Multi-Objective Optimization#
Solve multi-objective optimization problems with trade-offs between objectives, using saealib.
minimize() uses default components based on the number of objectives.
Use Optimizer.set_*() to configure the algorithm, surrogate model, or evaluation strategy individually.
This page covers what’s specific to problems with two or more objectives: choosing a comparator and extracting the Pareto front.
What you’ll be able to do
By the end of this page, you’ll be able to choose a Comparator for a multi-objective problem and extract the resulting Pareto front.
For the basic single-objective procedure, use Single-objective optimization; if you need a custom Comparator, see Custom components.
This page runs a multi-objective Problem, selects a Comparator, and extracts the Pareto front.
Set up the problem#
When multiple objective functions are in a trade-off relationship, there exist solutions where improving one worsens another.
Under this relationship, the set of solutions not dominated by any other solution in every objective is called the Pareto front.
Here, as an example, we minimize the ZDT1 function built into saealib.
from saealib.benchmarks import zdt1
problem = zdt1(n_var=10)
zdt1 is a Problem instance returning a two-objective benchmark problem with a convex Pareto front.
Run minimize#
Passing a Problem instance directly carries over the number of objectives from it.
from saealib import minimize
result = minimize(problem, max_fe=2000, seed=0)
print(result.x.shape) # (n_pareto, dim)
print(result.f.shape) # (n_pareto, n_obj)
Where result.x/result.f were a single point in the single-objective case, in the multi-objective case they become multiple solutions forming the Pareto front.
Configure an Optimizer#
With Optimizer(problem), you can configure each component independently.
set_*() returns the same Optimizer for chaining, and run() or iterate() executes the optimization.
from saealib import Optimizer, Termination, max_fe
optimizer = Optimizer(problem, seed=0).set_termination(Termination(max_fe(2000)))
ctx = optimizer.run()
pareto_f = ctx.pareto_archive.get_array("f")
Unconfigured components resolve to their defaults. Set multi-objective ranking by choosing one of the following Comparators.
Choose a comparator#
In multi-objective optimization, Comparator decides the relative superiority between candidate solutions.
If Problem’s comparator argument is omitted, one is auto-selected based on the number of objectives (SingleObjectiveComparator when n_obj == 1, NSGA2Comparator when n_obj > 1).
Class |
Behavior |
|---|---|
|
Diversity maintenance via non-dominated sorting and crowding distance (default) |
|
Fitness based on strength of dominance and neighborhood density |
|
Superiority judged by hypervolume contribution |
|
Superiority judged by epsilon-dominance |
|
Diversity maintenance via reference points. Requires |
|
Concentrates solutions near specified reference points. Requires |
comparator can be swapped as an attribute of the Problem instance.
from saealib.comparators import HypervolumeComparator
problem.comparator = HypervolumeComparator()
result = minimize(problem, max_fe=2000, seed=0)
Extract the Pareto front#
After running, result.ctx.pareto_archive holds the final Pareto front.
pareto_x = result.ctx.pareto_archive.get_array("x")
pareto_f = result.ctx.pareto_archive.get_array("f")
To compute the Pareto front from an arbitrary array of objective values, you can use non_dominated_sort directly.
from saealib.comparators import non_dominated_sort
archive_f = result.ctx.archive.get_array("f")
ranks, fronts = non_dominated_sort(archive_f, direction=problem.direction)
front0_f = archive_f[fronts[0]] # first non-dominated front