虽然scipy.optimize.minimize 中的SLSQP 算法很好，但它有很多限制。第一个是 QP 求解器，因此它适用于非常适合二次编程范式的方程。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，因此您通常需要在非常接近最终结果的情况下开始。
有一个约束非线性优化包（称为mystic）几乎与scipy.optimize 本身一样长——我建议它作为处理任何一般约束非线性优化的首选。
例如，你的问题，如果我理解你的伪代码，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。
我在上面将约束指定为“惩罚”，这是传统方式，因为当违反约束时，它们会对目标施加惩罚。
mystic 还提供非线性核变换，通过减少有效解的空间（即通过空间映射或核变换）来限制解空间。
例如，这里的mystic 解决了一个问题，该问题破坏了许多 QP 求解器，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 个类似的例子：https://github.com/uqfoundation/mystic。
我是作者，所以有点偏见。但是，偏差非常轻微。 mystic 既成熟又得到很好的支持，在解决硬约束非线性优化问题的能力方面无与伦比。