scipy.optimize的最小化功能无法给出正确的答案

问题描述

我正在尝试使用Scipy的最小化功能解决最小化问题。目标函数只是具有不同均值和方差的两个多元正态分布的比率。我希望找到函数g_func的最大值，这等效于找到函数g_optimization的最小值。另外，我添加了x [0] = 0的约束。在这里，x是具有8个元素的向量。目标函数g_optimization如下：

import numpy as np
from scipy.optimize import minimize
# Set up mean and variance for two MVN distributions
n_trait = 8
sigma = np.full((n_trait,n_trait),0.0005)
np.fill_diagonal(sigma,0.005)
omega = np.full((n_trait,0.0000236)
np.fill_diagonal(omega,0.0486)
sigma_pos = np.linalg.inv(np.linalg.inv(sigma)+np.linalg.inv(omega))
mu_pos = np.array([-0.01288244,0.08732091,0.01049617,0.0860966,0.10055626,0.07952922,0.04363669,-0.0061975])
mu_pri = 0
sigma_pri = omega

#objective function
def g_func(beta,mu_sim_pos):
    g1 = ((np.linalg.det(sigma_pri))**(1/2))/((np.linalg.det(sigma_pos))**(1/2))
    g2 = (-1/2)*np.linalg.multi_dot([np.transpose(beta-mu_sim_pos),np.linalg.inv(sigma_pos),beta-mu_sim_pos])
    g3 = (1/2)*np.linalg.multi_dot([np.transpose(beta-mu_pri),np.linalg.inv(sigma_pri),beta-mu_pri])
    g = g1*np.exp(g2+g3)
    return g
def g_optimization(beta,mu_sim_pos):
    return -1*g_func(beta,mu_sim_pos)

#optimization
start_point = np.full(8,0)
cons = ({'type': 'eq','fun' : lambda x: np.array([x[0]])})

anws = minimize (g_optimization,[start_point],args=(mu_pos),constraints=cons,options={'maxiter': 50},tol=0.001)
anws

优化在两次迭代后停止，并且函数给出的最小值为np.array（[0,10.32837891，-1.62396508,10.13790152,12.38752653,9.11615259,3.53201544，-4.22115517]）。这是不正确的，因为即使我们将起点np.zeros（8）插入g_optimization函数，给出的结果也是-657.0041125829354，它小于0。因此，提供的解决方案绝对不是最小的。

g_optimization(np.zeros(8),mu_pos) #gives solution of -657.0041125829354

我不确定我哪里出错了。

解决方法

我会尝试其他求解器。例如L-BFGS-B效果很好。 您可以查看所有选项here。

anws = minimize (g_optimization,[start_point],args=(mu_pos),method='L-BFGS-B',constraints=cons,options={'maxiter': 50},tol=0.001)

print(anws)

# success: True
# message: b'CONVERGENCE: REL_REDUCTION_OF_F_<=_FACTR*EPSMCH'
# fun: -21688.00879938617
# x: array([-0.0101048,0.09937778,0.01543875,0.0980401,0.11383878,0.09086455,0.05164822,-0.00280081])

---

编辑： L-BFGS-B无法处理一般约束h(x)=0，只能处理变量的边界框：

L-BFGS-B，TNC，SLSQP，Powell和trust-constr方法的变量界限。有两种方法可以指定范围： Bounds类的实例。 x中每个元素的（最小，最大）对的序列。 None用于指定无限制。

在您的情况下，您必须定义8对上下限。 对于x [0]，您必须进行严格限制，因为该方法无法处理x_low == x_high。

bounds = [(None,None)] * 8
bounds[0] = (0,0.00001)
anws = minimize (g_optimization,bounds=bounds,tol=0.001)
# fun: -21467.48153792194
# x: array([0.,0.10039832,0.01641271,0.0990599,0.11486735,0.09188037,0.05264228,-0.00183697])

另一种选择是从优化问题中排除值x [0]：

def g_optimization(beta,mu_sim_pos):
    beta2 = np.empty(8)
    beta2[0] = 0
    beta2[1:] = beta
    return -1*g_func(beta2,mu_sim_pos)

start_point = np.zeros(7)  # exclude x[0]

anws = minimize(g_optimization,tol=0.001)
# fun: -21467.47686079844
# x: array([0.10041797,0.01648995,0.09908046,0.11487707,0.09190585,0.05269467,-0.00174722])
#          ^ missing x[0]