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DIVERSITY PRESERVING EVOLUTIONARY MULTI-OBJECTIVE SEARCH
Brian Piper1, Hana Chmielewski2, Ranji Ranjithan1,2
1Operations Research2Civil EngineeringNorth Carolina State University
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)?
NEED FOR SOLUTION DIVERSITY
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, …; 1976-2005} -- Evolutionary algorithms {Zechman & Ranjithan, …; 1998-2011}
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
SEARCH FOR DIVERSE SOLUTIONS
Red = Non-dominated solutionsGreen = near-non-dominated solutions, but more diverse in decision space
PARETO SET & DECISION SPACE DIVERSITY
Red = Non-dominated solutionsGreen = near-non-dominated solutions, but more diverse in decision space
PARETO SET & DECISION SPACE DIVERSITY
Initialize population (P) of n individuals for generation i:
Evaluate fitness of solutions
Update Archive (A)
Select parents from (P U A) consideringPareto 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
GENERAL STEPS OF THE ALGORITHMS
"TRIPLE RANK" ALGORITHM SELECTION PROCEDURE
• Non-dominance-based Ranking
o primary criterion: Pareto front ranko Secondary criterion: OS hypervolume
contribution within each rank
• Decision space Diversity-based Rankings 1.distance to closest non-dominated point in
the DS
2.sum of the distances to the two closest neighbors in the DS
SELECTION PROCEDURE… A solution may become a parent by: 1. DOMINATING - being in the current Pareto front
2. EXCELLING over 50% of solutions in the Pareto ranking and at least one of the two diversity rankings
"TRIPLE RANK" ALGORITHM
SELECTION PROCEDURE… A solution may become a parent by:1. DOMINATING - being in the Pareto front
2. EXCELLING over 50% of solutions in the fitness ranking and at least one of the two diversity rankings
3. 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
"TRIPLE RANK" ALGORITHM
SELECTION PROCEDURE… A solution may become a parent by:1. DOMINATING - being in the Pareto front
2. EXCELLING over 50% of solutions in the fitness ranking and at least one of the two diversity rankings
3. 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
4. SPECIALIZING - performing well in just one of the three rankings
"TRIPLE RANK" ALGORITHM
"TRIPLE RANK" ALGORITHMARCHIVING 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
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)
SELECTION PROCEDURE: Binary Tournament Step
A solution may become a parent through: • R(ank)H(ypervolume)C(luster) Selection
o 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
CLUSTER SELECTION ALGORITHM
ARCHIVING PROCEDURE • Add to the archive non-dominated solution from current population
o Solution added if non-dominant and far in DS and OS from non-dominated solutions already added
• For each non-dominated solution in the archiveo Add from own cluster solutions different in DS
CLUSTER SELECTION ALGORITHM
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
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
DOMINATED PROMOTION ALGORITHM
SELECTION PROCEDURE: Binary Tournament Step
A solution may become a parent through: • Binary Tournament
o Both feasible, largest hypervolume winso Feasible beats infeasibleo Both infeasible, minimum sum of infeasibilities wins
DOMINATED PROMOTION ALGORITHM
DOMINATED PROMOTION ALGORITHMARCHIVING PROCEDURE
• Update non-dominated solutions• Within a neighborhood of each non-dominated solution in OS
o Select solutions that are sufficiently far in DS• Trim archive set by removing nearest neighbor solutions in DS
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
• Average pairwise distance of all solutions (normalized by decision space diameter), (Shir, et. al, 2009)
• Average nearest neighbor distance
• Minimum nearest neighbor distance
DIVERSITY METRICS
𝐷𝑠=( 1𝐷𝑆𝐷
) 2𝑛(𝑛+1)∑𝑖=1
𝑛− 1
∑𝑗=𝑖+1
𝑛
𝑑𝑖𝑗
𝐷2=𝑚𝑖𝑛 {𝑑𝑖𝑗 ,∀ 𝑖 , 𝑗∨𝑖≠ 𝑗 }
𝐷1=1𝑛∑
𝑖=1
𝑛
𝑚𝑖𝑛 {𝑑𝑖𝑗 , ∀ 𝑗 ≠𝑖 }
TESTING AND COMPARISON • Two test problems:
• Lamé Superspheres• Omni-Test
• GA Settings:• Population size: 200• Number of generations: 50-100• 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
• n = 4 Decision Variables• 2 Objective Functions
where and and
,
TEST PROBLEM: 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 Non-dominated 3.206 ± 0.001 0.340 ± 0.025 0.309 ± 0.044 0.028 ± 0.011
Archive Min. distances 3.194 ± 0.008 0.349 ± 0.017 0.591 ± 0.065 0.197 ± 0.066
Neighborhood Min distances
3.197 ± 0.006 0.357 ± 0.018 0.478 ± 0.058 0.094 ± 0.035
Dominated Promotion
Non-dominated 3.208 ± 0.001 0.363 ± 0.008 0.286 ± 0.020 0.062 ± 0.008
Archive Min. distances 3.198 ± 0.004 0.372 ± 0.011 0.707 ± 0.047 0.289 ± 0.069
Neighborhood Min distances
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
RESULTS: LAMÉ SUPERSPHERES
TR TR TR CS CS CS DP DP DP Niche CMA NSGA NSGA OmniND All Near ND All Near ND All Near CMA MO II II-Agg
RESULTS: LAMÉ SUPERSPHERES
COMPARISION OF SHIR DIVERSITY METRICON LAMÉ SUPERSPHERES TEST RUNS
(30 random trials)
• n = 5 Decision Variables• 2 Objective Functions
TEST PROBLEM: 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 30.056 ± 0.267 0.186 ± 0.070 0.182 ± 0.076 0.004 ± 0.005
Archive Min. distances 30.040 ± 0.267 0.199 ± 0.073 0.268 ± 0.109 0.012 ± 0.005
Neighborhood Min distances
30.040 ± 0.267 0.201 ± 0.073 0.256 ± 0.104 0.011 ± 0.005
Cluster Selection Non-dominated 30.315 ± 0.122 0.109 ± 0.064 0.117 ± 0.079 0.012 ± 0.004
Archive Min. distances 30.1723 ± 0.130
0.129 ± 0.073 0.246 ± 0.162 0.033 ± 0.012
Neighborhood Min distances
30.185 ± 0.127 0.129 ± 0.071 0.212 ± 0.126 0.027 ± 0.008
Dominated Promotion
Non-dominated 29.649 ± 0.151 0.272 ± 0.046 0.570 ± 0.159 0.0385 ± 0.007
Archive Min. distances 29.646 ± 0.151 0.281 ± 0.044 0.740 ± 0.192 0.050 ± 0.015
Neighborhood Min distances
29.649 ± 0.151 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 0.0301 ± 0.002
RESULTS: OMNI-TEST
TR TR TR CS CS CS DP DP DP Niche CMA NSGA NSGA OmniND All Near ND All Near ND All Near CMA MO II II-Agg
RESULTS: OMNI-TEST
COMPARISION OF SHIR DIVERSITY METRICON OMNI-TEST RUNS
(30 random trials)
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
TEST PROBLEM: LAMÉ SUPERSPHERES
CLUSTER SELECTION PARETO FRONT
TEST PROBLEM: LAMÉ SUPERSPHERES
CLUSTER SELECTION ALL-ARCHIVE NEAREST NEIGHBOR DISTANCE SELECTION
TEST PROBLEM: LAMÉ SUPERSPHERES
CLUSTER SELECTION HYPERVOLUME CONVERGENCE
TEST PROBLEM: LAMÉ SUPERSPHERES
LAMÉ SUPERSPHERES OBJECTIVE AND DECISION SPACES
TEST PROBLEM: OMNI-TEST
CLUSTER SELECTION PARETO FRONT
TEST PROBLEM: OMNI-TEST
CLUSTER SELECTION ALL-ARCHIVE NEAREST NEIGHBOR DISTANCE SELECTION
TEST PROBLEM: OMNI-TEST
CLUSTER SELECTION HYPERVOLUME CONVERGENCE