Principle A solution i constraineddominates a solution j, i any is true: 0. Solution i is easible and solution j is not.. Solutions i and j are both ineasible, but solution i has a smaller overall constraint violation. 3. Solutions i and j are easible and solution i dominates solution j Front Front

44 Constrained NSGA-II Simulation Results

45 Applications o MOEAs Space-crat trajectory optimization Engineering component design Microwave absorber design Ground-water monitoring Extruder screw design Airline scheduling VLSI circuit design Other applications (reer Deb, 00 and EMO-0 proceedings) 45

46 Spacecrat Trajectory Optimization Coverstone-Carroll et al. (000) with JPL Pasadena Three objectives or inter-planetary trajectory design Minimize time o light Maximize payload delivered at destination Maximize heliocentric revolutions around the Sun NSGA invoked with SEPTOP sotware or evaluation 46

47 Earth Mars Rendezvous Mass Delivered to Target (kg.) Mars Earth Individual 44 Mars Mars Earth Individual Transer Time (yrs.) Earth Earth Individual 73 Mars Individual 36 47

48 Salient Research Tasks Scalability o MOEAs to handle more than two objectives Mathematically convergent algorithms with guaranteed spread o solutions Test problem design Perormance metrics and comparative studies Controlled elitism Developing practical MOEAs Hybridization, parallelization Application case studies 49

49 Hybrid MOEAs Combine EAs with a local search method Better convergence Faster approach Two hybrid approaches Local search to update each solution in an EA population (Ishubuchi and Murata, 998; Jaskiewicz, 998) First EA and then apply a local search 97

50 Posteriori Approach in an MOEA Problem MOEA Multiple local searches Clustering Non domination check Which objective to use in local search? 98

51 Proposed Local Search Method Weighted sum strategy (or a Tchebyche metric) F = i w i i i is scaled Weight w i chosen based on location o i in the obtained ront w j = ( max j M k= ( max k Weights are normalized j (x))/( max j k (x))/( max k w i = i min j ) min k ) 99

52 Fixed Weight Strategy Extreme solutions are assigned extreme weights Linear relation between weight and itness max A a b MOEA solution set Many solution can converge to same solution ater local search min min set ater local search max 00

53 Design o a Cantilever Plate mm Base plate 60 mm P Scaled delection Scaled delection 6 NSGA II Weight 8 6 Clustered solutions Scaled delection Scaled delection 6 Local search Weight 8 6 Non dominated solutions Weight Weight Nine trade-o solutions are chosen 0

54 Trade-o Solutions (.00, 0.00) (0.60, 0.40) (0.50, 0.50) (0.43, 0.57) (0.38, 0.6) (0.35, 0.65) (0.3, 0.77) (0.4, 0.86) (0.00,.00) 03

55 Conclusions Ideal multi-objective optimization is generic and pragmatic Evolutionary algorithms are ideal candidates Many eicient algorithms exist, more eicient ones are needed With some salient research studies, MOEAs will revolutionize the act o optimization EAs have a deinite edge in multi-objective optimization and should become more useul in practice in coming years 06