Multi-objective

MOEA/D-DE

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

Metaheuristics.MOEAD_DE — Type

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
moead_de = MOEAD_DE(weights, options=Options(debug=false, iterations = 250))

# optimize
status_moead = optimize(f, bounds, moead_de)

# show results
display(status_moead)

source

NSGA-II

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

Metaheuristics.NSGA2 — Type

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.NSGA3 — Type

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_EMOA — Type

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.SPEA2 — Type

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.CCMO — Type

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