BilevelHeuristics.jl

This package implements different heuristic and metaheuristic algorithms for Bilevel Optimization (BO).

BilevelHeuristics.jl is still in early development so please send feedback or open issues.

Installation

Open the Julia REPL (Julia 1.6 or later) and press ] to open the Pkg prompt. To add this package, use the add command:

Type ]

pkg> add BilevelHeuristics

Or, equivalently, via the Pkg API:

julia> import Pkg; Pkg.add("BilevelHeuristics")

Algorithms

Current implemented Bilevel Metaheuristics include:

• BCA: Bilevel Centers Algorithm
• QBCA: BCA with a lower-level Quasi-Newton optimization method.
• QBCA2: Improved QBCA (implements conditions to avoid pseudo-feasible solutions)
• SABO: Surrogate-assisted Bilevel Optimization.
• SMS-MOBO: S-metic-selection-based Multi-objective Bilevel Optimization.
• BLEMO: Bilevel Evolutionary Multi-objective Optimization

Example

The following example illustrates the usage of BilevelHeuristics.jl. Here, BCA is used, but this example works for the other optimizers.

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
+============================+

x, y = minimizer(res) # upper (x) and lower (y) level decision vectors
Fmin, fmin = minimum(res) # upper and lower objective values.

TO DO

• Parallelize the lower-level optimization procedures.