Functional Decomposition of NSGA-II and Various Problem-Solving Strategies

1
Functional Decomposition of NSGA-II and Various
Problem-Solving Strategies

• Kalyanmoy Deb
• Professor of Mechanical Engineering
• Indian Institute of Technology Kanpur
• Director, Kanpur Genetic Algorithms Laboratory
  (KanGAL)
• Email deb_at_iitk.ac.in http//www.iitk.ac.in/kangal
  /deb.htm

2
Overview

• Essentials of multi-objective optimization
• NSGA-II platform
• Different multi-objective problem-solving tasks
• Omni-optimizer
• Degeneracy to various single and multi-objective
  tasks
• Conclusions

3
Multi-Objective OptimizationHandling multiple
conflicting objectives

• We often face them

4
Which Solutions are Optimal?

• Relates to the concept of domination
• x(1) dominates x(2), if
• x(1) is no worse than x(2) in all objectives
• x(1) is strictly better than x(2) in at least one
  objective
• Examples
• 3 dominates 2
• 3 does not dominate 5

5
Pareto-Optimal Solutions

• PNon-dominated(P)
• Solutions which are
• not dominated by any member of the set P
• O(N log N)
• algorithms exist
• Pareto-Optimal set
• Non-dominated(S)
• A number of solutions are optimal

6
Pareto-Optimal Fronts

• Depends on the type of objectives
• Definition of domination takes care of
  possibilities
• Always on the boundary of feasible region

7
Local Versus Global Pareto-Optimal Fronts

• Local Pareto-optimal Front Domination check is
  restricted within a neighborhood (in decision
  space) of P

8
Some Terminologies

• Ideal point (z)
• nonexistent, lower bound on Pareto-optimal set
• Utopian point (z)
• nonexistent
• Nadir point (znad)
• Upper bound on Pareto-optimal set
• Normalization

9
Differences with Single-Objective Optimization

• One optimum versus multiple optima
• Requires search and decision-making
• Two spaces of interest, instead of one

10
Ideal Multi-Objective Optimization
Step 1 Find a set of Pareto-optimal
solutions Step 2 Choose one from the set
11
Two Goals in Ideal Multi-Objective Optimization

• Converge to the Pareto-optimal front
• Maintain as diverse a distribution as possible

12
Elitist Non-dominated Sorting Genetic Algorithm
(NSGA-II)

• NSGA-II can extract Pareto-optimal frontier
• Also find a well-distributed set of solutions
• iSIGHT and modeFrontier adopted NSGA-II

• Fast-Breaking Paper in Engineering by ISI Web of
  Science (Feb04)

13
Functional Decomposition

• Convergence
• Emphasize non-dominated solutions
• Diversity
• Prefer less-crowded solutions
• Elite-preservation
• For ensuring convergence properties

14
An Iteration of NSGA-II
Convergence
Diversity-maintenance
Elite-preservation
15
NSGA-II Crowding Distance
Diversity is maintained
Overall Complexity O(N logM-1N)

• Improve diversity by
• k-mean clustering
• Euclidean distance
• measure
• Other techniques

16
Simulation on ZDT1
17
Simulation on ZDT3
18
Changing Dominance Relation

• Alter the meaning of Pareto-optimal points
• Constrained optimization (Fonseca and Fleming,
  1996, Deb et al., 2000)
• Cone dominance (guided dominance, Branke et al.,
  2000)
• Distributed EMO (Deb et al., 2003)
• Epsilon-MOEA (Laumanns et al., 2003 Deb et al.,
  2005)
• Robust and reliability-based EMO (Deb and Gupta,
  2005)

19
Constraint-Domination Principle
A solution i constraint-dominates a solution j,
if any is true

• i is feasible and j is not
• i and j are both infeasible, but i has a smaller
  overall constraint violation
• i and j are feasible and i dominates j

20
Constrained NSGA-II Simulation Results
Minimize
Minimize
Where
Where
21
Simulation on TNK
22
Simulation on CTP5
23
Cone Dominance

• Using a DMs preference (not a solution but a
  region)
• Guided domination principle Biased niching
  approach
• Weighted domination approach

24
Distributed Computing of Pareto-Optimal Set

• Guided domination concept to search different
  parts of Pareto-optimal region
• Distributed computing of different parts

25
Distributed computing A Three-Objective Problem

• Spatial computing, not temporal

NSGA-II Simulations
Theory
26
e-MOEA Using e-Dominance

• EA and archive populations evolve
• One EA and one archive member are mated
• Archive update using e-dominance
• EA update using usual dominance

27
Comparative Study on Three-Objective DTLZ
Problems
28
Test Problem DTLZ2
29
Multi-Objective Robust Solutions

• Not all Pareto-optimal points may be robust
• A is robust, but B is not
• Decision-makers will be interested in knowing
  robust part of the front

30
Domination Based on Aggregate Functions

• Functions averaged over a delta-neighborhod
• Alternate Strategy (Type II Robustness)

31
Effect of d-neighborhood Size

• Theory and NSGA-II simulation
• Larger d, more shift from original front
• Some part is more sensitive than others

32
Effect of d-neighborhood Size

• Theory and NSGA-II simulation
• Larger d, more shift from original front
• Some part is no more robust

33
Robust Front as Partial Global and Partial Local
Theory For global front
34
Simulation Using NSGA-II
Simulation
35
Reliability-Based Optimization

• Deterministic optimum often not reliable
• Due to uncertainities in decision
  variables/problem parameters
• Find the reliable solution for a specified
  Reliability

36
Constrained Domination for Reliability
Consideration

• Chance constraints P(g(x)0) ß
• ß depends on chosen reliability
• Prefer reliable solutions
• Indicates how
• P-O front moves away with ß

37
Goal Programming Using EMO

• Target function values are specified
• Convert them to objectives and perform domination

38
Goal Programming Using EMO

• Target function values are specified
• Convert them to objectives and perform domination
  check with them

39
Preferred Diversity

• Find a subset of Pareto-optimal points dictated
  by preference information
• Biased EMO (Branke and Deb, 2005)
• Reference-point based EMO (Deb and Sundar, 2006)
• Knee-based EMO (Branke et al., 2004)
• Nadir point and EMO (Deb and Chaudhuri, 2006)
• Multi-modal EMO (Deb and Reddy, 2003)
• Variable versus objective space niching

40
Preference-Based EMO

• EMO (NSGA-II) not efficient for many objectives
• Large number of points needed
• Domination-based methods are slow

41
EMO for a Biased Distribution

• Choose a hyper-plane
• Project points on it
• Compute two distances d and d
• Compute Dd(d/d)a
• Point b has small D
• Point a has large D

42
Biased Distribution in NSGA-II
ZDT2 a100
ZDT1
43
Biased NSGA-II (cont.)
Three-objective Problems a0 and a500
44
Reference Point Based EMO

• Wierzbicki, 1980
• A P-O solution closer to a reference point
• Multiple runs
• Too structured
• Extend for EMO
• Multiple reference points in one run
• A distribution of solutions around each reference
  point

45
Reference Point Based EMO (cont.)

• Ranking based on closeness to each reference
  point
• Clearing within each niche with e

46
More Results

• Five-objective with two reference points
  (z1-50.5
• z1-40.2, z50.8)
• A engineering design problem with three reference
  points

47
Knee Based EMO

• Find only the knee or near-knee solutions
• Knees are important solutions
• Not much motivation to move out from knees
• A large gain for a small loss in any pair of
  objectives
• Non-convex front
• No knee point
• Extreme solutions are attractors

48
Finding Knee Solutions

• Branke et al. (2004) for more details

49
Nadir Point and EMO

• Important for knowing range and normalization of
  objectives
• Difficult to find using classical method
• Pay-off table method does not work

50
EMO for Finding Nadir Point

• Emphasize only extreme points
• M3 find complete front, else use extremized
  crowded NSGA-II

51
EMO for Finding Nadir Point (cont.)

• DTLZ problems extended up to 20 objectives

52
Multi-Modal EMOs

• Different solutions having identical objective
  values
• Multi-modal Pareto-optimal solutions Design,
  Bioinformatics

53
Multiple Gene Subsets for Leukemia Samples

• Deb and Reddy (BioSystems, 2003)
• Multiple (26) four-gene combinations for 100
  classification
• Discovery of some common genes

54
Parameter Versus Objective-space Niching

• Distribution depends on the space niching is
  performed

55
Redefining Elites

• To aid in better diversity
• Controlled Elitist EMO (Deb and Goel, 2001)

56
Controlled Elitism

• Keep solutions from dominated fronts in GP

57
Controlled Elitism (cont.)

• ZDT4 has many local P-O fronts
• g()1 is global
• Controlled elitism can come closer to global P-O
  front

58
Omni-OptimizerMotivation from Computation

• Multiple is a generic case, single is specific
• Single objective as a degenerate case
  multi-objective case
• One algorithm for single and multi-objective
  problem solving (Deb and Tiwari, 2005)
• Accommodating NFL theorem, not violating it
• Single-objective, uni-optimum problems
• Single-objective, multi-optima problems
• Multi-objective, uni-optimal front problems
• Multi-objective, multi-optimal front problems

59
Structure of Omni-optimizer

• Very much like NSGA-II
• Epsilon-dominance
• Variable-space and objective space niching
• Use maximum
• of both crowding
• distances

60
Single-Objective, Uni-Optimum

• Dominance reduced to simple lt
• Epsilon-dominance to falt fb-e
• Allows multiple solutions within e to exist
• Elite-preservation is similar to CHC and (µ?)-ES

61
Shinn et al.s 12 Problems

• 12 problems

62
Single-Objective, Multi-Optima

• Variable-space niching help find multiple
  solutions
• Weierstrass function
• 16 minima with f0

63
104Sin2(x) 20 Minima
64
Himmelblaus Function 4 Minima
65
Multi-Objective, Uni-Pareto front

• Constrained and unconstrained test problems

66
More Results

• Comparable performance to existing EMO methods

67
Multi-Objective, Multi-Optima

• Nine regions leading to the same Pareto-optimal
  front
• Multiple solutions cause a single Pareto-optimal
  point

68
Nine Optimal Regions
Omni-optimizer
NSGA-II
69
Nine Optimal Fronts
70
Conclusions

• Functional decomposition of NSGA-II
• Non-domination for convergence
• Niching for diverse set of solutions
• Elite-preservation for reliable convergence
• For a new problem-solving, find the suitable
  place to change
• Many different problem-solving tasks achieved
  with NSGA-II
• Omni-optimizer provides a holistic approach for
  optimization

71
Thank You for Your Attention

• Acknowledgement
• KanGAL students, staff and collaborators

• For further information

http//www.iitk.ac.in/kangal Email
deb_at_iitk.ac.in