Question

python中约束非线性优化的推荐包是什么？

我想解决的具体问题是：

我有一个未知X（Nx1），我有M（Nx1）u个向量和M（NxN）s矩阵。

max [5th percentile of (ui_T*X), i in 1 to M]
st 
0<=X<=1 and
[95th percentile of (X_T*si*X), i in 1 to M]<= constant

当我开始解决问题时，我只对u和s进行了一次点评估，并且我能够使用cvxpy解决上述问题。

我意识到，除了对u和s的一个估计，我得到了整个值的分布，所以我想改变我的目标函数，以便我可以使用整个分布。上面的问题描述是我尝试以有意义的方式包含该信息。

cvxpy不能用来解决这个问题，我已经尝试了scipy.optimize.anneal，但我似乎无法设置未知值的界限。我也看了pulp，但它不允许非线性约束。

Answer 1

scipy为受约束的非线性优化提供了一个壮观的包。

您可以通过阅读optimize doc开始使用，但这是SLSQP的示例：

minimize(func, [-1.0,1.0], args=(-1.0,), jac=func_deriv, constraints=cons, method='SLSQP', options={'disp': True})

Answer 2

虽然SLSQP中的scipy.optimize.minimize算法很好，但它有一些限制。第一个是QP解算器，所以它适用于适合二次规划范式的方程。但是如果你有功能限制会发生什么？此外，scipy.optimize.minimize不是全局优化器，因此您通常需要非常接近最终结果。

有一个受约束的非线性优化包（称为mystic）已经存在了几乎与scipy.optimize本身一样长 - 我建议它作为处理任何一般约束的首选非线性优化。

例如，如果我理解您的伪代码，您的问题看起来像这样：

import numpy as np

M = 10
N = 3
Q = 10
C = 10

# let's be lazy, and generate s and u randomly...
s = np.random.randint(-Q,Q, size=(M,N,N))
u = np.random.randint(-Q,Q, size=(M,N))

def percentile(p, x):
    x = np.sort(x)
    p = 0.01 * p * len(x)
    if int(p) != p:
        return x[int(np.floor(p))]
    p = int(p)
    return x[p:p+2].mean()

def objective(x, p=5): # inverted objective, to find the max
    return -1*percentile(p, [np.dot(np.atleast_2d(u[i]), x)[0] for i in range(0,M-1)])


def constraint(x, p=95, v=C): # 95%(xTsx) - v <= 0
    x = np.atleast_2d(x)
    return percentile(p, [np.dot(np.dot(x,s[i]),x.T)[0,0] for i in range(0,M-1)]) - v

bounds = [(0,1) for i in range(0,N)]

因此，要在mystic中处理您的问题，您只需要指定边界和约束。

from mystic.penalty import quadratic_inequality
@quadratic_inequality(constraint, k=1e4)
def penalty(x):
  return 0.0

from mystic.solvers import diffev2
from mystic.monitors import VerboseMonitor
mon = VerboseMonitor(10)

result = diffev2(objective, x0=bounds, penalty=penalty, npop=10, gtol=200, \
                 disp=False, full_output=True, itermon=mon, maxiter=M*N*100)

print result[0]
print result[1]

结果如下所示：

Generation 0 has Chi-Squared: -0.434718
Generation 10 has Chi-Squared: -1.733787
Generation 20 has Chi-Squared: -1.859787
Generation 30 has Chi-Squared: -1.860533
Generation 40 has Chi-Squared: -1.860533
Generation 50 has Chi-Squared: -1.860533
Generation 60 has Chi-Squared: -1.860533
Generation 70 has Chi-Squared: -1.860533
Generation 80 has Chi-Squared: -1.860533
Generation 90 has Chi-Squared: -1.860533
Generation 100 has Chi-Squared: -1.860533
Generation 110 has Chi-Squared: -1.860533
Generation 120 has Chi-Squared: -1.860533
Generation 130 has Chi-Squared: -1.860533
Generation 140 has Chi-Squared: -1.860533
Generation 150 has Chi-Squared: -1.860533
Generation 160 has Chi-Squared: -1.860533
Generation 170 has Chi-Squared: -1.860533
Generation 180 has Chi-Squared: -1.860533
Generation 190 has Chi-Squared: -1.860533
Generation 200 has Chi-Squared: -1.860533
Generation 210 has Chi-Squared: -1.860533
STOP("ChangeOverGeneration with {'tolerance': 0.005, 'generations': 200}")
[-0.17207128  0.73183465 -0.28218955]
-1.86053344078

mystic非常灵活，可以处理任何类型的约束（例如，等式，不等式），包括符号和功能约束。我将约束指定为上面的“惩罚”，这是传统方式，因为它们在违反约束时对目标应用惩罚。 mystic还提供非线性内核转换，通过减少有效解的空间（即通过空间映射或内核转换）来约束解空间。

作为一个例子，这里的mystic解决了一个打破很多QP求解器的问题，因为约束不是约束矩阵的形式。它正在优化压力容器的设计。

"Pressure Vessel Design"

def objective(x):
    x0,x1,x2,x3 = x
    return 0.6224*x0*x2*x3 + 1.7781*x1*x2**2 + 3.1661*x0**2*x3 + 19.84*x0**2*x2

bounds = [(0,1e6)]*4
# with penalty='penalty' applied, solution is:
xs = [0.72759093, 0.35964857, 37.69901188, 240.0]
ys = 5804.3762083

from mystic.symbolic import generate_constraint, generate_solvers, simplify
from mystic.symbolic import generate_penalty, generate_conditions

equations = """
-x0 + 0.0193*x2 <= 0.0
-x1 + 0.00954*x2 <= 0.0
-pi*x2**2*x3 - (4/3.)*pi*x2**3 + 1296000.0 <= 0.0
x3 - 240.0 <= 0.0
"""
cf = generate_constraint(generate_solvers(simplify(equations)))
pf = generate_penalty(generate_conditions(equations), k=1e12)


if __name__ == '__main__':

    from mystic.solvers import diffev2
    from mystic.math import almostEqual
    from mystic.monitors import VerboseMonitor
    mon = VerboseMonitor(10)

    result = diffev2(objective, x0=bounds, bounds=bounds, constraints=cf, penalty=pf, \ 
                     npop=40, gtol=50, disp=False, full_output=True, itermon=mon)

    assert almostEqual(result[0], xs, rel=1e-2)
    assert almostEqual(result[1], ys, rel=1e-2)

在这里找到这个以及大约100个这样的例子：https://github.com/uqfoundation/mystic。

我是作者，所以我有点偏颇。但是，偏见很小。 mystic既成熟又受到良好支持，在解决硬约束非线性优化问题方面具有无与伦比的能力。

Answer 3

正如其他人所评论的那样，SciPy最小化包是一个很好的起点。我在下面列举了一个例子（Hock Schittkowski＃71基准），其中包括目标函数，等式约束和不等式约束。

import numpy as np
from scipy.optimize import minimize

def objective(x):
    return x[0]*x[3]*(x[0]+x[1]+x[2])+x[2]

def constraint1(x):
    return x[0]*x[1]*x[2]*x[3]-25.0

def constraint2(x):
    sum_eq = 40.0
    for i in range(4):
        sum_eq = sum_eq - x[i]**2
    return sum_eq

# initial guesses
n = 4
x0 = np.zeros(n)
x0[0] = 1.0
x0[1] = 5.0
x0[2] = 5.0
x0[3] = 1.0

# show initial objective
print('Initial SSE Objective: ' + str(objective(x0)))

# optimize
b = (1.0,5.0)
bnds = (b, b, b, b)
con1 = {'type': 'ineq', 'fun': constraint1} 
con2 = {'type': 'eq', 'fun': constraint2}
cons = ([con1,con2])
solution = minimize(objective,x0,method='SLSQP',\
                    bounds=bnds,constraints=cons)
x = solution.x

# show final objective
print('Final SSE Objective: ' + str(objective(x)))

# print solution
print('Solution')
print('x1 = ' + str(x[0]))
print('x2 = ' + str(x[1]))
print('x3 = ' + str(x[2]))
print('x4 = ' + str(x[3]))

如果SLSQP无法解决您的问题，nonlinear programming solvers for Python上还有一个更全面的讨论主题。如果您需要有关求解器方法的其他信息，可以使用Engineering Design Optimization上的课程材料。

Answer 4

通常，对于拟合，您可以使用scipy.optimize函数或lmfit，它只是扩展了scipy.optimize包，以便更容易传递边界之类的东西。我个人喜欢使用kmpfit，它是kapteyn库的一部分，基于MPFIT的C实现。

scipy.optimize.minimize()可能是最容易获得并且常用的。

Python约束非线性优化

4 个答案: