Question

我使用COBYLA对带有约束的线性目标函数进行成本最小化。我通过为每个包含约束来实现下限和上限。

import numpy as np
import scipy.optimize

def linear_cost(factor_prices):
    def cost_fn(x):
        return np.dot(factor_prices, x)
    return cost_fn


def cobb_douglas(factor_elasticities):
    def tech_fn(x):
        return np.product(np.power(x, factor_elasticities), axis=1)
    return tech_fn

def mincost(targets, cost_fn, tech_fn, bounds):

    n = len(bounds)
    m = len(targets)

    x0 = np.ones(n)  # Do not use np.zeros.

    cons = []

    for factor in range(n):
        lower, upper = bounds[factor]
        l = {'type': 'ineq',
             'fun': lambda x: x[factor] - lower}
        u = {'type': 'ineq',
             'fun': lambda x: upper - x[factor]}
        cons.append(l)
        cons.append(u)

    for output in range(m):
        t = {'type': 'ineq',
             'fun': lambda x: tech_fn(x)[output] - targets[output]}
        cons.append(t)

    res = scipy.optimize.minimize(cost_fn, x0,
                                  constraints=cons,
                                  method='COBYLA')

    return res

COBYLA并不尊重上限或下限，但它确实尊重技术约束。

>>> p = np.array([5., 20.])
>>> cost_fn = linear_cost(p)

>>> fe = np.array([[0.5, 0.5]])
>>> tech_fn = cobb_douglas(fe)

>>> bounds = [[0.0, 15.0], [0.0, float('inf')]]

>>> mincost(np.array([12.0]), cost_fn, tech_fn, bounds)
       x: array([ 24.00010147,   5.99997463])
 message: 'Optimization terminated successfully.'
   maxcv: 1.9607782064667845e-10
    nfev: 75
  status: 1
 success: True
     fun: 239.99999999822359

为什么COBYLA不会尊重第一个因素约束（即上限@ 15）？

Answer 1

COBYLA 实际上是尊重您提供的所有界限。

问题在于构建cons列表。也就是说，在lambda和Python（和Javascript）中的其他内部范围函数中绑定变量是词法，并且不会按照您的假设行事：http://eev.ee/blog/2011/04/24/gotcha-python-scoping-closures/循环结束后，变量lower upper的值为0和inf，变量factor的值为1，这些值是所有lambda函数使用的值。

一种解决方法是将变量的特定值显式绑定到伪关键字参数：

for factor in range(n):
    lower, upper = bounds[factor]
    l = {'type': 'ineq',
         'fun': lambda x, a=lower, i=factor: x[i] - a}
    u = {'type': 'ineq',
         'fun': lambda x, b=upper, i=factor: b - x[i]}
    cons.append(l)
    cons.append(u)

for output in range(m):
    t = {'type': 'ineq',
         'fun': lambda x, i=output: tech_fn(x)[i] - targets[i]}
    cons.append(t)

第二种方法是添加一个生成lambdas的工厂函数。

scipy - 为什么COBYLA不尊重约束？

1 个答案: