R-NSGAII

Reference Point Based MultiReference Point Based Multi--objective objective Optimization Using Evolutionary AlgorithmsOptimization Using Evolutionary Algorithms

Reviewed by

Paskorn [email protected]://dssg.cs.umb.edu

K. Deb and J. SundarGECCO 2006

mailto:[email protected]

http://dssg.cs.umb.edu/

http://dssg.cs.umb.edu/

December 2, 08 DSSG Group Meeting 2/31

Outline• Introduction• Reference Based EMO Approaches• Proposed Reference Point Based EMO Approach

R-NSGA-II: Reference Pointed Based NSGA-II• Simulation Results

– Two-objective Problem– Three-objective Problem– Five-objective Problem– 10-objective Problem– Engineering Design Problem (weld-beam problem)

• Conclusions


NSGA-II Problems• NSGA-II

– elitism Non-dominated sorting genetic algorithms-II was proposed by K.Deb in 2002

– NSGA-II has problems in solving problems with a large number of objectives*

• Problems with a large number of objectives, the most of individuals in the population are non-dominated solutions

-> the individuals (solutions) may not move towards the Pareto-optimal region.

-> the size of population may be increased to overcome this issuebut 1) this makes the algorithm work very slow 2) how large the population size should be?

* NSGA-II paper in 2002


ObservationThe nature of decision makers: DM (--who wants to find the

optimal solutions for their problems)• In large-objective problem solving, decision makers have some

clues in their mind.• For example,

– In the problem of maximizing throughput and minimizing latency, DM may have a clue that throughput should be about 99.9%

• The EMOA (evolutionary multi-objective optimization algorithms) can provide the Pareto-optimal solutions close to the point that the throughput is 99.9% instead of the entire frontier.

• The DM can concentrate on only the regions on the Pareto-optimal frontier which are of interest to her/him.


Research Approach• In large-objective problem solving, if DM

provide a clue, EMOA can be put to benefit in finding a preferred and smaller set of Pareto-optimal solutions instead of the entire frontier.

• The proposed algorithm uses the concept of reference point methodology– DM provides a clue as some points on objective

domain that the DM interests• The proposed algorithm attempts to find a set of

preferred Pareto-optimal solutions near the reference points called regions of interest to a decision maker.


Design Principles1) Multiple preference conditions can be specified

simultaneously2) For each provided reference point, a set of Pareto-

optimal solutions close to the provided reference point is the target set of solutions

3) The algorithm can be used for any shape of Pareto optimal frontier (e.g., convex, non-convex, continuous, discrete, connected, disconnected.)

4) The algorithm can be used to a large number of objectives (e.g., 10 or more objectives), a large number of variables and linear or non-linear constraints


R-NSGA-IIThe proposed reference point-based NSGA-II

• R-NSGA-II provides a set of Pareto optimal solutions near a provided set of reference points. So that, the DM can have an idea about the regions that the DM interests.

• R-NSGA-II is implemented based on NSGA-II– DM provides one or more

reference points.


Rank 4

Pt

Qt

Crowding distancesorting

Pt+1: The parent population that will generate offspring to the next generation

NSGA2: NSGA2: MainloopMainloop

Step 3: Apply non-dominated sorting to Rt population. All non-dominated fronts of Pt+Qt are copied to the parent population rank by rank.

Pt: Selected Parents at generation t

Qt: the offspring that are generated from Pt

Step 4: Stop adding the individuals in the rank when the size of parent population is larger than the population size (N)

Individuals in the last accepted rank, that make the parent population size larger than N (in example, rank 4), are sorted by crowding distance sorting.

Rank 1

Rank 2

Rank 3

Rank 4

Rt

Population size = N

rejected

Rank 1

Rank 2

Rank 3

Rank 4 Pt1

Step 1: Tournament Each individual is compared with another randomly selected individual. (niche comparison) The copy of the winner is placed in the mating pool

Individual 1

Individual 2

Individual N

Mating pool

Step 2: Apply crossover rate for each individual in a mating pool, and select a parent (s). Two parents perform crossover and generate two offspring.Two offspring will be placed in the offspring population Qt+1

Non-dominated sorting

tournamentQtcrossover


Crowding distance Assignment

F1

∞Crowding Distance

1 5 8

8 - 2

Example (objective F1)

20

∞

Individuals are sorted in each objective domain.

The first individual and the last individual in the rank are assigned the crowding distance = infinity.

For other individuals, the crowding distance is calculated by the different of the objective value of two closet neighbors.

2

5 - 1 20 - 5

Objective value


Niche comparison

Between two solutions with differing non-domination ranks the solution with the better rank is preferred.

Otherwise if both solutions belong to the same front then the solution which is located in lesser crowded region (has lower crowding distance value) is preferred.


R-NSGA-IIThe proposed reference point-based NSGA-II

• The crowding distance in NSGA-II is modified in R-NSGA-II.

• In R-NSGA-II, crowding distance represents how the solutions are close to the reference points.


Crowding Distance in R-NSGA-II1) For each reference point, the normalization

Euclidean distance (dIR) of each solution of the front is calculated and the solutions are sorted in ascending order of distance.

dIR =qPM

i=1(fi(x)−Rifmaxi −fmin

i)2

dIR : the normalization Euclidean distance from individual I to reference RM is the number of objectivesfi

max and fimin are the population maximum and minimum objective value of i-th

objective

Normalization is used to avoid the problem that the objectives are in the different scale. (e.g., one objective value is ~1000 and another objective is ~0.01).


• This way, the solution closest to the reference point is assigned a best rank for the reference point.

R1 R2

Min F1

Min F21 3

1 3

2 2

3 14 1

Rank to reference point R1



Crowding Distance in R-NSGA-II2) After such computations (distance to reference

points ranking) are performed, the minimum of the assigned ranks is assigned as the crowding distance to a solution.

R1 R2

Min F1

Min F21

1

2

11



The solutions with a smaller crowding distance are preferred. This is used in binary tournament. If the two randomly selected solutions are in

the same front, the one that has smaller crowding distance is the winner.


Crowding Distance in R-NSGA-II3) To control the number of the solutions, all solutions having a sum of normalized difference

in objective values (Dxy) of ε or less between them are grouped.

A randomly picked solution from each group is retained and rest all group members are assigned a large crowding distance in order to discourage them to remain in the population.

Objective Value 2 3 9

(3-2)/8 = 1/8 (9-2)/8 = 7/8

(9-3)/8 = 6/8

0

(10-9)/8 =1/8

0

9 (9-2)/8 = 7/8 (9-3)/8 = 6/8 (10-9)/8 =1/8

(10-3)/8 = 7/8

10

2 0 (10-2)/8 = 1

3 (3-2)/8 = 1/8 (10-3)/8=7/8

10 (10-2)/8 = 1 0

Dxy =qPM

i=1;(fi(x)−fi(y)fmaxi −fmin

i)2

E.g., in one objective problem, ε = 2/8

2 3 9 102/8 2/8


ε ε


R-NSGA-II• If the decision maker is interested in biasing

some objectives more than others, a suitable weight vector can be used with each reference point.

• The solutions with a shortest weighted Euclidean distance from the reference point can be emphasized.

dIR =qPM

i=1wi(fi(x)−Rifmaxi −fmin

i)2

wi : the weight value for i-th objective


Simulation Results

• Two to 10 objectives optimization problems– ZDT1, ZDT2, ZDT3, DTLZ2

• Weld Beam Problem– 2 Objectives

• R-NSGA-II– SBX with nc = 10– Polynomial mutation nm =20

• Population size = 100• Max generations = 500 generations


ZDT1

• 30 varaible• f1 in [0,1]• f2 = 1 – sqrt(f1)

When ε is large, the range of obtained solutions is also large


• When some of the reference points are infeasible, the obtained solutions are on the Pareto front and close to the reference points.


With three different weight vectors,

Weight vector = (0.2,0.8) (more emphasize on f2) the obtained solutions are close to f2


ZDT2• 30 varaible• f1 and f2 in [0,1]• f2 = 1 – f12

• Non-convex Pareto front.

The result is similar to the convex Pareto front


ZDT3

• 30 variable• Disconnected set of

Pareto fronts

• Problem:Point A is not on Pareto front but it is not dominated by any solutions.

A is obtained!


DTLZ2

• 14 variable• Three objectives

A good distribution of solutions near the two reference points are obtained.


DTLZ2 five objective• 14 variable problem• Five objectives• Two reference points

(0.5,0.5,0.5,0.5,0.5)(0.2,0.2,0.2,0.2,0.8)PM

i=1 f2i = 1

Results the obtained solutions are in [1.000,1.044]

Solutions are very close to the true Pareto optimal front


10-Objective DTLZ2 Problem• 19 variable• Reference point fi = 0.25 for all i

The obtained results concentrates near fi = 1/sqrt(10) = 0.316P10i=1 f

2i = 1

For all obtained solutions.Results are on true Pareto front


The welded beam design Problem


• Three reference points

• The result shows that a given reference point is not an optimal solution (i.e., 20,0.002) but the obtained results are on the Pareto Front.

• Thus, if the DM is interested in knowing trade-off optimal solutions in three major areas (min cost, intermediate cost and deflection, and min deflection), the proposed algorithm is able to find solutions near the given reference points instead of finding solution on the entire Pareto-optimal fornt, thereby allowing the DM to consider only a few solutions that lie on the regions of her/his interest.


Conclusions• The paper proposed R-NSGA-II

– R-NSGA-II applied preference based strategy to obtain a preferred set solutions near the reference points.

– R-NSGA-II is designed on NSGA-II by changing crowding distance calculation.

– R-NSGA-II works well on many-objective optimization problem.

– R-NSGA-II provide the decision-maker with a set of solutions near her/his preference so that a better and more reliable decision can be made.

R-NSGAII

Education

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