Docs for MOO

docs/src/optimization_packages/blackboxoptim.md

@@ -67,3 +67,21 @@ prob = Optimization.OptimizationProblem(f, x0, p, lb = [-1.0, -1.0], ub = [1.0,
 sol = solve(prob, BBO_adaptive_de_rand_1_bin_radiuslimited(), maxiters = 100000,
     maxtime = 1000.0)
 ```
+
+## Multi-objective optimization
+The optimizer for Multi-Objective Optimization is `BBO_borg_moea()`. Your fitness function should return a tuple of the objective values and you should indicate the fitness scheme to be (typically) Pareto fitness and specify the number of objectives. Otherwise, the use is similar, here is an example:
+
+```@example MOO-BBO
+using OptimizationBBO, Optimization, BlackBoxOptim
+using SciMLBase: MultiObjectiveOptimizationFunction
+u0 = [0.25, 0.25]
+opt = OptimizationBBO.BBO_borg_moea()
+function multi_obj_func_2(x, p)
+    f1 = (1.0 - x[1])^2 + 100.0 * (x[2] - x[1]^2)^2  # Rosenbrock function
+    f2 = -20.0 * exp(-0.2 * sqrt(0.5 * (x[1]^2 + x[2]^2))) - exp(0.5 * (cos(2π * x[1]) + cos(2π * x[2]))) + exp(1) + 20.0  # Ackley function
+    return (f1, f2)
+end
+mof = MultiObjectiveOptimizationFunction(multi_obj_func_2)
+prob = Optimization.OptimizationProblem(mof_2, u0; lb = [0.0, 0.0], ub = [2.0, 2.0])
+sol = solve(prob_2, opt, NumDimensions=2, FitnessScheme=ParetoFitnessScheme{2}(is_minimizing=true))
+```

docs/src/optimization_packages/evolutionary.md

@@ -41,3 +41,23 @@ f = OptimizationFunction(rosenbrock)
 prob = Optimization.OptimizationProblem(f, x0, p, lb = [-1.0, -1.0], ub = [1.0, 1.0])
 sol = solve(prob, Evolutionary.CMAES(μ = 40, λ = 100))
 ```
+
+## Multi-objective optimization
+The Rosenbrock and Ackley functions can be optimized using the `Evolutionary.NSGA2()` as follows:
+
+```@example MOO-Evolutionary
+using Optimization, OptimizationEvolutionary
+function func(x, p=nothing)::Vector{Float64}
+  f1 = (1.0 - x[1])^2 + 100.0 * (x[2] - x[1]^2)^2  # Rosenbrock function
+  f2 = -20.0 * exp(-0.2 * sqrt(0.5 * (x[1]^2 + x[2]^2))) - exp(0.5 * (cos(2π * x[1]) + cos(2π * x[2]))) + exp(1) + 20.0  # Ackley function
+  return [f1, f2]
+end
+initial_guess = [1.0, 1.0]
+function gradient_multi_objective(x, p=nothing)
+    ForwardDiff.jacobian(func, x)
+end
+obj_func = MultiObjectiveOptimizationFunction(func, jac=gradient_multi_objective)
+algorithm = OptimizationEvolutionary.NSGA2()
+problem = OptimizationProblem(obj_func, initial_guess)
+result = solve(problem, algorithm)
+```

docs/src/optimization_packages/metaheuristics.md

@@ -70,3 +70,54 @@ sol = solve(prob, ECA(), use_initial = true, maxiters = 100000, maxtime = 1000.0
 ### With Constraint Equations
 
 While `Metaheuristics.jl` supports such constraints, `Optimization.jl` currently does not relay these constraints.
+
+
+## Multi-objective optimization
+The zdt1 functions can be optimized using the `Metaheuristics.jl` as follows:
+
+```@example MOO-Evolutionary
+using Optimization, OptimizationEvolutionary
+function zdt1(x)
+    f1 = x[1]
+    g = 1 + 9 * mean(x[2:end])
+    h = 1 - sqrt(f1 / g)
+    f2 = g * h
+    # In this example, we have no constraints
+    gx = [0.0]  # Inequality constraints (not used)
+    hx = [0.0]  # Equality constraints (not used)
+    return [f1, f2], gx, hx
+end
+multi_obj_fun = MultiObjectiveOptimizationFunction((x, p) -> zdt1(x))
+
+# Define the problem bounds
+lower_bounds = [0.0, 0.0, 0.0]
+upper_bounds = [1.0, 1.0, 1.0]
+
+# Define the initial guess
+initial_guess = [0.5, 0.5, 0.5]
+
+# Create the optimization problem
+prob = OptimizationProblem(multi_obj_fun, initial_guess; lb = lower_bounds, ub = upper_bounds)
+
+nobjectives = 2
+npartitions = 100
+
+# reference points (Das and Dennis's method)
+weights = gen_ref_dirs(nobjectives, npartitions)
+
+# Choose the algorithm as required.
+alg1 = NSGA2()
+alg2 = NSGA3()
+alg3 = SPEA2()
+alg4 = CCMO(NSGA2(N=100, p_m=0.001))
+alg5 = MOEAD_DE(weights, options=Options(debug=false, iterations = 250))
+alg6 = SMS_EMOA()
+
+# Solve the problem
+sol1 = solve(prob, alg1; maxiters = 100, use_initial = true)
+sol2 = solve(prob, alg2; maxiters = 100, use_initial = true)
+sol3 = solve(prob, alg3; maxiters = 100, use_initial = true)
+sol4 = solve(prob, alg4)
+sol5 = solve(prob, alg5; maxiters = 100, use_initial = true)
+sol6 = solve(prob, alg6; maxiters = 100, use_initial = true)
+```