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假设我有一个非凸目标函数loss，它采用形状为 (n,)的np.ndarray命名并返回一个数字。目标有很多很多局部最小值，因为它本质上是另一个形状 (n,) 的常量数组的函数。Xfloat np.round(X * c, 2)c

你可以想象这样的事情：

def loss(X: np.ndarray) -> float:
    c = np.array([0.1, 0.5, -0.8, 7.0, 0.0])
    X_rounded = np.round(X * c, 2)
    return rosen(X_rounded)

线性约束用两个常数矩阵（也存储为 numpy's ndarray）表示，A其形状为 (m, n) b，形状为 (m,)。我需要在保持的同时loss最小化。另外，每个元素都必须服从，我有一个不错的初步猜测。XA.dot(X) == bX0 <= X_i <= 2X0 = [1, 1, ..., 1]

我不需要全局最小值，搜索可以立即停止loss(X) <= 1。目标主要是用 SQL 编写的，因此速度非常慢，所以我还希望优化在loss被评估约 200 次时终止。（这不是硬性要求，您也可以在优化运行 5 分钟后终止。）

使用 scipy，我可以做到

rv = minimize(loss,
              initial_guess,
              method='SLSQP',
              bounds=[(0, 2)] * n,
              constraints={
                  'type': 'eq',
                  'fun': lambda x: A.dot(x) - b
              },
              options={
                  'maxiter': 5
              })

我对这个解决方案不满意，因为结果比人为的初始猜测（作为冒烟测试在全局最小值附近采样）更糟糕，这可能是由于局部最小值的丰富和一些数值问题？此外，我无法估计每次迭代的目标调用次数（否则我可以通过设置来限制创新次数maxiter）。

我怎样才能更好地使用mystic，这可能更灵活？

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1 回答 1

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由于我不知道什么A和b是什么，我将即兴创作。因此，它不会与您的问题完全相同，但应该足够接近。

让我们通过构建损失函数和约束来设置问题。可能有更好的方法来构建约束，但以下内容非常笼统（虽然有点难看）：

>>> import mystic as my
>>> import numpy as np
>>> from mystic.models import rosen
>>>
>>> A = np.array([[9., 0., 0., 8., -1],
...               [1., 1., -1., 0., 0.],
...               [2., -2., 6., 0., 5.]])
>>> b = np.array([18., .75, 11.5])
>>> c = np.array([0.1, 0.5, -0.8, 7.0, 0.0])
>>>
>>> def loss(x):
...     x_rounded = np.round(x * c, 2)
...     return rosen(x_rounded)
...
>>> cons = my.symbolic.linear_symbolic(A, b)
>>> cons = my.symbolic.solve(cons)
>>> cons = my.symbolic.generate_constraint(my.symbolic.generate_solvers(cons))
>>> bounds = [(0,2)] * len(c)

然后尝试求解全局最小值：

>>> stepmon = my.monitors.VerboseMonitor(1)
>>> rv = my.solvers.diffev2(loss, x0=bounds, bounds=bounds, constraints=cons, itermon=stepmon, disp=1, npop=20)
Generation 0 has ChiSquare: 15478.596962
Generation 1 has ChiSquare: 1833.714503
Generation 2 has ChiSquare: 1833.714503
Generation 3 has ChiSquare: 270.601079
Generation 4 has ChiSquare: 160.690618
Generation 5 has ChiSquare: 160.690618
Generation 6 has ChiSquare: 127.289639
Generation 7 has ChiSquare: 127.289639
Generation 8 has ChiSquare: 127.289639
Generation 9 has ChiSquare: 123.054668
Generation 10 has ChiSquare: 123.054668
Generation 11 has ChiSquare: 123.054668
Generation 12 has ChiSquare: 122.561794
Generation 13 has ChiSquare: 121.069338
Generation 14 has ChiSquare: 120.828279
Generation 15 has ChiSquare: 117.732442
Generation 16 has ChiSquare: 117.732442
Generation 17 has ChiSquare: 117.340042
Generation 18 has ChiSquare: 117.340042
Generation 19 has ChiSquare: 117.340042
Generation 20 has ChiSquare: 117.340042
Generation 21 has ChiSquare: 117.340042
Generation 22 has ChiSquare: 116.750933
Generation 23 has ChiSquare: 116.750933
Generation 24 has ChiSquare: 116.750933
Generation 25 has ChiSquare: 116.750933
Generation 26 has ChiSquare: 116.750933
Generation 27 has ChiSquare: 116.750933
Generation 28 has ChiSquare: 116.750933
Generation 29 has ChiSquare: 116.750933
Generation 30 has ChiSquare: 116.750933
Generation 31 has ChiSquare: 116.750933
Generation 32 has ChiSquare: 116.750933
Generation 33 has ChiSquare: 116.750933
Generation 34 has ChiSquare: 116.750933
Generation 35 has ChiSquare: 116.750933
Generation 36 has ChiSquare: 116.750933
Generation 37 has ChiSquare: 116.750933
Generation 38 has ChiSquare: 116.750933
Generation 39 has ChiSquare: 116.750933
Generation 40 has ChiSquare: 116.750933
Generation 41 has ChiSquare: 116.750933
Generation 42 has ChiSquare: 116.750933
Generation 43 has ChiSquare: 116.750933
Generation 44 has ChiSquare: 116.750933
Generation 45 has ChiSquare: 116.750933
Generation 46 has ChiSquare: 116.750933
Generation 47 has ChiSquare: 116.750933
Generation 48 has ChiSquare: 116.750933
Generation 49 has ChiSquare: 116.750933
Generation 50 has ChiSquare: 116.750933
Generation 51 has ChiSquare: 116.750933
STOP("VTRChangeOverGeneration with {'ftol': 0.005, 'gtol': 1e-06, 'generations': 30, 'target': 0.0}")
Optimization terminated successfully.
         Current function value: 116.750933
         Iterations: 51
         Function evaluations: 1040

>>> A.dot(rv)
array([18.  ,  0.75, 11.5 ])

这行得通（它可能仍然不是全球最小值）......但这需要一些时间。所以，让我们尝试一个更快的本地求解器。

>>> stepmon = my.monitors.VerboseMonitor(1)
>>> rv = my.solvers.fmin_powell(loss, x0=[1]*len(c), bounds=bounds, constraints=cons, itermon=stepmon, disp=1)
Generation 0 has ChiSquare: 244559.856997
Generation 1 has ChiSquare: 116357.59447400003
Generation 2 has ChiSquare: 121.23445799999999
Generation 3 has ChiSquare: 117.635447
Generation 4 has ChiSquare: 117.59764200000001
Generation 5 has ChiSquare: 117.59764200000001
Optimization terminated successfully.
         Current function value: 117.597642
         Iterations: 5
         Function evaluations: 388
STOP("NormalizedChangeOverGeneration with {'tolerance': 0.0001, 'generations': 2}")

>>> A.dot(rv)
array([18.  ,  0.75, 11.5 ])

不错。但是，您想限制 的评估次数loss，并且还希望能够在loss接近最小值时停止......所以让我们说什么时候停止loss(x) <= 120。我还将函数评估的数量限制为200.

>>> stepmon = my.monitors.VerboseMonitor(1)
>>> rv = my.solvers.fmin_powell(loss, x0=[1]*len(c), bounds=bounds, constraints=cons, itermon=stepmon, disp=1, maxfun=200, gtol=None, ftol=120)
Generation 0 has ChiSquare: 244559.856997
Generation 1 has ChiSquare: 116357.59447400003
Generation 2 has ChiSquare: 121.23445799999999
Generation 3 has ChiSquare: 117.635447
Optimization terminated successfully.
         Current function value: 117.635447
         Iterations: 3
         Function evaluations: 175
STOP("VTRChangeOverGeneration with {'ftol': 120, 'gtol': 1e-06, 'generations': 30, 'target': 0.0}")

>>> A.dot(rv)
array([18.  ,  0.75, 11.5 ])
>>> rv
array([1.93873933, 0.00381084, 1.19255017, 0.0807893 , 0.0949684 ])

如果您使用解算器的类接口，则具有更大的灵活性，但我将把它留到下一次。

于 2020-12-30T17:15:55.590 回答