如何评估信任构造拉格朗日？

Question

我正在使用 scipy.optimize.minimize 中的 trust-constr 算法和间隔约束（下界 < g(x) < 上界）。 我想在找到的解决方案周围的区域中绘制拉格朗日函数，以分析收敛行为。

据我所知，拉格朗日定义为：

与：

在返回的 OptimizeResult 对象中，我可以找到屏障参数，但缺少松弛变量。存在拉格朗日乘子，但每个区间约束只有一个，而我期望有两个，因为每个区间约束都转换为两个规范不等式约束：

显然，我错过了一些东西，所以任何帮助将不胜感激。

最小可重现示例：

import scipy.optimize as so
import numpy as np

# Problem definition:
# Five 2D points are given, with equally spaced x coordinates.
# The y coordinate of the first point is zero, while the last point has value 10.
# The goal is to find the smallest y coordinate of the other points, given the
# difference between the y coordinates of two consecutive points has to lie within the
# interval [-3, 3].

xs = np.linspace(0, 4, 5)
y0s = np.zeros(xs.shape)
y0s[-1] = 10

objective_fun = lambda y: np.mean(y**2)

def constraint_fun(ys):
    '''
    Calculates the signed squared consecutive differences of the input vector, augmented
    with the first and last element of y0s.
    '''
    full_ys = y0s.copy()
    full_ys[1:-1] = ys
    consecutive_differences = full_ys[1:] - full_ys[:-1]
    return np.sign(consecutive_differences) * consecutive_differences**2

constraint = so.NonlinearConstraint(fun=constraint_fun, lb=-3**2, ub=3**2)

result = so.minimize(method='trust-constr', fun=objective_fun, constraints=[constraint], x0=y0s[1:-1])

# The number of interval constraints is equal to the size of the output vector of the constraint function.
print(f'Nr. of interval constraints: {len(constraint_fun(y0s[1:-1]))}')

# Expected nr of Lagrange multipliers: 2x number of interval constraints.
print(f'Nr. of Lagrange multipliers: {len(result.v[0])}')

输出：

Nr. of interval constraints: 4
Nr. of Lagrange multipliers: 4

预期输出：

Nr. of interval constraints: 4
Nr. of Lagrange multipliers: 8

Answer 1

你是对的，确实应该有 8 个拉格朗日乘数。作为解决方法，您可以使用旧的字典约束而不是 NonlinearConstraint 对象。

lb, ub = -3**2, 3**2
# Note that lb <= g(x) <= ub is equivalent to g(x) - lb >= 0, ub - g(x) >= 0
cons = [{'type': 'ineq', 'fun': lambda ys: constraint_fun(ys) - lb},
        {'type': 'ineq', 'fun': lambda ys: ub - constraint_fun(ys)}]

res = minimize(objective_fun, method="trust-constr", x0=y0s[1:-1], constraints=cons)

这里，“fun”预计是一个满足 fun(x) >= 0 的函数。这给了我 8 个预期的拉格朗日乘数。尽管如此，它也应该与 NonlinearConstraint 一起使用，因此可能值得在 Github 上的 scipy 存储库中提出一个问题。

关于你的第二个问题：res.constr包含解决方案中的约束值列表，即g(x) - lb和ub - g的值(x)。由于我们有 g(x) - lb - s = 0 和 ub-g(x)-s=0 ，因此立即得出 res.constr code> 只是您正在寻找的松弛变量的值（当我们使用字典约束时）。