toward a natural genetic / evolutionary algorithm for multiobjective optimization

Toward a NaturalGenetic/Evolutionary Algorithm for

Multiobjective OptimizationHariharane Ramasamy

Evolutionary Systems Inc., Cupertino, CA.

Toward a Natural Genetic/Evolutionary Algorithm for Multiobjective Optimization – p. 1/58

Outline

1. Problem Description

2. Practical Examples

3. Classical Algorithms

4. Genetic Algorithms

5. Multiobjective Genetic Algorithms

6. Extended Genetic Algorithms

7. Results

8. Protein Folding—Lattice Model

9. Future Research

10. Conclusion


Problem Description

1. In most practical optimization problems, one deals withsimultaneous optimization of multiple objectives that mayconflict with one another.

2. We seek efficient ways to optimize systems specified by atuple of parameters (not necessarily numerical). Eachtuple determines a system with fitnesses for the individualobjectives, this maps the parameter space into the fitnessspace.

3. The best compromise solutions are called the tradeoff ornondominated solutions and form the Pareto front. Theparameter set associated with the Pareto front is calledthe Pareto set


An Example

Figure 1: Multiobjective problem example with two objectives

The plot is an example of a biobjective problem (f1, f2) that depends on two parameters (x1,x2). Assuming minimization, the thick dark line represents the best tradeoff solutions andconstitute the Pareto front. In the above figure, point O is dominated by all points in theinterior of the quadrant OAB. The lines ~OA, ~OB, ~OC, and ~OD define the boundary pointswhere f1 or f2 gets better or worse by moving either horizontally or vertically with respect topoint O. However, the points inside the region OAC and ODB are neither dominated nornondominated by the point O. Each one of them has either f1 or f2 optimal, but not both withrespect to point O.


Local and Global Pareto Front

A local Pareto front dominates every point in its neighborhood and a global Pareto frontdominates every point in the objective space. In the left plot, A and B are two disjointobjective regions. Assuming minimization of objectives, B contains the global Pareto frontand A the local Pareto front. In the right plot, the curve AOBCD contains the entire objectivespace; OB is A local Pareto front, and the global Pareto front is the union of AO and CD.


Multiobjective problem

Thus a multiobjective problem can be stated as:

Minimize F (x) = (f1(x), f2(x), ..., fm(x))

such that x ∈ S

x = (x1, x2, ..., xn)

S is the feasible parameter space

m is the number of objectives

we seek feasible configurations, tuples (x1, x2,...,xn) that maps into points(f1, f2,...,fm) on objective space, which can’t be improved w.r.t anyobjective without worsening them w.r.t. some other objective.


Practical Examples

1. Protein folding problem

2. Multiple knapsack problem


Protein Folding Problem

Protein Fold

The primary sequence of a protein (left), based on its composition of amino acids, folds intoa unique three dimensional structure (right) under certain physiological conditions. Thefolding is driven by multiple physio-chemical properties like hydrogen bonds, protectinghydrophobes from water- a multiobjective problem.


Protein Folding—Lattice Model

Allowed Moves Buried Hydrophobe Rule

in in

.

The amino acids in the primary sequence of a protein is connected to each other by theformation of peptide bond that resembles a rectangular lattice. Using a three dimensionalrectangular lattice, protein folding is simulated with move rules defined by the left picture.The right picture defines the condition under which a side chain in the rectangular lattice isburied. The mutiobjective optimization is to find the three dimensional conformation that


Multiple Knapscack Problem

Problems such as bin packing, cutting stock, and financial managementcan all be modeled as multiple knapsack problems, in which;

1. we are given a set of items with specific weights and profits andmultiple knapsacks with fixed capacity.

2. We have to fill up each knapsack, maximizing its profits and spaceutilization.

We present results for two variations of multiple knapsack problems.

1. In one problem, when an item is selected, it is included in allknapsacks.

2. In the second, the selected item is included in only one knapsack.


Existing Methods

1. Traditional methods

2. Genetic algorithm methods


Summary of Traditional Methods

Name Advantages Disadvantages

WeightedMethods

Simple, works well when Paretofront is simple.

Computationally expensive, failswith complicated Pareto fronts.

ǫ-constraintMethod

Works with complicated regions. Success largely depends on theinitial solution, which might beselected in an infeasible region.

LexicographicApproach

Simple, optimizes objectives se-quentially by predefine priority.

A limited number of Pareto opti-mal points are found.

Normal Bound-ary Intersec-tion (NBI)

Finds a well-spread Paretopoints.

Fails with complicated land-scapes in objective space.


Genetic Algorithm

Genetic algorithm, motivated from natural evolution, repeatedly apply genetic operators on apool of solutions called populations.

generate initial population P randomlyset new population Pn

while desired convergence is achieved in the population doperform crossover with probability pc

if the fitness of offspring are better then parents thenadd the offspring to Pn

elsedecide with very low probability to include offspring in Pn

end ifperform mutation with probability pm

perform reproduction with probability pr

if size of(Pn) ≥ N thenreplace P with N members from Pn

reduce size of Pn by Nend if

end while


Genetic Algorithm—Metrics

1. Diversity—Genetic Algorithm often converge to a few dominant solutions andlose diversity. Any further optimization is ineffective since repeatedapplication of operators yield the few converged dominant solutions. Diversityis essential for multiobjective problems.

2. Elitism—selecting individuals with a bias to create better individuals is calledelitism; selecting the best parents to replace the less fit members drives thepopulation to converge to a few best parents.

3. Scalability—Performance of the algorithm should not deteriorate with anincrease in the number of objectives and parameters.

4. Exploration—Ability of an algorithm to find new solutions or reproduce lostsolutions is called exploration.

5. Exploitation—Ability to retain the current best solutions in the population iscalled exploitation.


Niching

1. Goldberg and Richardson proposed the initial method to promote thediversity within the population.

2. The method used a distance function, called sharing function, whichcalculates similarities within the population members using aparameter called niche radius in parameter or objective space.

3. The members in a crowded neighborhood get their fitness degraded,thus preventing their selection for next generation.

4. The sharing function helps to keep the solutions diverse.


Niching

1. Niche radius is difficult to determine. Too large a radius will ignore good tradeoffsolutions; too small can consider points that are already dominated in the set.

2. When there are more than two objectives, the number of solutions on the Pareto frontincreases, along with the complexity of the niche calculation.

3. In practical problems with more than two objectives, the sharing function has met littlesuccess.


Multiobjective Genetic Algorithms


ParetoArchivedEvolutionStrategy

Gives a clear description forclass of problems that genetic al-gorithm should excel.

Does not scale well with problemdimension.

Vector Evalu-ated GeneticAlgorithm

Executes by population reshuf-fling using each objective func-tion.

Loses good solutions and doesnot have a good spread of Paretooptimal set.

Weight-BasedGenetic Algo-rithm

Similar to weighted methods,evolves the weight along withthe optimization of the objectivefunction.

Fails when the objective functionis nonconvex.


Multiobjective Genetic Algorithms


Niched ParetoGenetic Algo-rithm

Builds a small nondominated setalong with the algorithm.

Success largely depends on thesmall nondominated subset.

NondominatedSorting Ge-netic Algorithm

Niching is performed in the deci-sion space.

Niching function is too complex,and scales poorly as number ofobjectives increases.

StrengthPareto Evo-lutionaryAlgorithm

Finds well-spread Pareto pointsby maintaining an external non-dominated set of fixed size.

clustering is performed to main-tain the size of the externalPareto set. This clustering pro-cess scales poorly with the size.


Extended Genetic Algorithms

Will be published in paper, only results are presented here


Results

We’ll see that our versions outperformed a large number of existing algorithms, ona battery of tests commonly used in the literature to evaluate optimizationmethods. Specifically, do our proposed methods :

1. find more Pareto front points with ease?

2. effectively move towards Pareto front?

3. explore and exploit the search space effectively?

4. scale with increase in number of objectives and parameters?

5. find different Pareto set points that map to one point in objective space?

6. scale well with scattered Pareto set?

7. find more that one distinct points in the Pareto set that map to the same pointin the Pareto front?

8. have a framework for flexible extension?


Results











Schaeffer-1

0

5

10

15

20

25

0 5 10 15 20 25

f2

f1

Table 1: Summary of Results—Schaeffer 1Name Pop. Size Total - Pareto front Points Duration

NSGA 100 ≤ 30

sec

SPEA 100 100 ≤ 30 sec

MOEAD 100 100 ≤ 30 sec

NSGA 1000 1000 2 min

SPEA 1000 1000 8 min

MOEAD 1000 300 11 sec

MOEAD 10000 300 11 sec

SEGA 100 2541 ≤ 30 sec

SEGA 100 18500+ 3 min

PEGA 100 608 ≤ 30 sec

PEGA 100 900+ 3 min


Schaeffer-2

0

5

10

15

20

25

30

35

-1 0 1 2 3 4

f2

f1

Table 2: Summary of Results—Schaeffer 2Name Pop. Size Total - Pareto front Points Duration

NSGA 100 100 ≤ 30

sec

SPEA 100 100 ≤ 30

sec

MOEAD 100 100 ≤ 30

sec

SEGA 100 3767 ≤ 30

sec

PEGA 100 x 2 409 ≤ 30

sec

SEGA 100 114000+ 3 min

PEGA 100 x 2 2300+ 3 min


Viennet

0

10

20

30

40

50

60

0 1 2 3 4 5 6 7 8 9

f2

f1

0

0.05

0.1

0.15

0.2

0 1 2 3 4 5 6 7 8 9

f3

Table 3: Viennet—Summary of ResultsName Unique PF points Unique PS points

NSGA 100 100

SPEA 1 1

MOEAD 40 65

SEGA 2394 2405

PEGA 3522 3605


Kalyanamoy

0 0.2

0.4 0.6

0.8 1 0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

f3

DTLZ4 - SEGA Population Plot

SEGAPEGA

f1

f2

f3

Table 4: Kalayanomoy Scalable Multiobjective ProblemDTLZ4—Seven Objectives—Summary of Results

Name Unique Pareto front points Unique Pareto set points

NSGA2 999 999

SEGA 744 773

PEGA 3728 4815


Results











Pareto front—Progress

20000

25000

30000

35000

20000 25000 30000 35000

f 2 P

rofit

s

f1 Profits

Knapsack 750-3 - F1 Versus F2 after 500 Generations

1000



20000

25000

30000

35000

20000 25000 30000 35000

f 2 P

rofit

s

f1 Profits


1000



20000

25000

30000

35000

20000 25000 30000 35000

f 2 P

rofit

s

f1 Profits


1500



20000

25000

30000

35000

20000 25000 30000 35000

f 2 P

rofit

s

f1 Profits


2000



20000

25000

30000

35000

20000 25000 30000 35000

f 2 P

rofit

s

f1 Profits


2500



20000

25000

30000

35000

20000 25000 30000 35000

f 2 P

rofit

s

f1 Profits


3000



20000

25000

30000

35000

20000 25000 30000 35000

f 2 P

rofit

s

f1 Profits


3300


Pareto front

20000

25000

30000

35000

20000 25000 30000 35000

f 3 P

rofit

s

f1 Profits

Knapsack 750-3 - PF F1 Versus F2 after 3300 Generations

500100015002000250030003300


Pareto front

20000

25000

30000

35000

20000 25000 30000 35000

f 3 P

rofit

s

f1 Profits


500100015002000250030003300


Results











Exploration And Exploitation

20000

25000

30000

35000

20000 25000 30000 35000

f 2 P

rofit

s

f1 Profits

PEGA - F1 Vs F2 Population

F1F2F3

F1-F2F1-F3F2-F3

F1-F2-F3



20000

25000

30000

35000

20000 25000 30000 35000

f 3 P

rofit

s

f2 Profits


F1F2F3

F1-F2F1-F3F2-F3

F1-F2-F3



20000

25000

30000

35000

20000 25000 30000 35000

f 3 P

rofit

s

f1 Profits


F1F2F3

F1-F2F1-F3F2-F3

F1-F2-F3


Results










Scaling with Objectives

Kalyanamoy Scalable Objective Test function - Five



DTLZ4—f1 vs f2—NSGA (left), SEGA (middle), PEGA(right)

0

0.5

1

1.5

2

0 0.5 1 1.5 2

f 2

f1

DTLZ4 - 5 Objectives - PF f1 vs f2 - NSGA2

NSGA2

0

0.5

1

1.5

2

0 0.5 1 1.5 2

f 2

f1

DTLZ4 - 5 Objectives - PF f1 vs f2 - SEGA

SEGA

0

0.5

1

1.5

2

0 0.5 1 1.5 2

f 2

f1

DTLZ4 - 5 Objectives - PF f1 vs f2 - PEGA

PEGA

Figure 2: Results of DTLZ4 with five objectives

Among the tests we did, only the NSGA results can be compared qualitatively with SEGAand PEGA. The graph clearly show the superior performance of EGAs.




0

0.5

1

1.5

2

0 0.5 1 1.5 2

f 4

f3


NSGA2

0

0.5

1

1.5

2

0 0.5 1 1.5 2

f 4

f3


SEGA

0

0.5

1

1.5

2

0 0.5 1 1.5 2

f 4

f3


PEGA






0

0.5

1

1.5

2

0 0.5 1 1.5 2

f 5

f1


NSGA2

0

0.5

1

1.5

2

0 0.5 1 1.5 2

f 5

f1


SEGA

0

0.5

1

1.5

2

0 0.5 1 1.5 2

f 5

f1


PEGA




Scaling with objectives

Kalyanamoy Scalable Objective Test function - Seven




0

0.5

1

1.5

2

0 0.5 1 1.5 2

f 2

f1


NSGA2

0

0.5

1

1.5

2

0 0.5 1 1.5 2

f 2

f1


SEGA

0

0.5

1

1.5

2

0 0.5 1 1.5 2

f 2

f1


PEGA

Figure 5: Results of DTLZ4 with seven objectives





0

0.5

1

1.5

2

0 0.5 1 1.5 2

f 4

f3


NSGA2

0

0.5

1

1.5

2

0 0.5 1 1.5 2

f 4

f3


SEGA

0

0.5

1

1.5

2

0 0.5 1 1.5 2

f 4

f3


PEGA






0

0.5

1

1.5

2

0 0.5 1 1.5 2

f 6

f5


NSGA2

0

0.5

1

1.5

2

0 0.5 1 1.5 2

f 6

f5


SEGA

0

0.5

1

1.5

2

0 0.5 1 1.5 2

f 6

f5


PEGA






0

0.5

1

1.5

2

0 0.5 1 1.5 2

f 7

f1


NSGA2

0

0.5

1

1.5

2

0 0.5 1 1.5 2

f 7

f1


SEGA

0

0.5

1

1.5

2

0 0.5 1 1.5 2

f 7

f1


PEGA




Results










Scaling with Parameter Space

Kalyanamoy Scalable Objective Test function - Five



DTLZ4—Pareto set ( x1, x2, x3)—NSGA (left), SEGA (middle), PEGA(right)

0 0.2

0.4 0.6

0.8 1 0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

x3

DTLZ4 - 5 Objectives - PS (x1,x2,x3) - NSGA2

NSGA2

x1

x2

x3

0 0.2

0.4 0.6

0.8 1 0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

x3

DTLZ4 - 5 Objectives - PS (x1,x2,x3) - SEGA

SEGA

x1

x2

x3

0 0.2

0.4 0.6

0.8 1 0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

x3

DTLZ4 - 5 Objectives - PS (x1,x2,x3) - PEGA

PEGA

x1

x2

x3





Kalyanamoy Scalable Objective Test function - Seven



DTLZ4—Pareto set ( x1, x2, x3)—NSGA (left), SEGA (middle), PEGA(right)

0 0.2

0.4 0.6

0.8 1 0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

x3

DTLZ4 - 7 Objectives - PS (x1, x2, x3) - NSGA2

NSGA2

x1

x2

x3

0 0.2

0.4 0.6

0.8 1 0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

x3

DTLZ4 - 7 Objectives - PS (x1, x2, x3) - SEGA

SEGA

x1

x2

x3

0 0.2

0.4 0.6

0.8 1 0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

x3

DTLZ4 - 7 Objectives - PS (x1, x2, x3) - PEGA

PEGA

x1

x2

x3




Multiple Knapsack Problem(Exclusive)

We present results for 750 items and 3 knapsacks


Three Knapsacks, 750 Items—Profits

SEGA f1 vs f3 PEGA f1 vs f3

0

5000

10000

15000

20000

25000

30000

35000

0 5000 10000 15000 20000 25000 30000 35000

f 3

f1

Knapsack 750-3 - f1 vs f3 Profits - SEGA

0

5000

10000

15000

20000

25000

30000

35000

0 5000 10000 15000 20000 25000 30000 35000

f 3

f1

Knapsack 750-3 - f1 vs f3 Profits - SEGA

Figure 11: Profits KP 750-3. The left and right columns show the profits-to-weightsratio obtained by the sequential and parallel extended genetic algorithms for f1 vsf3. The conflicting nature of the objective functions can be easily seen in the plots


Three Knapsacks, 750 Items—Items

SEGA f1 vs f3 PEGA f1 vs f3

0

250

500

750

0 250 500 750

Ite

ms in

KS

3

Items in KS1

Knapsack 750-3 - f1 vs f3 Items - SEGA

0

250

500

750

0 250 500 750

Ite

ms in

KS

3

Items in KS1

Knapsack 750-3 - f1 vs f3 Items - PEGA

Figure 12: Profits KP 750-3. The plots show the items obtained by the sequentialand parallel extended genetic algorithms for f1 vs f3. The conflicting nature of theobjective functions can be easily seen in the plots


Parameter Space—Exploration

Solutions that Map to Similar Sum

600

650

700

750

800

0 25 50 75 100

Tota

l Item

s Inc

luded

in a

ll Kna

psac

ks(S

EGA+

PEGA

)

Number of Occurrences

Knapsack 750-3 - Number of unique solutions

Figure 13: Item KP 750-3. The plots show the number of solutions assigned to thethree knapsacks that map to the same sum


Results










Scattered Pareto Set

Non Dominated Points/Non Dominated Set

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

f 2

f1

F1 versus F2 Corresponding x1, x2, x3 - Set

0 0.2 0.4 0.6 0.8 1x1 -1

-0.5 0

0.5 1

x2-1

-0.5

0

0.5

1

x3

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

f 2

f1

F1 versus F2 Corresponding x1, x2, x3 - Set

0 0.2 0.4 0.6 0.8 1x1 -1

-0.5 0

0.5 1

x2-1

-0.5

0

0.5

1

x3


Results










Flexibility

The results presented so far were ran with the same algorithmwithout altering any parameters.

Niching methods require suitable parameter selection to reachPareto front which is totally absent here.

Population size is not increased to obtain a larger Pareto set. This isa great advantage as we cannot provide infinite source to hold thepopulation.


Protein Folding - Interesting Results

Helix Beta Sheets

Figure 14: Two interesting results obtained by extended genetic algorithms. The red and bluerepresents hydrophobic and hydrophilic amino acids. The two green lattices, connecting thetwo helices, represent the glycine, which has four additional lattice movements. Glycine isoften found near sharp turns in the protein due to its small size. In the right result, every othermember in the sequence is hydrophobic, and hence the algorithm produced a beta sheet thatfolds against itself to bury the hydrophobes


Conclusion

Pseudo-parallelism is a feature in the genetic algorithm in which parameter space isexplored simultaneously by different members in the population. Extended geneticalgorithms successfully extended pseudo-parallelism not only in parameter space butalso in objective space.

The new algorithms leave much room for extension and improvements. A few keyenhancements are mentioned in Future research.


Conclusion


Parallel and sequential extended genetic algorithms represent two different extensionsof the genetic algorithm. Both the algorithms were applied for the first time to theprotein folding problem and were presented in several conferences in the year 1996.



Conclusion



Extended genetic algorithms have better performance in exploring objective andparameter space than do existing multiobjecitve genetic algorithms.



Conclusion




Duplication and transposon operators were introduced for the first time in the geneticalgorithm.



Conclusion




Duplication and transposon operators were introduced for the first time in the geneticalgorithm.

Extended genetic algorithms’ performed better with increase in the number ofobjectives and variables than were the multiobjective genetic algorithms.



Future Research

Natural Genetic Algorithm—described in the next slide represents the combination ofparallel and sequential extended genetic algorithm.


Future Research


Coding—binary coding is used in most of the problems. The selection of coding dependson the type of problem and the parameters associated with the problem.


Future Research


Coding—binary coding is used in most of the problems. The selection of coding dependson the type of problem and the parameters associated with the problem.

Operators—duplication and transposon operators used an arbitrary method. There aredifferent ways the operators can be performed.


Future Research

Toward a Natural Genetic / Ramasamy Algorithm


Questions

please email sundar_hariharane @ yahoo.com


toward a natural genetic / evolutionary algorithm for multiobjective optimization

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