多维 scipy.optimize.LinearConstraint？

我建议编写一个循环，一次一行地形成约束。

如果要添加对应于 的约束a_ij < x_i - x_j < a_ji，可以通过向约束矩阵添加一个向量来实现，该向量全为零，但第 i 个位置为 1，第 j 个位置为负 1。当 SciPy 将此向量与 x 进行点积时，结果将为x_i - x_j。

您可能还想避免重复约束。如果您同时约束值x_i - x_j和值x_j - x_i，那么这些值是相同的，只是乘以 -1。由于约束矩阵中的秩较低，重复约束可能会导致某些最小化方法出现问题。在下面的例子中，我任意选择了所有约束的 i < j。

这是一个例子。

import numpy as np
import scipy

something = np.arange(9).reshape(3, 3).T
constraints_list = []
lb = []
ub = []
N = 3
for i in range(N):
    for j in range(N):
        if i == j:
            # If i == j, then x_i - x_j = 0, and constraint is always satisfied or always violated
            continue
        if j > i:
            # If j > i, then this constraint duplicates another constraint.
            continue
        new_cons = np.zeros(N)
        new_cons[i] = 1
        new_cons[j] = -1
        constraints_list.append(new_cons)
        lb.append(something[i, j])
        ub.append(something[j, i])
constraints_array = np.array(constraints_list)
constraint = scipy.optimize.LinearConstraint(constraints_array, lb, ub)


# Compute constraint values in old way
x = np.array([1.1, 1.2, 1.3])
print("Old constraint values")
ones = np.ones_like(x)
print(np.outer(x, ones) - np.outer(ones, x))

# Compute constraint values in new way
print("New constraint values (same as lower triangle of above array)")
print(constraint.A @ x)

print("Old constraint lower bound")
print(something)
print("New constraint lower bound (same as lower triangle of above array)")
print(constraint.lb)

输出：

Old constraint values
[[ 0.  -0.1 -0.2]
 [ 0.1  0.  -0.1]
 [ 0.2  0.1  0. ]]
New constraint values (same as lower triangle of above array)
[0.1 0.2 0.1]
Old constraint lower bound
[[0 3 6]
 [1 4 7]
 [2 5 8]]
New constraint lower bound (same as lower triangle of above array)
[1 2 5]

这实际上将您的约束“展平”为二维线性矩阵。