Scipy优化中多重线性约束的正确实现与性能优化

numbers = [1, 2, 3]
funcs = []
for n in numbers:
    funcs.append(lambda: n) # n在此处并未立即绑定
for func in funcs:
    print(func())

3
3
3

# 示例：错误的约束定义方式
cons = []
groups = [[0, 1, 2, 3], [4, 5], [6, 7, 8, 9]]
z_group = [0.25, 0.55, 0.2]

for idx, select in enumerate(groups):
    # 此处的idx和select存在延迟绑定问题
    cons.append({'type': 'eq', 'fun': lambda x: z_group[idx] - x[select].sum()})

def create_group_constraint(idx, select_indices, target_value):
    """
    创建一个用于特定组求和约束的内部函数。
    idx: 组的索引
    select_indices: 该组包含的变量索引
    target_value: 该组变量之和的目标值
    """
    def inner_constraint(x):
        return target_value - x[select_indices].sum()
    return inner_constraint

# 应用到约束列表中
cons = []
groups = [[0, 1, 2, 3], [4, 5], [6, 7, 8, 9]]
z_group = [0.25, 0.55, 0.2]

for idx, select in enumerate(groups):
    cons.append({'type': 'eq', 'fun': create_group_constraint(idx, select, z_group[idx])})

cons = []
groups = [[0, 1, 2, 3], [4, 5], [6, 7, 8, 9]]
z_group = [0.25, 0.55, 0.2]

for idx, select in enumerate(groups):
    # idx=idx 和 select=select 会在每次循环迭代时绑定当前值
    cons.append({'type': 'eq', 'fun': lambda x, current_idx=idx, current_select=select: z_group[current_idx] - x[current_select].sum()})

import numpy as np
from scipy.optimize import minimize, LinearConstraint, Bounds

utility_vector = np.array([0.10, 0.08, 0.05, 0.075, 0.32, 
                           0.21, 0.18, 0.05, 0.03, 0.12])
x0 = np.zeros((10,)) 
groups = [[0, 1, 2, 3], [4, 5], [6, 7, 8, 9]]
z_group = [0.25, 0.55, 0.2]

def opt_func(x, u, target):
    utility = (x * u).sum()
    return (utility - target)**2

n_variables = len(x0)

# 1. 定义总和约束：x.sum() = 1.0
# 对应 A @ x = 1.0，其中 A 是一个全为1的行向量
A_total_sum = np.ones((1, n_variables))
lb_total_sum = 1.0
ub_total_sum = 1.0
total_sum_constraint = LinearConstraint(A_total_sum, lb_total_sum, ub_total_sum)

# 2. 定义分组和约束：x[selection].sum() = Z_i
# 这需要构建一个矩阵 A_group_sum，其中每行对应一个组的约束
# 例如，对于 groups[0] = [0, 1, 2, 3]，对应的行在索引0,1,2,3处为1，其余为0
A_group_sum = np.zeros((len(groups), n_variables))
lb_group_sum = np.array(z_group)
ub_group_sum = np.array(z_group)

for idx, select in enumerate(groups):
    A_group_sum[idx, select] = 1

group_sum_constraint = LinearConstraint(A_group_sum, lb_group_sum, ub_group_sum)

# 3. 定义变量边界：x >= 0
# Scipy的minimize函数通常通过Bounds参数处理简单的变量边界
bounds = Bounds(0, np.inf, keep_feasible=True) # 所有x >= 0

# 4. 执行优化
res_linear = minimize(fun=opt_func, 
                      x0=x0, 
                      method='trust-constr', # 'trust-constr'方法支持LinearConstraint
                      bounds=bounds, 
                      constraints=[total_sum_constraint, group_sum_constraint], 
                      args=(utility_vector, 0.16),
                      tol=1e-4)

print(res_linear)

print(f'\nTotal allocation sum: {res_linear.x.sum():.4f}')
for idx, select in enumerate(groups):
    print(f'Group {idx} ({select}) sum: {res_linear.x[select].sum():.4f}, target: {z_group[idx]}')
    print(f'  Difference: {z_group[idx] - res_linear.x[select].sum():.4e}')