1. DIVERSITY PRESERVING EVOLUTIONARY MULTI-OBJECTIVE SEARCH Brian Piper1, Hana Chmielewski2, Ranji Ranjithan1,2 1Operations Research 2Civil Engineering North Carolina State University

2. NEED FOR SOLUTION DIVERSITY
The optimal solution to a modeled system is not necessarily optimal for the real system
For complex engineering problems, usually modeled system ≈ (but ≠) real system
A set of alternative solutions (optimal and near-optimal) with maximally different solution characteristics can be useful in decision-making
Can we systematically search for maximally different solutions that are “good” (near-optimal)?

3. SEARCH FOR DIVERSE SOLUTIONS
Formal search methods for solutions that are diverse, i.e., maximally different in decision space:
Prior bodies of work on methods based on
-- Mathematical programming methods {Brill et al, …; }
-- Evolutionary algorithms {Zechman & Ranjithan, …; }
Recent interests in Evolutionary Multi-objective
Optimization for solution diversity
-- E.g., Ulrich et al. 2010; Shir et al. 2009, 2010
Goal: new EMO-methods for generating Pareto solutions that are diverse in decision space

4. PARETO SET & DECISION SPACE DIVERSITY
Finds Pareto front with efficiency comparable to an algorithm with only fitness criteria
Grants higher reproductive probability to some slightly less fit solutions with high decision space diversity
Produces subsequent populations and a final solution set with high fitness and input parameter diversity
Red = Non-dominated solutions
Green = near-non-dominated solutions, but more diverse in decision space

5. PARETO SET & DECISION SPACE DIVERSITY
The final set of solutions can be a mix of the Pareto and near-Pareto solutions
Red = Non-dominated solutions
Green = near-non-dominated solutions, but more diverse in decision space

6. GENERAL STEPS OF THE ALGORITHMS
Initialize population (P) of n individuals
for generation i:
    Evaluate fitness of solutions
    
    Update Archive (A)
    
Select parents from (P U A) considering
Pareto optimality metric in objective space (OS)
    Diversity metric in decision space (DS)
Perform crossover and mutation
if stopping criterion is unmet, proceed to generation i + 1
Select from archive the final solution set

7. "TRIPLE RANK" ALGORITHM
SELECTION PROCEDURE
 Non-dominance-based Ranking
primary criterion: Pareto front rank
Secondary criterion: OS hypervolume contribution within each rank
Decision space Diversity-based Rankings 
distance to closest non-dominated point in the DS
sum of the distances to the two closest neighbors in the DS

8. "TRIPLE RANK" ALGORITHM
SELECTION PROCEDURE…
 A solution may become a parent by: 
DOMINATING - being in the current Pareto front
EXCELLING over 50% of solutions in the Pareto ranking and at least one of the two diversity rankings
     

9. "TRIPLE RANK" ALGORITHM
SELECTION PROCEDURE…
 A solution may become a parent by:
DOMINATING - being in the Pareto front
EXCELLING over 50% of solutions in the fitness ranking and at least one of the two diversity rankings
REPLACING one of its nearest neighbors already in the parent set if one of its rankings greatly exceeds its neighbor's, and the other two are within a range

10. "TRIPLE RANK" ALGORITHM
SELECTION PROCEDURE…
 A solution may become a parent by:
DOMINATING - being in the Pareto front
EXCELLING over 50% of solutions in the fitness ranking and at least one of the two diversity rankings
REPLACING one of its nearest neighbors already in the parent set if one of its rankings greatly exceeds its neighbor's, and the other two are within a range
SPECIALIZING - performing well in just one of the three rankings

11. "TRIPLE RANK" ALGORITHM
ARCHIVING PROCEDURE
Non-dominated points are added to a Pareto archive (AP)
Near-Pareto optimal solutions with high diversity ranks are added to a diversity archive (AD)
The AP is trimmed (< population size) by keeping the non-dominated solutions with the highest diversity ranks within a neighborhood

12. CLUSTER SELECTION ALGORITHM
SELECTION PROCEDURE: Clustering Step
Assign rank based on constrained non-dominated sort
Calculate hypervolume contribution within each rank
Cluster solutions in Objective Space (OS) 
K-means, hierarchical, etc.
Calculate distance to cluster centroid in Decision Space (DS)

13. SELECTION PROCEDURE: Binary Tournament Step
CLUSTER SELECTION ALGORITHM   
SELECTION PROCEDURE: Binary Tournament Step
 A solution may become a parent through: 
R(ank)H(ypervolume)C(luster) Selection
Binary Tournament
Lower Pareto rank wins
If equal rank & hypervolume difference is not within a threshold
then: hypervolume contribution wins
else: larger distance to centroid of cluster in decision space wins

14. CLUSTER SELECTION ALGORITHM
ARCHIVING PROCEDURE
Add to the archive non-dominated solution from current population
Solution added if non-dominant and far in DS and OS from non-dominated solutions already added
For each non-dominated solution in the archive
Add from own cluster solutions different in DS

15. DOMINATED PROMOTION ALGORITHM
SELECTION PROCEDURE: Hypervolume-based Fitness Assignment 
For each dominated solution assign: 
the hypervolume of the new non-dominated front if that solution is considered as a non-dominated solution
remove all solutions dominating the one being considered
For each non-dominated solution, assign the hypervolume of the non-dominated set

16. DOMINATED PROMOTION ALGORITHM
SELECTION PROCEDURE: Hypervolume-based Fitness Assignment 
For each dominated solution assign: 
the hypervolume of the new non-dominated front if that solution is considered as a non-dominated solution
remove all solutions dominating the one being considered
For each non-dominated solution, assign the hypervolume of the non-dominated set

17. SELECTION PROCEDURE: Binary Tournament Step
DOMINATED PROMOTION ALGORITHM
SELECTION PROCEDURE: Binary Tournament Step
 A solution may become a parent through: 
Binary Tournament
Both feasible, largest hypervolume wins
Feasible beats infeasible
Both infeasible, minimum sum of infeasibilities wins

18. DOMINATED PROMOTION ALGORITHM
ARCHIVING PROCEDURE
Update non-dominated solutions
Within a neighborhood of each non-dominated solution in OS
Select solutions that are sufficiently far in DS
Trim archive set by removing nearest neighbor solutions in DS

19. FINAL SOLUTION SELECTION
The final solution set is constructed two different ways by searching the "neighborhood" around each non-dominated solution and selecting the solution with the largest minimum distance in the DS:  
1. to other points in
   the neighborhood
            OR
2. to all other points in the archive

20. Average pairwise distance of all solutions (normalized by decision
DIVERSITY METRICS
Average pairwise distance of all
solutions (normalized by decision
space diameter), (Shir, et. al, 2009)
Average nearest neighbor distance
Minimum nearest neighbor distance
𝐷𝑠= 1 𝐷𝑆 𝐷 2 𝑛(𝑛+1) 𝑖=1 𝑛−1 𝑗=𝑖+1 𝑛 𝑑 𝑖𝑗
𝐷 1 = 1 𝑛 𝑖=1 𝑛 𝑚𝑖𝑛 𝑑 𝑖𝑗 ,∀𝑗≠𝑖
𝐷 2 =𝑚𝑖𝑛 𝑑 𝑖𝑗 ,∀𝑖,𝑗|𝑖≠𝑗

21. TESTING AND COMPARISON
Two test problems:
Lamé Superspheres
Omni-Test
GA Settings:
Population size: 200
Number of generations:
Simulated Binary Crossover
Gaussian Mutation
30 random trials
Three candidates for diversity front
Best Pareto front found
Diversity front chosen by nearest neighborhood solution
Diversity front chosen by nearest archive solution
Performance comparisons based on
Hypervolume metric
Three diversity metrics

22. min 𝑓 1 = 1+𝑟 ⋅ cos 2 (𝑥 1 ) min 𝑓 2 = 1+𝑟 ⋅ sin 2 𝑥 1
TEST PROBLEM: LAMÉ SUPERSPHERES
n = 4 Decision Variables
2 Objective Functions
min 𝑓 1 = 1+𝑟 ⋅ cos 2 (𝑥 1 ) min 𝑓 2 = 1+𝑟 ⋅ sin 2 𝑥 1
where 𝜀= 1 𝑛 −1 𝑖=1 𝑛 𝑥 𝑖 and 𝑟= ( sin (𝜋⋅𝜀 )) 2 and
𝑥 1 ∈ 0, 𝜋 2 , 𝑥 𝑖 ∈ 1,5 ∀𝑖=2,…, 𝑛

23. RESULTS: LAMÉ SUPERSPHERES
Algorithm
Archive Sorting for Final Solution Set Selection Method
Hypervolume
Shir Diversity Metric
Avg. Dist. To Nearest Neighbor
Min. of Min. Dist. To Nearest Neighbor
Triple Rank
Non-dominated
3.206 ± 0.003
0.353 ± 0.014
0.345 ± 0.059
0.033 ± 0.027
Archive Min. distances
3.190 ± 0.009
0.375 ± 0.017
0.752 ± 0.065
0.335 ± 0.080
Neighborhood Min distances
3.194 ± 0.006
0.398 ± 0.017
0.592 ± 0.065
0.113 ± 0.051
Cluster Selection
3.206 ± 0.001
0.340 ± 0.025
0.309 ± 0.044
0.028 ± 0.011
3.194 ± 0.008
0.349 ± 0.017
0.591 ± 0.065
0.197 ± 0.066
3.197 ± 0.006
0.357 ± 0.018
0.478 ± 0.058
0.094 ± 0.035
Dominated Promotion
3.208 ± 0.001
0.363 ± 0.008
0.286 ± 0.020
0.062 ± 0.008
3.198 ± 0.004
0.372 ± 0.011
0.707 ± 0.047
0.289 ± 0.069
3.201 ± 0.004
0.393 ± 0.008
0.462 ± 0.035
0.078 ± 0.016
Niching-CMA
3.172 ± 0.037
0.412 ± 0.061
CMA-MO
3.205 ± 0.007
0.115 ± 0.019
NSGA-II
3.203 ± 0.001
0.224 ± 0.046
NSGA-II-Agg.
3.109 ± 0.108
0.307 ± 0.049
Omni-Opt.
2.481 ± 0.375
0.029 ± 0.060
best for the metric value and the robustness (i.e., low standard deviation)
red bold font for best (if it occurs in other algorithms besides ours)
black bold font for best occurring in our algorithms

24. COMPARISION OF SHIR DIVERSITY METRIC ON LAMÉ SUPERSPHERES TEST RUNS
RESULTS: LAMÉ SUPERSPHERES
COMPARISION OF SHIR DIVERSITY METRIC
ON LAMÉ SUPERSPHERES TEST RUNS
(30 random trials)
TR     TR     TR   CS   CS    CS    DP   DP    DP  Niche  CMA NSGA NSGA Omni
ND    All     Near  ND   All    Near   ND   All    Near CMA    MO    II     II-Agg

25. min 𝑓 1 = 𝑖=1 𝑛 sin (𝜋 𝑥 𝑖 ) min 𝑓 2 = 𝑖=1 𝑛 cos (𝜋 𝑥 𝑖 )
TEST PROBLEM: OMNI-TEST
n = 5 Decision Variables
2 Objective Functions
min 𝑓 1 = 𝑖=1 𝑛 sin (𝜋 𝑥 𝑖 ) min 𝑓 2 = 𝑖=1 𝑛 cos (𝜋 𝑥 𝑖 )
𝑥 𝑖 ∈ 0,6 , ∀𝑖

26. RESULTS: OMNI-TEST Algorithm
Archive Sorting for Final Solution Set Selection Method
Hypervolume
Shir Diversity Metric
Avg. Dist. To Nearest Neighbor
Min. of Min. Dist. To Nearest Neighbor
Triple Rank
Non-dominated
± 0.267
0.186 ± 0.070
0.182 ± 0.076
0.004 ± 0.005
Archive Min. distances
± 0.267
0.199 ± 0.073
0.268 ± 0.109
0.012 ± 0.005
Neighborhood Min distances
0.201 ± 0.073
0.256 ± 0.104
0.011 ± 0.005
Cluster Selection
± 0.122
0.109 ± 0.064
0.117 ± 0.079
0.012 ± 0.004
± 0.130
0.129 ± 0.073
0.246 ± 0.162
0.033 ± 0.012
± 0.127
0.129 ± 0.071
0.212 ± 0.126
0.027 ± 0.008
Dominated Promotion
± 0.151
0.272 ± 0.046
0.570 ± 0.159
± 0.007
± 0.151
0.281 ± 0.044
0.740 ± 0.192
0.050 ± 0.015
0.281 ± 0.043
0.712 ± 0.180
0.049 ± 0.015
Niching-CMA
30.27 ± 0.05
0.247 ± 0.061
CMA-MO
30.43 ± 0.002
0.042 ± 0.028
NSGA-II
30.17 ± 0.034
0.191 ± 0.085
NSGA-II-Agg.
29.81 ± 0.2
0.207 ± 0.065
Omni-Opt.
29.72 ± 0.20
± 0.002
best for the metric value and the robustness (i.e., low standard deviation)
red bold font for best (if it occurs in other algorithms besides ours)
black bold font for best occurring in our algorithms

27. COMPARISION OF SHIR DIVERSITY METRIC
RESULTS: OMNI-TEST
COMPARISION OF SHIR DIVERSITY METRIC
ON OMNI-TEST RUNS
(30 random trials)
TR     TR     TR   CS   CS    CS    DP   DP    DP  Niche  CMA NSGA NSGA Omni
ND    All     Near  ND   All    Near   ND   All    Near CMA    MO    II     II-Agg

28. - Relaxation margin for near-optimality
OBSERVATIONS & OUTLOOK
Dominated Promotion algorithms perform consistently well in Pareto optimality and DS diversity metrics
While most algorithms are robust, some sensitivity to internal parameters is observed. There is a need for improvement in parameter selections for:
- Relaxation margin for near-optimality
- Neighborhood size in final solution selection
Next steps also include
- Improving the adaptive mechanism for archive trimming and updating
- Applying and testing the methods on more test problems and engineering applications

30. CLUSTER SELECTION PARETO FRONT
TEST PROBLEM: LAMÉ SUPERSPHERES
CLUSTER SELECTION PARETO FRONT
Lame_cluster_selection (_best and _nearest_all)

31. CLUSTER SELECTION ALL-ARCHIVE NEAREST NEIGHBOR DISTANCE SELECTION
TEST PROBLEM: LAMÉ SUPERSPHERES
CLUSTER SELECTION ALL-ARCHIVE
NEAREST NEIGHBOR DISTANCE SELECTION
Lame_cluster_selection (_nearest_all)

32. TEST PROBLEM: LAMÉ SUPERSPHERES
CLUSTER SELECTION HYPERVOLUME CONVERGENCE
Lame_sluster_selection

33. LAMÉ SUPERSPHERES OBJECTIVE AND DECISION SPACES
TEST PROBLEM: LAMÉ SUPERSPHERES
LAMÉ SUPERSPHERES OBJECTIVE AND DECISION SPACES

34. CLUSTER SELECTION PARETO FRONT
TEST PROBLEM: OMNI-TEST
CLUSTER SELECTION PARETO FRONT
Cluster_selection_best and _nearest_all

35. CLUSTER SELECTION ALL-ARCHIVE NEAREST NEIGHBOR DISTANCE SELECTION
TEST PROBLEM: OMNI-TEST
CLUSTER SELECTION ALL-ARCHIVE
NEAREST NEIGHBOR DISTANCE SELECTION
CLUSTER_SELECTION_best and _nearest_all

36. CLUSTER SELECTION HYPERVOLUME CONVERGENCE
TEST PROBLEM: OMNI-TEST
CLUSTER SELECTION HYPERVOLUME CONVERGENCE
Cluster selection _best and _nearest_all