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Performance Scaling of Multi-objective Evolutionary Algorithms Vineet R.Khare1, Xin Yao1 and Kalyanmoy Deb2 1 Natural

Computation Group School of Computer Science The University of Birmingham Edgbaston, Birmingham B15 2TT, UK {V.R.Khare, X.Yao}@cs.bham.ac.uk 2 Kanpur

Genetic Algorithms Laboratory (KanGAL) Indian Institute of Technology Kanpur Kanpur 208016, INDIA [email protected]

Overview • • • • • • •

Introduction Algorithms used Test problems Performance metrics Experimental study Results Conclusions

Introduction • MOEA studies restricted to 2-3 objectives. – NSGA II (with PAES) – on five 2 and 3 objective test problems [1]. – PESA (with PAES and SPEA) – on six 2 objective test finctions T1 to T6 [2]. – SPEA2 (with SPEA, NSGA-II, and PESA) – on 2 objective test problems except for 3 and 4 objective knapsack problem [3].

• Scalability with respect to: – Number of objectives. – Number of variables.

Algorithms Used • • • • •

PESA, SPEA2 and NSGA-II Four scalable test problems 2 to 8 objectives Real encoding Variational operators – Simulated Binary Crossover[4] – Polynomial Mutation[5]

Test Problems • Four test problems (DTLZ 1, 2, 3 and 6) proposed in [6] – Ease of construction – Scalability – Knowledge of exact shape and location of resulting PO front – Ability to control difficulties

Test Problems • Construction – an example (DTLZ3)

Test Problems • PO front for 3 objectives

Test Problems • PO front for 3 objectives

Test Problems • PO front for 3 objectives

Performance Metrics • Functionally independent [7] set of variables – Convergence to the reference set – Diversity – Running time

• Convergence metric – Average orthogonal distance from the PO front

• Diversity metric 0 ≤ D(P) ≤ 1

Performance Metrics • Calculating diversity metric – – – –

Obtained points projected on a hyper-plane Plane divided into M-1 dimensional boxes Assume each box contains one reference point For each such box, • Calculate h(box) (= 1 or 0) for final non-dominated set • Calculate H(box) (= 1 or 0) for reference set • Calculate m(h(box)) and m(H(box)) using: h(… j-1… ) 0 0 1 0 1 0 1

h(… j… ) 0 0 0 1 1 1 1

h(… j+1… ) 0 1 0 1 0 0 1

m(h(… j… )) 0.00 0.50 0.50 0.67 0.67 0.75 1.00

Performance Metrics • Calculating diversity metric (contd.) ∑ D( P) = ∑

m(h(i, j ,...))

i , j ,... H ( i , j ,...) ≠ 0 i , j ,...

m( H (i, j ,...))

H ( i , j ,...) ≠ 0

• To avoid boundary effects ∑ D( P) = ∑

i , j ,... H ( i , j ,...) ≠ 0 i , j ,...

H ( i , j ,...) ≠ 0

•

m(h(i, j ,...)) −∑

i , j ,...

m ( 0)

H ( i , j ,...) ≠ 0

m( H (i, j ,...)) − ∑

i , j ,...

m(0)

H ( i , j ,...) ≠ 0

Two different diversity metrics: 1. Based on PO front (Diversity metric1 – DPO(P) ) 2. Based on converged front (Diversity metric2 – DC(P))

Experimental Study • Tuned parameter values Parameter Crossover probability Distribution index for SBX Mutation probability (if n = # of variables) Distribution index for polynomial mutation Ratio of internal population size to archive size # of grid cells per dimension (PESA)

PESA 0.8 15 1/n 15 1:1 10

SPEA 2 0.7 15 1/n 15 1:1 -

•Equal function evaluations •Number of generations # of generations For M = 2, 3 and 4 DTLZ1 & DTLZ2 300 DTLZ3 & DTLZ6 500

For M = 6 and 8 600 1000

NSGA-II 0.7 15 1/n 20 1:1 -

Experimental Study • Population size [4, pages 404-405]: – Increase in number of objectives ⇒More solutions in first non-dominated front ⇒Similar fitness, low selection advantage

• Population scheme used - quadratic M 2 3 4 5 6 7 8 9 10

Population Size 20 50 100 150 250 400 600 850 1150

Maximum proportion of non-dominated solutions 0.2 0.22 0.28 0.36 0.45 0.52 0.6 0.68 ~0.75

Results – Convergence Metric

DTLZ1

DTLZ2

DTLZ3 DTLZ6 Convergence Metric Vs Number of Objectives

Results – Diversity Metric1

DTLZ1

DTLZ2

DTLZ3 DTLZ6 Diversity Metric1 (DPO(P)) Vs Number of Objectives

Results – Diversity Metric2

DTLZ1

DTLZ2

DTLZ3 DTLZ6 Diversity Metric2 (DC(P)) Vs Number of Objectives

Results – Running Time

DTLZ1

DTLZ2

DTLZ3 DTLZ6 Running Time (sec.) Vs Number of Objectives

Conclusion • PESA scales very well in terms of convergence, but is poor in diversity maintenance and running time. • SPEA2 scales well in terms of diversity maintenance but suffers in converging to global PO front and in running time. • NSGA II scales well in running time and diversity maintenance but suffers in converging to global PO front.

Conclusion • Grid size in PESA • Conclusions drawn from 2-3 objectives cannot be generalized for higher objectives

References 1. 2.

3.

4. 5. 6.

7.

Kalyanmoy Deb, Amrit Pratap, Sameer Agarwal, and T. Meyarivan. A Fast and Elitist Multiobjective Genetic Algorithm: NSGA–II. IEEE Transactions on Evolutionary Computation, 6(2):182–197, April 2002. David W. Corne, Joshua D. Knowles, and Martin J. Oates. The Pareto Envelope-based Selection Algorithm for Multiobjective Optimization. In Marc Schoenauer, Kalyanmoy Deb, Gunter Rudolph, Xin Yao, Evelyne Lutton, Juan Julian Merelo, and Hans-Paul Schwefel, editors, Proceedings of the Parallel Problem Solving from Nature VI Conference, pages 839–848, Paris, France, 2000. Springer. Lecture Notes in Computer Science No. 1917. Eckart Zitzler, Marco Laumanns, and Lothar Thiele. SPEA2: Improving the Strength Pareto Evolutionary Algorithm. Technical Report 103, Computer Engineering and Networks Laboratory (TIK), Swiss Federal Institute of Technology (ETH) Zurich, Gloriastrasse 35, CH-8092 Zurich, Switzerland, May 2001. Kalyanmoy Deb. Multi-Objective Optimization using Evolutionary Algorithms. John Wiley & Sons, Chichester, UK, 2001. ISBN 0-471-87339-X. Kalyanmoy Deb and Mayank Goyal. A Combined Genetic Adaptive Search (geneAS) for Engineering Design. Computer Science and Informatics, 26(4):30–45, 1996. Kalyanmoy Deb, Lothar Thiele, Marco Laumanns, and Eckart Zitzler. Scalable Test Problems for Evolutionary Multi-Objective Optimization. Technical Report 112, Computer Engineering and Networks Laboratory (TIK), Swiss Federal Institute of Technology (ETH), Zurich, Switzerland, 2001. Kalyanmoy Deb and Sachin Jain. Running Performance Metrics for Evolutionary Multi-objective Optimization. Technical Report 2002004, KanGAL, Indian Institute of Technology, Kanpur 208016, India, 2002.