Question

我有一个最小二乘最小化问题受到不等式约束的影响，我试图用scipy.optimize.minimize来解决这个问题。看起来不平等约束有两种选择：COBYLA和SLSQP。

我首先尝试了SLSQP，因为它允许最小化函数的显式偏导数。根据问题的缩放程度，它会失败并显示错误：

Positive directional derivative for linesearch    (Exit mode 8)

每当施加间隔或更一般的不等式约束时。

此前已经观察到，例如，here。手动缩放要最小化的函数（以及相关的偏导数）似乎摆脱了问题，但我无法通过更改选项中的ftol来实现相同的效果。

总的来说，这整件事让我对日常工作有了怀疑。这是一个简化的例子：

import numpy as np
import scipy.optimize as sp_optimize

def cost(x, A, y):

    e = y - A.dot(x)
    rss = np.sum(e ** 2)

    return rss

def cost_deriv(x, A, y):

    e = y - A.dot(x)
    deriv0 = -2 * e.dot(A[:,0])
    deriv1 = -2 * e.dot(A[:,1])

    deriv = np.array([deriv0, deriv1])

    return deriv


A = np.ones((10,2)); A[:,0] = np.linspace(-5,5, 10)
x_true = np.array([2, 2/20])
y = A.dot(x_true)
x_guess = x_true / 2

prm_bounds = ((0, 3), (0,1))

cons_SLSQP = ({'type': 'ineq', 'fun' : lambda x: np.array([x[0] - x[1]]),
               'jac' : lambda x: np.array([1.0, -1.0])})

# works correctly
min_res_SLSQP = sp_optimize.minimize(cost, x_guess, args=(A, y), jac=cost_deriv, bounds=prm_bounds, method='SLSQP', constraints=cons_SLSQP, options={'disp': True})
print(min_res_SLSQP)

# fails
A = 100 * A
y = A.dot(x_true)
min_res_SLSQP = sp_optimize.minimize(cost, x_guess, args=(A, y), jac=cost_deriv, bounds=prm_bounds, method='SLSQP', constraints=cons_SLSQP, options={'disp': True})
print(min_res_SLSQP)

# works if bounds and inequality constraints removed
min_res_SLSQP = sp_optimize.minimize(cost, x_guess, args=(A, y), jac=cost_deriv,
method='SLSQP', options={'disp': True})
print(min_res_SLSQP)

如何设置ftol以避免失败？更一般地说，COBYLA会出现类似的问题吗？ COBYLA对于这种不等式约束最小二乘优化问题是更好的选择吗？

发现在成本函数中使用平方根可以提高性能。然而，对于问题的非线性重新参数化（更简单但更接近我在实践中需要做的事情），它再次失败。以下是详细信息：

import numpy as np
import scipy.optimize as sp_optimize


def cost(x, y, g):

    e = ((y - x[1]) / x[0]) - g

    rss = np.sqrt(np.sum(e ** 2))

    return rss


def cost_deriv(x, y, g):

    e = ((y- x[1]) / x[0]) - g

    factor = 0.5 / np.sqrt(e.dot(e))
    deriv0 = -2 * factor * e.dot(y - x[1]) / (x[0]**2)
    deriv1 = -2 * factor * np.sum(e) / x[0]

    deriv = np.array([deriv0, deriv1])

    return deriv


x_true = np.array([1/300, .1])
N = 20
t = 20 * np.arange(N)
g = 100 * np.cos(2 * np.pi * 1e-3 * (t - t[-1] / 2))
y = g * x_true[0] + x_true[1]

x_guess = x_true / 2
prm_bounds = ((1e-4, 1e-2), (0, .4))

# check derivatives
delta = 1e-9
C0 = cost(x_guess, y, g)
C1 = cost(x_guess + np.array([delta, 0]), y, g)
approx_deriv0 = (C1 - C0) / delta
C1 = cost(x_guess + np.array([0, delta]), y, g)
approx_deriv1 = (C1 - C0) / delta
approx_deriv = np.array([approx_deriv0, approx_deriv1])
deriv = cost_deriv(x_guess, y, g)

# fails
min_res_SLSQP = sp_optimize.minimize(cost, x_guess, args=(y, g), jac=cost_deriv,
bounds=prm_bounds, method='SLSQP', options={'disp': True})
print(min_res_SLSQP)

Answer 1

而不是最小化np.sum(e ** 2)，最小化sqrt(np.sum(e ** 2))或更好（在计算方面）：np.linalg.norm(e)！

此修改：

不会更改有关x
将需要后处理
更强大

有了这个改变，所有的情况都有效，甚至使用数值微分（我懒得修改渐变，这需要反映这个！）。

示例输出（func-evals的数量给出了num-diff）：

Optimization terminated successfully.    (Exit mode 0)
            Current function value: 3.815547437029837e-06
            Iterations: 16
            Function evaluations: 88
            Gradient evaluations: 16
     fun: 3.815547437029837e-06
     jac: array([-6.09663382, -2.48862544])
 message: 'Optimization terminated successfully.'
    nfev: 88
     nit: 16
    njev: 16
  status: 0
 success: True
       x: array([ 2.00000037,  0.10000018])
Optimization terminated successfully.    (Exit mode 0)
            Current function value: 0.0002354577991007501
            Iterations: 23
            Function evaluations: 114
            Gradient evaluations: 23
     fun: 0.0002354577991007501
     jac: array([ 435.97259208,  288.7483819 ])
 message: 'Optimization terminated successfully.'
    nfev: 114
     nit: 23
    njev: 23
  status: 0
 success: True
       x: array([ 1.99999977,  0.10000014])
Optimization terminated successfully.    (Exit mode 0)
            Current function value: 0.0003392807206384532
            Iterations: 21
            Function evaluations: 112
            Gradient evaluations: 21
     fun: 0.0003392807206384532
     jac: array([ 996.57340243,   51.19298764])
 message: 'Optimization terminated successfully.'
    nfev: 112
     nit: 21
    njev: 21
  status: 0
 success: True
       x: array([ 2.00000008,  0.10000104])

虽然SLSQP可能存在一些问题，但考虑到广泛的应用范围，它仍然是经过最严格测试和最强大的代码之一！

与COBYLA相比，我还期望SLSQP更好，因为后者主要基于线性化。（但只是把它作为一个猜测;给定最小化界面很容易尝试！）

<强>替代

通常，基于内点的凸二次规划解算器将是最佳方法。但为此，你需要离开scipy。（或者SOCP解算器会更好......我不确定）。

cvxpy带来了一个漂亮的建模系统和一个优秀的开源解算器（ECOS;虽然从技术上来说是一个圆锥解算器 - >更通用且不那么健壮;但应该击败SLSQP。 / p>

使用cvxpy和ECOS，如下所示：

import numpy as np
import cvxpy as cvx

""" Problem data """
A = np.ones((10,2)); A[:,0] = np.linspace(-5,5, 10)
x_true = np.array([2, 2/20])
y = A.dot(x_true)
x_guess = x_true / 2

prm_bounds = ((0, 3), (0,1))

# problematic case
A = 100 * A
y = A.dot(x_true)

""" Solve """
x = cvx.Variable(len(x_true))
constraints = [x[0] >= x[1]]
for ind, (lb, ub) in enumerate(prm_bounds):  # ineffecient -> matrix-based expr better!
    constraints.append(x[ind] >= lb)
    constraints.append(x[ind] <= ub)

objective = cvx.Minimize(cvx.norm(A*x - y))
problem = cvx.Problem(objective, constraints)
problem.solve(solver=cvx.ECOS, verbose=False)
print(problem.status)
print(problem.value)
print(x.value.T)

# optimal
# -6.67593652593801e-10
# [[ 2.   0.1]]

使用SLSQP的Python的scipy.optimize.minimize失败了“linesearch的正方向导数”

1 个答案: