案例一：复杂系统多目标协同优化

# 宝鸡文理学院计算机学院 ·《最优化方法》案例库代码规范
import numpy as np

def dominates(a, b):
    # Pareto 支配判据：全维不劣且至少一维严格优
    return np.all(a <= b) and np.any(a < b)

def fast_non_dominated_sort(F):
    """
    快速非支配排序：输入目标矩阵 F (N x M)
    返回分层前沿索引列表 [F1, F2, ...]
    """
    N = F.shape[0]
    S = [[] for _ in range(N)]
    n = np.zeros(N, dtype=int)      # 被支配计数 n_p
    fronts = [[]]
    for p in range(N):
        for q in range(N):
            if p != q and dominates(F[p], F[q]):
                S[p].append(q)
            elif p != q and dominates(F[q], F[p]):
                n[p] += 1
        if n[p] == 0:
            fronts[0].append(p)     # 第一层前沿 F1
    k = 0
    while fronts[k]:
        next_front = []
        for p in fronts[k]:
            for q in S[p]:
                n[q] -= 1
                if n[q] == 0:
                    next_front.append(q)
        k += 1
        fronts.append(next_front)
    return [f for f in fronts if len(f) > 0]

def crowding_distance(F, front):
    # 拥挤度距离：边界解取 inf，内部解累加归一化邻域间距
    l = len(front)
    d = np.zeros(l)
    for m in range(F.shape[1]):
        order = np.argsort(F[front, m])
        d[order[0]] = d[order[-1]] = np.inf
        span = F[front[order[-1]], m] - F[front[order[0]], m] + 1e-12
        for i in range(1, l - 1):
            d[order[i]] += (F[front[order[i + 1]], m] - F[front[order[i - 1]], m]) / span
    return d

def tournament_selection(rank, crowd):
    # 锦标赛选择：先比非支配层级，同层再比拥挤度距离
    a, b = np.random.randint(0, len(rank), 2)
    if (rank[a], -crowd[a]) <= (rank[b], -crowd[b]):
        return a
    return b