虽然scipy.optimize.minimize
中的SLSQP
算法很好,但它有许多限制。首先是它是一个QP
求解器,因此适用于适合于二次规划范例的方程。但如果您具有函数约束会发生什么?此外,scipy.optimize.minimize
不是全局优化器,因此您通常需要非常靠近最终结果才能开始。
有一个受限非线性优化包(名为mystic
)几乎存在于scipy.optimize
本身出现的时间——我建议将其作为处理任何一般受限非线性优化的首选。
例如,如果我理解您的伪代码正确,您的问题大致如下:
import numpy as np
M = 10
N = 3
Q = 10
C = 10
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):
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):
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,要处理您在 mystic
中的问题,您只需指定边界和约束条件。
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]
结果看起来像这样:
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条件(例如相等性、不等式),包括符号和函数约束。
上面我将约束条件指定为“惩罚”,这是一种传统方法,当约束条件被违反时,它们会对目标函数施加惩罚。
mystic
还提供了非线性内核变换,通过减少有效解空间的空间映射或内核变换来约束解空间。
以下是一个示例,展示了mystic
如何解决许多二次规划求解器无法处理的问题,因为这些约束条件不是以约束矩阵的形式给出的。该问题是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
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)
在这里找到这个例子以及大约100个类似的例子:https://github.com/uqfoundation/mystic.
我是作者,所以我有点偏见。但是,这种偏见非常微小。mystic
既成熟又得到良好支持,并且在解决困难的受限非线性优化问题方面无与伦比。