Examples

This package provides different tools for optimization. Hence, this section gives different examples for using the implemented Metaheuristics.

Single-Objective Optimization

Firstly import this package

using Metaheuristics

Now, let us define the objective function to be minimized:

f(x) = 10length(x) + sum( x.^2 - 10cos.(2π*x) )
f (generic function with 1 method)

The search space (a.k.a. box-constraints) can be defined as follows:

bounds = boxconstraints(lb = -5ones(10), ub = 5ones(10))
BoxConstrainedSpace{Float64}([-5.0, -5.0, -5.0, -5.0, -5.0, -5.0, -5.0, -5.0, -5.0, -5.0], [5.0, 5.0, 5.0, 5.0, 5.0, 5.0, 5.0, 5.0, 5.0, 5.0], [10.0, 10.0, 10.0, 10.0, 10.0, 10.0, 10.0, 10.0, 10.0, 10.0], 10, true)
boxconstraints in a Matrix format.

You can also define the bounds using bounds = [-5ones(10) 5ones(10)]'; however this is not longer recommended.

It is possible to provide some information on the minimization problem. Let's provide the true optimum to stop the optimizer when a tolerance f_tol is satisfied.

information = Information(f_optimum = 0.0)
Information(0.0, Float64[])

Generic options or settings (e.g. budget limitation, tolerances, etc) can be provided as follows:

options = Options(f_calls_limit = 9000*10, f_tol = 1e-5, seed=1)
Options
=======
  rng:             Random.TaskLocalRNG()
  seed:            1
  x_tol:           1.0e-8
  f_tol:           1.0e-5
  g_tol:           0.0
  h_tol:           0.0
  debug:           false
  verbose:         false
  f_tol_rel:       2.220446049250313e-16
  time_limit:      Inf
  iterations:      0
  f_calls_limit:   90000.0
  store_convergence: false
  parallel_evaluation: false

Now, we can provide the Information and Options to the optimizer (ECA in this example).

algorithm = ECA(information = information, options = options)
Algorithm Parameters
====================
  ECA(η_max=2.0, K=7, N=0, N_init=0, p_exploit=0.95, p_bin=0.02, ε=0.0, adaptive=false, resize_population=false)

Optimization Result
===================
  Empty status.

Now, the optimization is performed as follows:

result = optimize(f, bounds, algorithm)
Optimization Result
===================
  Iteration:       540
  Minimum:         1.98992
  Minimizer:       [4.9425e-09, -2.63681e-09, 5.26807e-10, …, -2.05997e-09]
  Function calls:  37800
  Total time:      1.6553 s
  Stop reason:     Due to Convergence Termination criterion.

The minimum and minimizer:

minimum(result)
1.9899181141865938
minimizer(result)
10-element Vector{Float64}:
  4.942500864198351e-9
 -2.6368055210774052e-9
  5.268065838332036e-10
 -2.221590569601794e-9
  2.268687428475678e-9
 -0.9949586341971358
  1.364299749866016e-9
  0.994958642923819
  8.33735329614517e-10
 -2.0599681591713486e-9
Second run is faster in Julia

As you may know, the second run can be faster.

Constrained Optimization

It is common that optimization models include constraints that must be satisfied. For example: The Rosenbrock function constrained to a disk

Minimize:

\[{\displaystyle f(x,y)=(1-x)^{2}+100(y-x^{2})^{2}}\]

subject to:

\[{\displaystyle x^{2}+y^{2}\leq 2}\]

where $-2 \leq x,y \leq 2$.

In Metaheuristics.jl, a feasible solution is such that $g(x) \leq 0$ and $h(x) \approx 0$. Hence, in this example the constraint is given by $g(x) = x^2 + y^2 - 2 \leq 0$. Moreover, the equality and inequality constraints must be saved into Arrays.

Constraints handling

In this package, if the algorithm was not designed for constrained optimization, then solutions with the lower constraint violation sum will be preferred.

julia> using Metaheuristics
julia> function f(x) x,y = x[1], x[2] fx = (1-x)^2+100(y-x^2)^2 gx = [x^2 + y^2 - 2] # inequality constraints hx = [0.0] # equality constraints # order is important return fx, gx, hx endf (generic function with 1 method)
julia> bounds = [-2.0 -2; 2 2]2×2 Matrix{Float64}: -2.0 -2.0 2.0 2.0
julia> optimize(f, bounds, ECA(N=30, K=3))Optimization Result =================== Iteration: 667 Minimum: 0.00184604 Minimizer: [0.957093, 0.915803] Function calls: 20010 Feasibles: 30 / 30 in final population Total time: 1.2678 s Stop reason: Maximum objective function calls exceeded.

Multiobjective Optimization

To implement a multiobjective optimization problem and solve it, you can proceed as usual. Here, you need to provide constraints if they exist, otherwise put gx = [0.0]; hx = [0.0]; to indicate an unconstrained multiobjective problem.

julia> using UnicodePlots # to visualize in console (optional)
julia> using Metaheuristics
julia> function f(x) # objective functions v = 1.0 + sum(x .^ 2) fx1 = x[1] * v fx2 = (1 - sqrt(x[1])) * v fx = [fx1, fx2] # constraints gx = [0.0] # inequality constraints hx = [0.0] # equality constraints # order is important return fx, gx, hx endf (generic function with 1 method)
julia> bounds = [zeros(30) ones(30)]';
julia> optimize(f, bounds, NSGA2())Optimization Result =================== Iteration: 251 Non-dominated: 100 / 100 ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀F space⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ┌────────────────────────────────────────┐ 2 ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ f₂ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠘⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠈⠓⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠁⠓⠒⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠠⠠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ 0 ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠤⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ └────────────────────────────────────────┘0⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀3⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀f₁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ Function calls: 50100 Feasibles: 100 / 100 in final population Total time: 3.3161 s Stop reason: Maximum objective function calls exceeded.

Bilevel Optimization

Bilevel optimization problems can be solved by using the package BilevelHeuristics.jl which extends Metaheuristics.jl for handling those hierarchical problems.

Defining objective functions corresponding to the BO problem.

Upper level (leader problem):

using BilevelHeuristics

F(x, y) = sum(x.^2) + sum(y.^2)
bounds_ul = [-ones(5) ones(5)] 

Lower level (follower problem):

f(x, y) = sum((x - y).^2) + y[1]^2
bounds_ll = [-ones(5) ones(5)];

Approximate solution:

res = optimize(F, f, bounds_ul, bounds_ll, BCA())

Output:

+=========== RESULT ==========+
  iteration: 108
    minimum: 
          F: 4.03387e-10
          f: 2.94824e-10
  minimizer: 
          x: [-1.1460768817533927e-5, 7.231706879604178e-6, 3.818596951258517e-6, 2.294324313691869e-6, 1.8770952450067828e-6]
          y: [1.998748659975197e-6, 9.479307908087866e-6, 6.180041276047425e-6, -7.642051857319683e-6, 2.434166021682429e-6]
    F calls: 2503
    f calls: 5062617
    Message: Stopped due UL function evaluations limitations. 
 total time: 26.8142 s
+============================+

See BilevelHeuristics documentation for more information.

Decision-Making

Although Metaheuristics is focused on the optimization part, some decision-making algorithms are available in this package (see Multi-Criteria Decision-Making).

The following example shows how to perform a posteriori decision-making.

julia> # load the problem
julia> f, bounds, pf = Metaheuristics.TestProblems.ZDT1();

julia> # perform multi-objective optimization
julia> res = optimize(f, bounds, NSGA2());

julia> # user preferences
julia> w = [0.5, 0.5];

julia> # set the decision-making algorithm
julia> dm_method = CompromiseProgramming(Tchebysheff())

julia> # find the best decision
julia> sol = best_alternative(res, w, dm_method)
(f = [0.38493217206706115, 0.38037042164979956], g = [0.0], h = [0.0], x = [3.849e-01, 7.731e-06, …, 2.362e-07])

Providing Initial Solutions

Sometimes you may need to use the starter solutions you need before the optimization process begins, well, this example illustrates how to do it.


julia> f(x) = abs(x[1]) + x[2] + x[3]^2 # objective functionf (generic function with 1 method)
julia> algo = ECA(N = 61); # optimizer
julia> # one solution can be provided x0 = [0.5, 0.5, 0.5];
julia> set_user_solutions!(algo, x0, f);
julia> # or multiple solutions can be given X0 = rand(30, 3); # 30 solutions with dim 3
julia> set_user_solutions!(algo, X0, f);
julia> optimize(f, [0 0 0; 1 1 1.0], algo)Optimization Result =================== Iteration: 251 Minimum: 5.8527e-47 Minimizer: [1.6089e-47, 3.31929e-47, 3.04058e-24] Function calls: 15280 Total time: 0.1482 s Stop reason: Due to Convergence Termination criterion.

Batch Evaluation

Evaluating multiple solutions at the same time can reduce computational time. To do that, define your function on an input N x D matrix and function values into matrices with outcomes in rows for all N solutions. Also, you need to put parallel_evaluation=true in the Options to indicate that your f is prepared for parallel evaluations.

f(X) = begin
    fx = sum(X.^2, dims=2)       # objective function ∑x²
    gx = sum(X.^2, dims=2) .-0.5 # inequality constraints ∑x² ≤ 0.5
    hx = zeros(0,0)              # equality constraints
    fx, gx, hx
end

options = Options(parallel_evaluation=true)

res = optimize(f, [-10ones(5) 10ones(5)], ECA(options=options))

See Parallelization tutorial for more details.

Modifying an Existing Metaheuristic

You may need to modify one of the implemented metaheuristics to improve the algorithm performance or test new mechanisms. This example illustrates how to do it.

Modifying algorithms could break stuff

Be cautious when modifying a metaheuristic due to those changes will overwrite the default method for that metaheuristic.

Let's assume that we want to modify the stop criteria for ECA. See Contributing for more details.

using Metaheuristics
import LinearAlgebra: norm

# overwrite method
function Metaheuristics.stop_criteria!(
        status,
        parameters::ECA, # It is important to indicate the modified Metaheuristic 
        problem,
        information,
        options,
        args...;
        kargs...
    )

    if status.stop
        # nothing to do
        return
    end

    # Diversity-based stop criteria

    x_mean = zeros(length(status.population[1].x))
    for sol in status.population
        x_mean += sol.x
    end
    x_mean /= length(status.population)
    
    distances_mean = sum(sol -> norm( x_mean - sol.x ), status.population)
    distances_mean /= length(status.population)

    # stop when solutions are close enough to the geometrical center
    new_stop_condition = distances_mean <= 1e-3

    status.stop = new_stop_condition

    # (optional and not recommended) print when this criterium is met
    if status.stop
        @info "Diversity-based stop criterium"
        @show distances_mean
    end


    return
end

f, bounds, opt = Metaheuristics.TestProblems.get_problem(:sphere);
optimize(f, bounds, ECA())
Metaheuristics.RestartType
Restart(optimizer, every=100)

Resets the optimizer every specified number of iterations (100 by default).

Example

julia> f, bounds, _ = Metaheuristics.TestProblems.rastrigin();

julia> optimize(f, bounds, Restart(ECA(), every=200))

Customization

The restart condition can be updated by overloading the restart_condition method:

function Metaheuristics.restart_condition(status, restart::Restart, information, options)
    st.iteration % params.every == 0
end
source