Python约束的非线性优化

而 SLSQP 中的算法 scipy.optimize.minimize 很好,它有很多局限性。第一个是 QP 因此,它将适用于符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呢?而且 科学优化最小化 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 )它已经存在了将近 scipy.optimize 我建议将其作为处理任何一般约束非线性优化的工具。

例如,如果我理解你的伪代码,你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)]

所以,在 神秘的 ,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:

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

神秘的 非常灵活,可以处理任何类型的约束(例如等式、不等式),包括符号和函数约束。 我将约束指定为上面的“惩罚”,这是传统的方式,因为当违反约束时,它们会对目标施加惩罚。 神秘的 还提供了非线性核变换,其通过减少有效解的空间(即通过空间映射或核变换)来约束解空间。

例如 神秘的 因为约束不是以约束矩阵的形式,所以解决了一个打破了许多QP求解器的问题。它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)

在这里找到这个和大约100个类似的例子: https://github.com/uqfoundation/mystic .

我是作者,所以我有点偏见。然而,这种偏差很小。 神秘的 成熟且支持良好,在解决硬约束非线性优化问题方面具有无与伦比的能力。

scipy 有一个壮观的包,用于约束非线性优化。

您可以通过阅读 optimize doc ,但这里有一个SLSQP示例:

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

正如其他人所评论的,SciPy最小化包是一个很好的开始。我们还回顾了 Python Gekko paper (见第4节)。我在下面的示例(Hock Schittkowski#71基准测试)中包含了一个目标函数、等式约束和不等式约束 Scipy.optimize.minimize .

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]))

Python Gekko也存在同样的问题:

from gekko import GEKKO
m = GEKKO()
x1,x2,x3,x4 = m.Array(m.Var,4,lb=1,ub=5)
x1.value = 1; x2.value = 5; x3.value = 5; x4.value = 1

m.Equation(x1*x2*x3*x4>=25)
m.Equation(x1**2+x2**2+x3**2+x4**2==40)
m.Minimize(x1*x4*(x1+x2+x3)+x3)

m.solve(disp=False)
print(x1.value,x2.value,x3.value,x4.value)

还有一个更全面的讨论主题 nonlinear programming solvers for Python 如果SLSQP不能解决您的问题。我的课程材料 Engineering Design Optimization 如果需要有关解算器方法的其他信息,可以使用。

通常用于您可以使用的配件 scipy.optimize 功能,或 lmfit 它只是扩展了scipy.optimize包,使其更容易传递边界之类的内容。就我个人而言,我喜欢使用 kmpfit ,是kapteyn库的一部分,基于MPFIT的C实现。

scipy.optimize.minimize() 可能是最容易获得且最常用的。