multi objective optimization and benchmark functions result

Project Incharge: Mr. Divya Kumar

Muti-objective optimization using NSGA II and SPEA2

Team(CS07):

1 Piyush Agarwal2 Saquib Aftab3 Ravi Ratan4 Ravi Shankar5 Pradhumna Mainali

Multi-objective optimization 1/31

What we did?

Studied genetic algorithm: single objective optimization,multi-objective optimization problems.Implemented NSGA II and Strength Pareto EvolutionaryAlgorithm (SPEA2) in MATLABTested SPEA2 algorithm on all benchmark functionTested NSGA II algorithm on all benchmark functionCompared the results


What is Evolutionary Algorithm?

Evolutionary algorithms (EAs) are often well-suited foroptimization problems involving several, often conflictingobjectivesEvolutionary algorithms typically generate sets ofsolutions, allowing computation of an approximation ofthe entire Pareto frontSPEA2 and NSGA II are two such EvolutionaryAlgorithms implemented on multi-objective functions


Life cycle of EA

Initialization : Initializing the population in the firstgeneration satisfying the bounds and constraints of theproblem.Parent Selection : Selection of fittest individuals for themating poolRecombination: Forming new individuals from themating pool. Crossover and Mutation is applied to theparents to produce new individuals.Survivor Selection: Fittest individuals of parents andchildren combined are selected as population for nextgeneration.


Life cycle of EA contd.

Figure 1: Life cycle of Evolutionary Algorithm


NSGA II Algorithm

Input1 N (Population Size)2 P (Population)3 Q (Offsprings)4 T (Maximum number of generations)

Output1 A (non-dominated set)


NSGA II Algorithm Contd.

1 Initialization: Generate an initial population P.2 Mating selection: Perform binary tournament selection

with replacement on P in order to fill the mating pool.3 Variation: Apply recombination and mutation operators to

the mating pool and set P to the resulting population andstore the result into Q.

4 Non dominated sort: Non dominated sort of P and Q5 Fronts division: Divide into fronts. Front 0 is

non-dominated.6 New generation: Selection of new population from fronts


Fast non dominated sorting

1 Each population i is compared with every population j.2 ni is the count of individuals which dominate the ith

population.3 Si is the set of individuals that i dominates.4 When ni = 0, that means the individual is the best solution

and is assigned first front.5 After getting the first front,for each individuals in Si, np is

decremented by 1 and the next front in obtained like inStep 4.


Generation of NSGA II

Figure 2: One generation of NSGA II algorithm


Evolution of individuals on ZDT 3 Function


SPEA2 Algorithm

Input1 N(Population Size)2 N(Archive size)3 T (Maximum number of generations)

Output1 A (non-dominated set)


SPEA2 Algorithm contd.

1 Initialization: Generate an initial population P0 and createthe empty archive P0 = φ. Set t = 0.

2 Fitness assignment: Calculate fitness values of individualsin Pt and Pt.

3 Environmental selection: Copy all non-dominatedindividuals in Pt and Pt to Pt+1 keeping the size N.

4 Termination: If t ≥ T or another stopping criterion issatisfied then set A to the set of decision vectors in Pt+1.Stop.

5 Mating selection: Perform binary tournament selectionwith replacement on Pt+1 in order to fill the mating pool.

6 Variation: Apply recombination and mutation operators tothe mating pool and set Pt+1 to the resulting population.Increment generation counter (t = t + 1) and go to Step 2


Evaluation of individuals on Viennet Function


Testing on Benchmark Functions

Both the algorithm, NSGA II and SPEA2 are tested on allbenchmark functions. Benchmark functions may be convex ornon-convex (un-constraint) or can have single or multipleconstraints.For all tests on benchmark, the Red graphrepresents NSGA II curve and Yellow represents SPEA 2 curve.The x-axis represents the First objective function and the y -axis represents Second objective function.


Schaffer function N. 1

Minimize =

f1(x) = x2

f2(x) = (x − 2)2

s.t =

−A ≤ x ≤ A10 ≤ A ≤ 105


ZDT1

Minimize =

f1(x) = x1

f2(x) = g(x)h(f1(x), g(x))g(x) = 1 + 9

29∑30

i=2 xi

h(f1(x), g(x)) = 1 −√

f1(x)g(x)

s.t =

0 ≤ xi ≤ 11 ≤ i ≤ 30


Schaffer function N. 2

Minimize =

f1(x) =

−x, if x ≤ 1x − 2, if 1 ≤ x < 34 − x, if 3 ≤ x < 4x − 4, if 4 ≤ x

f2(x) = (x − 5)2

s.t ={−5 ≤ x ≤ 10


ZDT3

Minimize =

f1(x) = x1

f2(x) = g(x)h(f1(x), g(x))g(x) = 1 + 9

29∑30

i=2 xi

h(f1(x), g(x)) = 1 −√

f1(x)g(x) − ( f1(x)

g(x) ) sin(10πf1(x))

s.t =

0 ≤ xi ≤ 11 ≤ i ≤ 30


ZDT2

Minimize =

f1(x) = x1

f2(x) = g(x)h(f1(x), g(x))g(x) = 1 + 9

29∑30

i=2 xi

h(f1(x), g(x)) = 1 − ( f1(x)g(x) )2

s.t =

0 ≤ xi ≤ 11 ≤ i ≤ 30


Fonseca and Fleming function

Minimize =

f1(x) = 1 − exp(−∑n

i=1(xi −1√

n)2)

f2(x) = 1 − exp(−∑n

i=1(xi −1√

n)2)

s.t =

−4 ≤ xi ≤ 41 ≤ i ≤ n


Kursawe function

Minimize =

f1(x) =∑2

i=1[−10 exp(−0.2√

x2i + x2

i+1)]

f2(x) =∑3

i=1[|xi|0.8 + 5 sin(x3

i )]√

n)2)

s.t =

−5 ≤ xi ≤ 51 ≤ i ≤ 3


Poloni’s two objective function (POL)

Minimize =

f1(x, y) = [1 + (A1 − B1(x, y))2 + (A2 − B2(x, y))2]f2(x, y) = (x + 3)2 + (y + 1)2

s.t =

A1 = 0.5 sin(1) − 2 cos(1) + sin(2) − 1.5 cos(2)A2 = 1.5 sin(1) − cos(1) + 2 sin(2) − 0.5 cos(2)B1(x, y) = 0.5 sin(x) − 2 cos(x) + sin(y) − 1.5 cos(y)B2(x, y) = 1.5 sin(x) − cos(x) + 2 sin(y) − 0.5 cos(y)−π ≤ x, y ≤ π


Viennet function

Minimize =

f1(x, y) = 0.5(x2 + y2) + sin(x2 + y2)

f2(x, y) =(3x−2y+4)2

8 +(x−y+1)2

27 + 15f3(x, y) = 1

x2+y2+1 − 1.1 exp(−(x2 + y2))

s.t ={−3 ≤ x, y ≤ 3


Binh and Korn function

Minimize =

f1(x, y

)= 4x2 + 4y2

f2(x, y

)= (x − 5)2 + (y − 5)2

s.t =

g1(x, y) = (x − 5)2 + y2

≤ 25g2(x, y) = (x − 8)2 + (y + 3)2

≥ 7.70 ≤ x ≤ 50 ≤ y ≤ 3


Chakong and Haimes function

Minimize =

f1(x, y) = 2 + (x − 2)2 + (y − 1)2

f2(x, y) = 9x − (y − 1)2

s.t =

g1(x, y) = x2 + y2

≤ 225g2(x, y) = x − 3y + 10 ≤ 0−20 ≤ x, y ≤ 20


Test Function

Minimize =

f1(x) = x2− y

f2(x) = −0.5x − y − 1

s.t =

g1(x, y) = 6.5 − x

6 − y ≥ 0g2(x, y) = 7.5 − 0.5x − y ≥ 0g3(x, y) = 30 − 5x − y ≥ 0−7 ≤ x, y ≤ 4


Osyczka and Kundu function

Minimize =

f1(x) = −25(x1 − 2)2− (x2 − 2)2

− (x3 − 1)2− (x4 − 4)2

− (x5 − 1)2

f2(x) =∑6

i=1 x2i

s.t = g1(x) = x1 + x2 − 2 ≥ 0g2(x) = 6 − x1 − x2 ≥ 0g3(x) = 2 − x2 + x1 ≥ 0g4(x) = 2 − x1 + 3x2 ≥ 0g5(x) = 4 − (x3 − 3)2

− x4 ≥ 0g6(x) = (x5 − 3)2 + x6 − 4 ≥ 00 ≤ x1, x2, x6 ≤ 101 ≤ x3, x5 ≤ 50 ≤ x4 ≤ 6


Constr-Ex problem function

Minimize =

f1(x, y) = xf2(x, y) =

1+yx

s.t = g1(x, y) = y + 9x ≥ 6g1(x, y) = −y + 9x ≥ 10.1 ≤ x ≤ 10 ≤ y ≤ 5


Work in progress

One of the application of multi-objective real life problems isPortfolio Optimization. In a portfolio problem with an assetuniverse of n securities, let xi (i = 1, 2, . . . , n) designate theproportion of initial capital to be allocated to security i. Andtypically there are two conflicting goals

Minimize risk.∑n

i=1∑n

j=1 xiσijxj

Maximize profit∑n

i=1 rixi

where ri is the expected return of ith security. σij is the covariance between ith and jth security.


Constraints of Portfolio Optimization

n∑i=1

xi = 1

α <= xi <= β

dmin <= d <= dmax

5/10/40 rule

0 <= α <= β <= 1

where, α and β are minimum and maximum capital proportionto be allocated to security i respectively. dmin and dmax is the

minimum and maximum number of non zero securities in theportfolio respectively.


Conclusion

It was derived from the project that the multi-objectiveevolutionary algorithm can solve multi-objective functionssatisfying given sets of constraints. Higher number ofgenerations will lead to better solutions until an upper bound isreached where all solutions tend to converge. Multi-objectiveoptimization algorithms can solve various real life applicationsby converting them into sets of objective functions andapplying constraints.


multi objective optimization and benchmark functions result

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