diversity preserving evolutionary multi-objective search brian piper 1, hana chmielewski 2, ranji...

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

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


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" 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


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


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


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 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



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



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



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


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



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


CLUSTER SELECTION PARETO FRONT


CLUSTER SELECTION ALL-ARCHIVE NEAREST NEIGHBOR DISTANCE SELECTION


CLUSTER SELECTION HYPERVOLUME CONVERGENCE


LAMÉ SUPERSPHERES OBJECTIVE AND DECISION SPACES


CLUSTER SELECTION PARETO FRONT


CLUSTER SELECTION ALL-ARCHIVE NEAREST NEIGHBOR DISTANCE SELECTION


CLUSTER SELECTION HYPERVOLUME CONVERGENCE

diversity preserving evolutionary multi-objective search brian piper 1, hana chmielewski 2, ranji...

Documents

pareto solutions

solution diversity slide

diverse solutions

different solutions

fitness of solutions

decision space search

decision space ds

ds slide