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Python 具有线性约束和有界的非凸函数在mystic中的极小化

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Python 具有线性约束和有界的非凸函数在mystic中的极小化,python,optimization,constraints,non-convex,mystic,Python,Optimization,Constraints,Non Convex,Mystic,假设我有一个非凸目标函数loss，它接受一个名为X的np.ndarray，其形状为（n，）并返回一个float数字。目标有许多许多局部极小值，因为它本质上是一个np的函数。圆形（X*c，2），其中c是另一个恒定的形状数组（n，） 你可以想象这样的情况： 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) re

假设我有一个非凸目标函数
loss
，它接受一个名为
X
的
np.ndarray
，其形状为（n，）并返回一个
float
数字。目标有许多许多局部极小值，因为它本质上是一个
np的函数。圆形（X*c，2）
，其中
c
是另一个恒定的形状数组（n，）

你可以想象这样的情况：

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)
线性约束用两个常量矩阵表示（也存储为numpy的
ndarray
），
A
的形状为（m，n）和
b
的形状为（m，）。我需要在保持
A.dot（X）=b
的同时，将
X的
损失降至最低。另外，
X
的每个元素都必须服从
0，因为我不知道
A
和
b
是什么，所以我要即兴创作。所以它不会和你的问题完全一样，但应该足够接近

让我们通过建立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，但以下内容相当普遍（尽管有点难看）：

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 ])


这是可行的（可能还不是全球最低值）。。。但这需要一些时间。那么，让我们试试更快的局部解算器


>>> 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，您希望限制
损失
的评估次数，并且在
损失
接近最小值时能够停止。。。因此，让我们假设当
loss（x）“比我最初的猜测严重得多”听起来很奇怪时停止。您最初的猜测是否满足约束条件？如果你能从分析上区分损失函数，那么如果你也提供梯度，你就可以大大减少调用的次数。@AskoldIlvento它绝对满足约束条件。事实上，我只是使用全局最小值作为初始猜测
x0
，这样
损失（x0）=1e-14
，但由此产生的
损失（rv.x）
通常在700左右。说到损失函数，它的很大一部分实际上是用SQL编写的（这使得它的速度慢得离谱），但我相信它是可微的，除非有几个不连续点。

>>> 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 ])



>>> 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 ])



>>> 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 ])