# Multi-objective

## MOEA/D-DE

Multiobjective optimization problems with complicated Pareto sets by (Li and Zhang, 2008).

Metaheuristics.MOEAD_DEType
MOEAD_DE(weights ;
F = 0.5,
CR = 1.0,
λ = Array{Vector{Float64}}[], # ref. points
η = 20,
p_m = -1.0,
T = round(Int, 0.2*length(weights)),
δ = 0.9,
n_r = round(Int, 0.02*length(weights)),
z = zeros(0),
B = Array{Int}[],
s1 = 0.01,
s2 = 20.0,
information = Information(),
options = Options())

MOEAD_DE implements the original version of MOEA/D-DE. It uses the contraint handling method based on the sum of violations (for constrained optimizaton): g(x, λ, z) = max(λ .* abs.(fx - z)) + sum(max.(0, gx)) + sum(abs.(hx))

To use MOEAD_DE, the output from the objective function should be a 3-touple (f::Vector, g::Vector, h::Vector), where f contains the objective functions, g and h are the equality and inequality constraints respectively.

A feasible solution is such that g_i(x) ≤ 0 and h_j(x) = 0.

Ref. Multiobjective Optimization Problems With Complicated Pareto Sets, MOEA/D and NSGA-II; Hui Li and Qingfu Zhang.

Example

Assume you want to solve the following optimizaton problem:

Minimize:

f(x) = (x_1, x_2)

subject to:

g(x) = x_1^2 + x_2^2 - 1 ≤ 0

x_1, x_2 ∈ [-1, 1]

A solution can be:


# Dimension
D = 2

# Objective function
f(x) = ( x, [sum(x.^2) - 1], [0.0] )

# bounds
bounds = [-1 -1;
1  1.0
]

nobjectives = 2
npartitions = 100

# reference points (Das and Dennis's method)
weights = gen_ref_dirs(nobjectives, npartitions)

# define the parameters

# optimize

# show results
display(status_moead)
source

## NSGA-II

A fast and elitist multiobjective genetic algorithm: NSGA-II by (Deb et al., 2002).

Metaheuristics.NSGA2Type
NSGA2(;
N = 100,
η_cr = 20,
p_cr = 0.9,
η_m = 20,
p_m = 1.0 / D,
information = Information(),
options = Options(),
)

Parameters for the metaheuristic NSGA-II.

Parameters:

• N Population size.
• η_cr η for the crossover.
• p_cr Crossover probability.
• η_m η for the mutation operator.
• p_m Mutation probability (1/D for D-dimensional problem by default).

To use NSGA2, the output from the objective function should be a 3-touple (f::Vector, g::Vector, h::Vector), where f contains the objective functions, g and h are inequality, equality constraints respectively.

A feasible solution is such that g_i(x) ≤ 0 and h_j(x) = 0.

using Metaheuristics

# Dimension
D = 2

# Objective function
f(x) = ( x, [sum(x.^2) - 1], [0.0] )

# bounds
bounds = [-1 -1;
1  1.0
]

# define the parameters (use NSGA2() for using default parameters)
nsga2 = NSGA2(N = 100, p_cr = 0.85)

# optimize
status = optimize(f, bounds, nsga2)

# show results
display(status)
source

## NSGA-III

An Evolutionary Many-Objective Optimization Algorithm Using Reference-Point-Based Nondominated Sorting Approach, Part I: Solving Problems With Box Constraints by (Deb and Jain, 2014).

Metaheuristics.NSGA3Type
NSGA3(;
N = 100,
η_cr = 20,
p_cr = 0.9,
η_m = 20,
p_m = 1.0 / D,
partitions = 12,
reference_points = Vector{Float64}[],
information = Information(),
options = Options(),
)

Parameters for the metaheuristic NSGA-III.

Parameters:

• N Population size.
• η_cr η for the crossover.
• p_cr Crossover probability.
• η_m η for the mutation operator.
• p_m Mutation probability (1/D for D-dimensional problem by default).
• reference_points reference points usually generated by gen_ref_dirs.
• partitions number of Das and Dennis's reference points if reference_points is empty.

To use NSGA3, the output from the objective function should be a 3-touple (f::Vector, g::Vector, h::Vector), where f contains the objective functions, g and h are inequality, equality constraints respectively.

A feasible solution is such that g_i(x) ≤ 0 and h_j(x) = 0.

using Metaheuristics

# Objective function, bounds, and the True Pareto front
f, bounds, pf = Metaheuristics.TestProblems.get_problem(:DTLZ2)

# define the parameters (use NSGA3() for using default parameters)
nsga3 = NSGA3(p_cr = 0.9)

# optimize
status = optimize(f, bounds, nsga3)

# show results
display(status)
source

## SMS-EMOA

An EMO algorithm using the hypervolume measure as a selection criterion by (Emmerich et al., 2005).

Metaheuristics.SMS_EMOAType
SMS_EMOA(;
N = 100,
η_cr = 20,
p_cr = 0.9,
η_m = 20,
p_m = 1.0 / D,
n_samples = 10_000,
information = Information(),
options = Options(),
)

Parameters for the metaheuristic SMS-EMOA.

Parameters:

• N Population size.
• η_cr η for the crossover.
• p_cr Crossover probability.
• η_m η for the mutation operator.
• p_m Mutation probability (1/D for D-dimensional problem by default).
• n_samples number of samples to approximate hypervolume in many-objective (M > 2).

To use SMS_EMOA, the output from the objective function should be a 3-touple (f::Vector, g::Vector, h::Vector), where f contains the objective functions, g and h are inequality, equality constraints respectively.

A feasible solution is such that g_i(x) ≤ 0 and h_j(x) = 0.

using Metaheuristics

# Dimension
D = 2

# Objective function
f(x) = ( x, [sum(x.^2) - 1], [0.0] )

# bounds
bounds = [-1 -1;
1  1.0
]

# define the parameters (use SMS_EMOA() for using default parameters)
sms_emoa = SMS_EMOA(N = 100, p_cr = 0.85)

# optimize
status = optimize(f, bounds, sms_emoa)

# show results
display(status)
source

## SPEA2

Improved strength Pareto evolutionary algorithm by (Zitzler et al., 2001).

Metaheuristics.SPEA2Type
SPEA2(;
N = 100,
η_cr = 20,
p_cr = 0.9,
η_m = 20,
p_m = 1.0 / D,
information = Information(),
options = Options(),
)

Parameters for the metaheuristic NSGA-II.

Parameters:

• N Population size.
• η_cr η for the crossover.
• p_cr Crossover probability.
• η_m η for the mutation operator.
• p_m Mutation probability (1/D for D-dimensional problem by default).

To use SPEA2, the output from the objective function should be a 3-touple (f::Vector, g::Vector, h::Vector), where f contains the objective functions, g and h are inequality, equality constraints respectively.

A feasible solution is such that g_i(x) ≤ 0 and h_j(x) = 0.

using Metaheuristics

# Dimension
D = 2

# Objective function
f(x) = ( x, [sum(x.^2) - 1], [0.0] )

# bounds
bounds = [-1 -1;
1  1.0
]

# define the parameters (use SPEA2() for using default parameters)
nsga2 = SPEA2(N = 100, p_cr = 0.85)

# optimize
status = optimize(f, bounds, nsga2)

# show results
display(status)
source

## CCMO

A Coevolutionary Framework for Constrained Multiobjective Optimization Problems proposed by (Tian et al., 2021).

Metaheuristics.CCMOType
CCMO(base_optimizer; infromation, options)

Parameters for CCMO algorithm. base_algorithm only supports NSGA2().

A feasible solution is such that g_i(x) ≤ 0 and h_j(x) = 0.

Example

julia> f, bounds, pf = Metaheuristics.TestProblems.MTP();

julia> ccmo = CCMO(NSGA2(N=100, p_m=0.001));

julia> optimize(f, bounds, ccmo)
+=========== RESULT ==========+
iteration: 500
population:        ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀F space⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
┌────────────────────────────────────────┐
2 │⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│
│⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│
│⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│
│⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│
│⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│
│⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│
│⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│
f₂   │⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│
│⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│
│⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│
│⠀⠀⠈⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│
│⠀⠀⠀⠀⠈⠑⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│
│⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│
│⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠢⠤⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│
0 │⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠒⠒⠒⠄⠤⠤⢠⢀⣀⢀⣀⠀⣀⣀⡀⣀⡀│
└────────────────────────────────────────┘
⠀0⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀1⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀f₁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
non-dominated solution(s):
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀F space⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
┌────────────────────────────────────────┐
2 │⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│
│⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│
│⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│
│⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│
│⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│
│⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│
│⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│
f₂   │⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│
│⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│
│⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│
│⠀⠀⠈⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│
│⠀⠀⠀⠀⠈⠑⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│
│⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│
│⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠢⠤⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│
0 │⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠒⠒⠒⠄⠤⠤⢠⢀⣀⢀⣀⠀⣀⣀⡀⣀⡀│
└────────────────────────────────────────┘
⠀0⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀1⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀f₁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
f calls: 50000
feasibles: 100 / 100 in final population
total time: 7.0616 s
stop reason: Maximum number of iterations exceeded.
+============================+
source