"""
Multiobjective Multifactorial Evolutionary Algorithm With Online Transfer Parameter Estimation (MO-MFEA-II)
This module implements MOMFEA for multi-objective multi-task optimization with knowledge transfer.
References
----------
[1] Bali, Kavitesh Kumar, et al. "Cognizant multitasking in multiobjective multifactorial evolution: MO-MFEA-II." IEEE transactions on cybernetics 51.4 (2020): 1784-1796.
Notes
-----
Author: Jiangtao Shen
Email: j.shen5@exeter.ac.uk
Date: 2025.12.16
Version: 1.0
"""
import time
from tqdm import tqdm
from scipy.stats import norm
from scipy.optimize import minimize_scalar
from ddmtolab.Algorithms.STMO.NSGA_II import nsga2_sort
from ddmtolab.Methods.Algo_Methods.algo_utils import *
[docs]
class MO_MFEA_II:
"""
Multiobjective Multifactorial Evolutionary Algorithm With Online Transfer Parameter Estimation.
Attributes
----------
algorithm_information : dict
Dictionary containing algorithm capabilities and requirements
"""
algorithm_information = {
'n_tasks': '[2, K]',
'dims': 'unequal',
'objs': 'unequal',
'n_objs': '[2, M]',
'cons': 'unequal',
'n_cons': '[0, C]',
'expensive': 'False',
'knowledge_transfer': 'True',
'n': 'equal',
'max_nfes': 'equal'
}
@classmethod
def get_algorithm_information(cls, print_info=True):
return get_algorithm_information(cls, print_info)
[docs]
def __init__(self, problem, n=None, max_nfes=None, save_data=True, save_path='./Data',
name='MO-MFEA-II', disable_tqdm=True):
"""
Initialize MO-MFEA-II.
Parameters
----------
problem : MTOP
Multi-task optimization problem instance
n : int, optional
Population size per task (default: 100)
max_nfes : int, optional
Maximum number of function evaluations per task (default: 10000)
save_data : bool, optional
Whether to save optimization data (default: True)
save_path : str, optional
Path to save results (default: './Data')
name : str, optional
Name for the experiment (default: 'momfea_test')
disable_tqdm : bool, optional
Whether to disable progress bar (default: True)
"""
self.problem = problem
self.n = n if n is not None else 100
self.max_nfes = max_nfes if max_nfes is not None else 10000
self.save_data = save_data
self.save_path = save_path
self.name = name
self.disable_tqdm = disable_tqdm
[docs]
def optimize(self):
"""
Execute the MO-MFEA-II algorithm.
Returns
-------
Results
Optimization results containing decision variables, objectives, constraints, and runtime
"""
start_time = time.time()
problem = self.problem
n = self.n
nt = problem.n_tasks
dims = problem.dims
max_nfes_per_task = par_list(self.max_nfes, nt)
max_nfes = self.max_nfes * nt
# Initialize population and evaluate for each task
decs = initialization(problem, n)
objs, cons = evaluation(problem, decs)
nfes = n * nt
# Skill factor indicates which task each individual belongs to
pop_sfs = [np.full((n, 1), fill_value=i) for i in range(nt)]
all_decs, all_objs, all_cons = init_history(decs, objs, cons)
pbar = tqdm(total=max_nfes, initial=nfes, desc=f"{self.name}", disable=self.disable_tqdm)
while nfes < max_nfes:
# Perform NSGA-II sorting to get dominance ranks for each task
rank = []
for i in range(nt):
rank_i, _, _ = nsga2_sort(objs[i], cons[i])
rank.append(rank_i.copy())
# Select parents using binary tournament selection
pop_decs = []
pop_objs = []
pop_cons = []
for i in range(nt):
matingpool_i = tournament_selection(2, n, rank[i])
pop_decs.append(decs[i][matingpool_i, :])
pop_objs.append(objs[i][matingpool_i, :])
pop_cons.append(cons[i][matingpool_i, :])
# Transform populations to unified search space for knowledge transfer
pop_decs, pop_objs, pop_cons = space_transfer(problem, pop_decs, pop_objs, pop_cons, type='uni')
# Learn RMP matrix online
rmpMatrix = learnRMP(pop_decs, dims)
# Merge populations from all tasks into single arrays
pop_decs, pop_objs, pop_cons, pop_sfs = vstack_groups(pop_decs, pop_objs, pop_cons, pop_sfs)
off_decs = np.zeros_like(pop_decs)
off_objs = np.zeros_like(pop_objs)
off_cons = np.zeros_like(pop_cons)
off_sfs = np.zeros_like(pop_sfs)
# Randomly pair individuals for assortative mating
shuffled_index = np.random.permutation(pop_decs.shape[0])
for i in range(0, len(shuffled_index), 2):
p1 = shuffled_index[i]
p2 = shuffled_index[i + 1]
sf1 = pop_sfs[p1].item()
sf2 = pop_sfs[p2].item()
rmp_value = rmpMatrix[sf1, sf2]
# Cross-task transfer: crossover if same task or rmp condition met
if sf1 == sf2 or np.random.rand() < rmp_value:
off_dec1, off_dec2 = crossover(pop_decs[p1, :], pop_decs[p2, :], mu=2)
off_decs[i, :] = off_dec1
off_decs[i + 1, :] = off_dec2
off_sfs[i] = np.random.choice([sf1, sf2])
off_sfs[i + 1] = sf1 if off_sfs[i] == sf2 else sf2
else:
# No transfer: randomly pick individuals from same task for crossover
for x, p in enumerate([p1, p2]):
sf = pop_sfs[p].item()
# Find all individuals with the same skill factor
same_sf_indices = np.where(pop_sfs.flatten() == sf)[0]
# Remove current individual from candidates
same_sf_indices = same_sf_indices[same_sf_indices != p]
# Randomly select another individual from the same task
idx = np.random.choice(same_sf_indices)
# Crossover with the selected individual
off_dec_curr, _ = crossover(pop_decs[p, :], pop_decs[idx, :], mu=2)
off_dec_curr = mutation(off_dec_curr, mu=5)
off_decs[i + x, :] = off_dec_curr
# Inherit skill factor from parent
off_sfs[i + x] = sf
# Trim to task dimensionality and evaluate offspring
task_idx1 = off_sfs[i].item()
task_idx2 = off_sfs[i + 1].item()
off_dec1_trimmed = off_decs[i, :dims[task_idx1]]
off_dec2_trimmed = off_decs[i + 1, :dims[task_idx2]]
off_objs[i, :], off_cons[i, :] = (
x[0] for x in evaluation_single(problem, off_dec1_trimmed, task_idx1, unified=True, fill_value=0.)
)
off_objs[i + 1, :], off_cons[i + 1, :] = (
x[0] for x in evaluation_single(problem, off_dec2_trimmed, task_idx2, unified=True, fill_value=0.)
)
# Merge parents and offspring populations
pop_decs, pop_objs, pop_cons, pop_sfs = vstack_groups((pop_decs, off_decs), (pop_objs, off_objs),
(pop_cons, off_cons), (pop_sfs, off_sfs))
# Environmental selection: keep best n individuals per task
pop_decs, objs, cons, pop_sfs = momfea_selection(pop_decs, pop_objs, pop_cons, pop_sfs, n, nt)
# Transform back to native search space
decs, objs, cons = space_transfer(problem, pop_decs, objs, cons, type='real')
nfes += n * nt
pbar.update(n * nt)
append_history(all_decs, decs, all_objs, objs, all_cons, cons)
pbar.close()
runtime = time.time() - start_time
# Save results
results = build_save_results(all_decs=all_decs, all_objs=all_objs, runtime=runtime, max_nfes=max_nfes_per_task,
all_cons=all_cons, bounds=problem.bounds, save_path=self.save_path,
filename=self.name, save_data=self.save_data)
return results
def momfea_selection(all_decs, all_objs, all_cons, all_sfs, n, nt):
"""
Environmental selection for MOMFEA using NSGA-II criteria.
Parameters
----------
all_decs : np.ndarray
Decision variable matrix of the combined population of shape (n_total, d_max)
all_objs : np.ndarray
Objective value matrix corresponding to all_decs of shape (n_total, n_obj)
all_cons : np.ndarray
Constraint value matrix corresponding to all_decs of shape (n_total, n_con)
all_sfs : np.ndarray
Skill factor array indicating task assignment for each individual of shape (n_total, 1)
n : int
Number of individuals to select per task (population size per task)
nt : int
Number of tasks in the multi-task optimization problem
Returns
-------
pop_decs : list[np.ndarray]
Selected decision variable matrices for each task, length nt, each of shape (n, d_max)
pop_objs : list[np.ndarray]
Selected objective value matrices for each task, length nt, each of shape (n, n_obj)
pop_cons : list[np.ndarray]
Selected constraint matrices for each task, length nt, each of shape (n, n_con)
pop_sfs : list[np.ndarray]
Selected skill factor arrays for each task, length nt, each of shape (n, 1)
Notes
-----
Selection is performed independently for each task using NSGA-II sorting based on
non-dominated rank and crowding distance. The top-n individuals with smallest rank
values are retained for each task.
"""
pop_decs, pop_objs, pop_cons, pop_sfs = [], [], [], []
# Process each task separately
for i in range(nt):
# Extract all individuals belonging to task i
indices = np.where(all_sfs.flatten() == i)[0]
current_decs, current_objs, current_cons, current_sfs = select_by_index(
indices, all_decs, all_objs, all_cons, all_sfs
)
# NSGA-II sorting: rank based on non-dominated sorting and crowding distance
rank, _, _ = nsga2_sort(current_objs, current_cons)
# Select top-n individuals with smallest rank values
indices_select = np.argsort(rank)[:n]
selected_decs, selected_objs, selected_cons, selected_sfs = select_by_index(
indices_select, current_decs, current_objs, current_cons, current_sfs
)
# Store selected individuals for this task
pop_decs, pop_objs, pop_cons, pop_sfs = append_history(
pop_decs, selected_decs,
pop_objs, selected_objs,
pop_cons, selected_cons,
pop_sfs, selected_sfs
)
return pop_decs, pop_objs, pop_cons, pop_sfs
def learnRMP(subpops, vars):
"""
Learn the relationship matrix (RMP) between multiple tasks.
Parameters
----------
subpops : list
List of subpopulations, either as numpy arrays or dicts with 'data' key.
Each subpopulation contains solution variables for one task.
vars : list or array-like
Dimensionality (number of variables) for each task.
Returns
-------
rmpMatrix : np.ndarray
Symmetric relationship matrix of shape (numtasks, numtasks).
rmpMatrix[i,j] indicates the similarity between task i and task j.
Diagonal elements are 1.0, off-diagonal values are in [0, 1].
Notes
-----
The RMP (Relationship Matrix of Problems) quantifies inter-task similarities
by computing probabilistic overlap between learned Gaussian models.
Higher RMP values indicate stronger task relationships, enabling better
knowledge transfer in multi-task optimization.
"""
# Convert to dict format if needed
if isinstance(subpops, list) and isinstance(subpops[0], np.ndarray):
subpops = [{'data': pop} for pop in subpops]
numtasks = len(subpops)
maxDim = max(vars)
rmpMatrix = np.eye(numtasks)
# Add noise and build probabilistic models
probmodel = []
for i in range(numtasks):
model = {}
model['nsamples'] = subpops[i]['data'].shape[0]
nrandsamples = int(np.floor(0.1 * model['nsamples']))
# Create random samples with maxDim columns
randMat = np.random.rand(nrandsamples, maxDim)
# Pad subpops data to maxDim with ZEROS (to match MATLAB behavior)
current_data = subpops[i]['data']
padded_data = current_data
# Combine original data with random samples
combined_data = np.vstack([padded_data, randMat])
model['mean'] = np.mean(combined_data, axis=0)
model['stdev'] = np.std(combined_data, axis=0, ddof=1)
probmodel.append(model)
# Compute pairwise similarities
for i in range(numtasks):
for j in range(i + 1, numtasks):
popdata = [
{'probmatrix': np.ones((probmodel[i]['nsamples'], 2))},
{'probmatrix': np.ones((probmodel[j]['nsamples'], 2))}
]
Dim = min(vars[i], vars[j])
# Compute probabilities for population i
for k in range(probmodel[i]['nsamples']):
for l in range(Dim):
popdata[0]['probmatrix'][k, 0] *= norm.pdf(
subpops[i]['data'][k, l],
probmodel[i]['mean'][l],
probmodel[i]['stdev'][l]
)
popdata[0]['probmatrix'][k, 1] *= norm.pdf(
subpops[i]['data'][k, l],
probmodel[j]['mean'][l],
probmodel[j]['stdev'][l]
)
# Compute probabilities for population j
for k in range(probmodel[j]['nsamples']):
for l in range(Dim):
popdata[1]['probmatrix'][k, 0] *= norm.pdf(
subpops[j]['data'][k, l],
probmodel[i]['mean'][l],
probmodel[i]['stdev'][l]
)
popdata[1]['probmatrix'][k, 1] *= norm.pdf(
subpops[j]['data'][k, l],
probmodel[j]['mean'][l],
probmodel[j]['stdev'][l]
)
# Optimize to find RMP value
result = minimize_scalar(
lambda x: loglik(x, popdata, numtasks),
bounds=(0, 1),
method='bounded'
)
rmp_value = max(0, result.x + np.random.normal(0, 0.01))
rmp_value = min(rmp_value, 1)
rmpMatrix[i, j] = rmp_value
rmpMatrix[j, i] = rmp_value
return rmpMatrix
def loglik(rmp, popdata, ntasks):
"""
Compute the negative log-likelihood for a given RMP value.
Parameters
----------
rmp : float
Relationship matrix parameter value in [0, 1] to evaluate.
Represents the strength of inter-task relationship.
popdata : list
List of dicts, each containing 'probmatrix' of shape (nsamples, 2).
probmatrix[:, 0] are probabilities under own task model,
probmatrix[:, 1] are probabilities under other task model.
ntasks : int
Total number of tasks in the multi-task problem.
Returns
-------
f : float
Negative log-likelihood value. Lower values indicate better fit
of the RMP parameter to the observed probability distributions.
Notes
-----
This function is used as the objective in optimization to find the optimal
RMP value that maximizes the likelihood of observing the population data
under a mixture model with inter-task knowledge transfer.
"""
f = 0
# Make a copy to avoid modifying the original
popdata_copy = [{'probmatrix': pop['probmatrix'].copy()} for pop in popdata]
for i in range(2):
for j in range(2):
if i == j:
popdata_copy[i]['probmatrix'][:, j] *= (1 - (0.5 * (ntasks - 1) * rmp / ntasks))
else:
popdata_copy[i]['probmatrix'][:, j] *= 0.5 * (ntasks - 1) * rmp / ntasks
# Compute negative log-likelihood
f += np.sum(-np.log(np.sum(popdata_copy[i]['probmatrix'], axis=1)))
return f