"""
Nondominated Sorting Genetic Algorithm II with Strengthened Dominance Relation (NSGA-II-SDR)
This module implements NSGA-II-SDR for multi-objective optimization problems.
References
----------
[1] Y. Tian, R. Cheng, X. Zhang, Y. Su, and Y. Jin. A strengthened dominance relation considering convergence and diversity for evolutionary many-objective optimization. IEEE Transactions on Evolutionary Computation, 2019, 23(2): 331-345.
Notes
-----
Author: Jiangtao Shen
Email: j.shen5@exeter.ac.uk
Date: 2025.12.14
Version: 1.0
"""
from tqdm import tqdm
import time
from scipy.spatial.distance import cdist
from ddmtolab.Methods.Algo_Methods.algo_utils import *
[docs]
class NSGA_II_SDR:
"""
Nondominated Sorting Genetic Algorithm II with Strengthened Dominance Relation for multi-objective optimization.
Attributes
----------
algorithm_information : dict
Dictionary containing algorithm capabilities and requirements
"""
algorithm_information = {
'n_tasks': '[1, K]',
'dims': 'unequal',
'objs': 'unequal',
'n_objs': '[2, M]',
'cons': 'unequal',
'n_cons': '[0, C]',
'expensive': 'False',
'knowledge_transfer': 'False',
'n': 'unequal',
'max_nfes': 'unequal'
}
@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, muc=20.0, mum=15.0, save_data=True, save_path='./Data',
name='NSGA-II-SDR', disable_tqdm=True):
"""
Initialize NSGA-II-SDR algorithm.
Parameters
----------
problem : MTOP
Multi-task optimization problem instance
n : int or List[int], optional
Population size per task (default: 100)
max_nfes : int or List[int], optional
Maximum number of function evaluations per task (default: 10000)
muc : float, optional
Distribution index for simulated binary crossover (SBX) (default: 20.0)
mum : float, optional
Distribution index for polynomial mutation (PM) (default: 15.0)
save_data : bool, optional
Whether to save optimization data (default: True)
save_path : str, optional
Path to save results (default: './TestData')
name : str, optional
Name for the experiment (default: 'NSGA-II-SDR_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.muc = muc
self.mum = mum
self.save_data = save_data
self.save_path = save_path
self.name = name
self.disable_tqdm = disable_tqdm
[docs]
def optimize(self):
"""
Execute the NSGA-II-SDR algorithm.
Returns
-------
Results
Optimization results containing decision variables, objectives, constraints, and runtime
"""
start_time = time.time()
problem = self.problem
nt = problem.n_tasks
n_per_task = par_list(self.n, nt)
max_nfes_per_task = par_list(self.max_nfes, nt)
# Initialize population and evaluate for each task
decs = initialization(problem, n_per_task)
objs, _ = evaluation(problem, decs)
nfes_per_task = n_per_task.copy()
all_decs, all_objs = init_history(decs, objs)
# Perform initial non-dominated sorting for each task using SDR
rank = []
for i in range(nt):
rank_i, _, _ = nsga2sdr_sort(objs[i])
rank.append(rank_i.copy())
pbar = tqdm(total=sum(max_nfes_per_task), initial=sum(n_per_task), desc=f"{self.name}",
disable=self.disable_tqdm)
while sum(nfes_per_task) < sum(max_nfes_per_task):
# Skip tasks that have exhausted their evaluation budget
active_tasks = [i for i in range(nt) if nfes_per_task[i] < max_nfes_per_task[i]]
if not active_tasks:
break
for i in active_tasks:
# Parent selection via binary tournament based on rank
matingpool = tournament_selection(2, n_per_task[i], rank[i])
# Generate offspring through crossover and mutation
off_decs = ga_generation(decs[i][matingpool, :], muc=self.muc, mum=self.mum)
off_objs, _ = evaluation_single(problem, off_decs, i)
# Merge parent and offspring populations
objs[i], decs[i] = vstack_groups((objs[i], off_objs), (decs[i], off_decs))
# Environmental selection: sort and keep best n individuals using SDR
rank[i], _, _ = nsga2sdr_sort(objs[i])
index = np.argsort(rank[i])[:n_per_task[i]]
objs[i], decs[i], rank[i] = select_by_index(index, objs[i], decs[i], rank[i])
nfes_per_task[i] += n_per_task[i]
pbar.update(n_per_task[i])
append_history(all_decs[i], decs[i], all_objs[i], objs[i])
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=nfes_per_task,
bounds=problem.bounds, save_path=self.save_path,
filename=self.name, save_data=self.save_data)
return results
def nsga2sdr_sort(objs):
"""
Sort solutions based on NSGA-II-SDR criteria using strengthened dominance relation and crowding distance.
Parameters
----------
objs : np.ndarray
Objective value matrix of shape (pop_size, n_obj)
Returns
-------
rank : np.ndarray
Ranking of each solution (0-based index after sorting) of shape (pop_size,).
rank[i] indicates the position of solution i in the sorted order
front_no : np.ndarray
Non-dominated front number of each solution of shape (pop_size,)
crowd_dis : np.ndarray
Crowding distance of each solution of shape (pop_size,)
Notes
-----
Solutions are sorted first by front number (ascending), then by crowding distance (descending).
Larger crowding distance values indicate better diversity preservation.
The strengthened dominance relation (SDR) is used for non-dominated sorting to improve
performance on many-objective optimization problems.
"""
pop_size = objs.shape[0]
# Perform non-dominated sorting using strengthened dominance relation
front_no, _ = nd_sort_sdr(objs, pop_size)
# Calculate crowding distance for diversity preservation
crowd_dis = crowding_distance(objs, front_no)
# Sort by front number (ascending), then by crowding distance (descending)
sorted_indices = np.lexsort((-crowd_dis, front_no))
# Create rank array: rank[i] gives the sorted position of solution i
rank = np.empty(pop_size, dtype=int)
rank[sorted_indices] = np.arange(pop_size)
return rank, front_no, crowd_dis
def nd_sort_sdr(pop_obj: np.ndarray, n_sort: int) -> Tuple[np.ndarray, int]:
"""
Do non-dominated sorting by strengthened dominance relation (SDR).
Parameters
----------
pop_obj : np.ndarray
Objective value matrix, shape (N, M)
n_sort : int
Number of solutions to sort
Returns
-------
front_no : np.ndarray
Non-dominated front number for each solution, shape (N,)
max_fno : int
Maximum front number assigned
"""
N = pop_obj.shape[0]
# Min-Max Normalization
obj_min = np.min(pop_obj, axis=0)
obj_max = np.max(pop_obj, axis=0)
obj_range = obj_max - obj_min
obj_range[obj_range == 0] = 1
pop_obj = (pop_obj - obj_min) / obj_range
# Calculate L1-norm (sum) of each solution for convergence measure
norm_p = np.sum(pop_obj, axis=1)
# Calculate cosine similarity for diversity measure
if N > 1:
cosine = 1 - cdist(pop_obj, pop_obj, metric='cosine')
np.fill_diagonal(cosine, 0)
else:
cosine = np.zeros((1, 1))
# Calculate angle (in radians) between solution vectors
angle = np.arccos(np.clip(cosine, -1, 1))
# Find minimum angle threshold for strengthened dominance
if N > 1:
# Get minimum angle for each solution
min_angle_per_sol = np.min(angle, axis=1)
# Get unique sorted minimum angles
unique_min_angles = np.sort(np.unique(min_angle_per_sol))
# Select the middle value (or the ceil(N/2)-th smallest unique value)
idx = min(np.ceil(50*N / 100).astype(int), len(unique_min_angles) - 1)
minA = unique_min_angles[idx]
else:
minA = np.pi / 4 # Default for single solution
# Calculate theta values for strengthened dominance relation
Theta = np.maximum(1, (angle / minA) ** 1)
# Build dominance matrix using strengthened dominance relation
# Solution i strengthened-dominates solution j if: norm_p[i] * Theta[i,j] < norm_p[j]
dominate = np.zeros((N, N), dtype=bool)
for i in range(N - 1):
for j in range(i + 1, N):
if norm_p[i] * Theta[i, j] < norm_p[j]:
dominate[i, j] = True
elif norm_p[j] * Theta[j, i] < norm_p[i]:
dominate[j, i] = True
# Non-dominated sorting based on strengthened dominance
front_no = np.full(N, np.inf)
max_fno = 0
while np.sum(front_no != np.inf) < min(n_sort, N):
max_fno += 1
# Find solutions not dominated by any others under SDR
current = ~np.any(dominate, axis=0) & (front_no == np.inf)
# Assign front number
front_no[current] = max_fno
# Remove current solutions from domination relationships
dominate[current, :] = False
return front_no, max_fno