python - Scipy 最小化成功终止，但不满足不等式约束

Question

我正在尝试根据 x a1+y a2+a3 z >= 1 形式的一系列 (N=20) 不等式约束来最小化函数 0.5*(x^2+y^2)。解决方案应在 x=0.50251、y=-0.5846、z=0.36787 左右。该例程以“优化成功终止”消息终止，但超过一半的约束未得到遵守。我也尝试了不同的求解器，结果相同。

缩放目标函数会改变解决方案，但不会收敛到预期的解决方案。

from scipy.optimize import minimize
import numpy as np


Pct=np.array([[-0.664,  3.179],[ 0.231, -2.044],[-2.493,  3.25 ],[ 0.497, -0.654],[-1.27,   1.248],[-1.185,  1.814],[-1.843,  4.386],[-1.616,  1.401],[ 0.052, -1.232],[-3.145,  0.404],[ 0.672, -1.655],[ 2.202, -1.888],[ 4.084, -1.067],[ 1.006, -1.671],[-2.255,  1.51 ],[-1.264,  1.663],[ 1.897, -2.217],[ 1.843, -1.276],[-1.693,  1.623],[ 2.297, -1.709]])
Sid=np.array([-1,  1, -1,  1, -1, -1, -1, -1,  1, -1,  1,  1,  1,  1, -1, -1,  1,  1, -1,  1])

# func to be minimized 
def OptFunc(x):
  return 0.5*(x[0]**2+x[1]**2)
def JacOptFunc(x):
  return np.array([x[0],x[1],0.0])  

# Constraints
c=[]
for i in range(len(Sid)):
  c+=[{'type': 'ineq', 'fun': lambda x:  Sid[i]*(x[0]*Pct[i,0]+x[1]*Pct[i,1]+x[2])-1 }]
cons=tuple(c)    

# start optimization
res = minimize(OptFunc,(0.3,-0.2,0.1),constraints=cons,method='SLSQP',jac=JacOptFunc)

#expected solution should be around
# [0.5025062702615434, -0.584685257866671, 0.36787016514022236]
print("-->",res.message)
print("solution ",res.x,flush=True)

print("Check Constraints")
cons=list(cons)
for i in range(len(cons)):
  lokfun=c[i]['fun']
  print("Constraint # ",i," value: ",lokfun(res.x))

预期结果就在附近x=0.50251, y=-0.5846, z=0.36787但我得到以下输出:

--> Optimization terminated successfully.
solution  [-1.14580677e-04 -1.16285140e-04  1.00006446e+00]

Check Constraints
Constraint #  0  value:  -1.9997708716077622
Constraint #  1  value:  0.0002756791862408292
Constraint #  2  value:  -1.999972183420499
Constraint #  3  value:  8.356438220613605e-05
Constraint #  4  value:  -2.0000648541023893
Constraint #  5  value:  -1.9999892973558606
Constraint #  6  value:  -1.9997656060620763
Constraint #  7  value:  -2.000086707390163
Constraint #  8  value:  0.00020176559401496874
Constraint #  9  value:  -2.0003778375289833
Constraint #  10  value:  0.00017991418852214558
Constraint #  11  value:  3.1700190727068644e-05
Constraint #  12  value:  -0.0002794107423930159
Constraint #  13  value:  0.00014350480474445426
Constraint #  14  value:  -2.000147249362345
Constraint #  15  value:  -2.0000159082853974
Constraint #  16  value:  0.00010490510804150865
Constraint #  17  value:  1.6681482228886324e-06
Constraint #  18  value:  -2.0000697148012767
Constraint #  19  value:  -1.354516498963676e-11

Answer 1

我对 scipy.optimize 知之甚少，但我可以发现问题

for i in range(len(Sid)):
  c+=[{'type': 'ineq', 'fun': lambda x:  Sid[i]*(x[0]*Pct[i,0]+x[1]*Pct[i,1]+x[2])-1 }]

问题在于 Python 闭包是后期绑定(bind)的，这意味着每个约束中 i 的值实际上是在循环完成后计算的。实际上，您实际上是在施加相同的(最后一个)约束 20 次。见 https://docs.python-guide.org/writing/gotchas/#late-binding-closures

可能的修复:

from scipy.optimize import minimize
import numpy as np


Pct=np.array([[-0.664,  3.179],[ 0.231, -2.044],[-2.493,  3.25 ],[ 0.497, -0.654],[-1.27,   1.248],[-1.185,  1.814],[-1.843,  4.386],[-1.616,  1.401],[ 0.052, -1.232],[-3.145,  0.404],[ 0.672, -1.655],[ 2.202, -1.888],[ 4.084, -1.067],[ 1.006, -1.671],[-2.255,  1.51 ],[-1.264,  1.663],[ 1.897, -2.217],[ 1.843, -1.276],[-1.693,  1.623],[ 2.297, -1.709]])
Sid=np.array([-1,  1, -1,  1, -1, -1, -1, -1,  1, -1,  1,  1,  1,  1, -1, -1,  1,  1, -1,  1])

# func to be minimized 
def OptFunc(x):
    return 0.5*(x[0]**2+x[1]**2)
def JacOptFunc(x):
    return np.array([x[0],x[1],0.0])  

# Constraints
def constraint_maker(i=0):  # i MUST be an optional keyword argument, else it will not work
    def constraint(x):
        return Sid[i]*(x[0]*Pct[i,0]+x[1]*Pct[i,1]+x[2])-1 
    return constraint

c=[]
for i in range(len(Sid)):
    c+=[{'type': 'ineq', 'fun': constraint_maker(i)}]
cons=tuple(c)

# start optimization
res = minimize(OptFunc,(0.3,-0.2,0.1),constraints=cons,method='SLSQP',jac=JacOptFunc)

#expected solution should be around
# [0.5025062702615434, -0.584685257866671, 0.36787016514022236]
print("-->",res.message)
print("solution ",res.x)

print("Check Constraints")
cons=list(cons)
for i in range(len(cons)):
    lokfun=c[i]['fun']
    print("Constraint # ",i," value: ",lokfun(res.x))

结果

--> Optimization terminated successfully.
solution  [ 0.52374351 -0.56495542  0.37021863]
Check Constraints
Constraint #  0  value:  0.7735403550593944
Constraint #  1  value:  0.6459722649608017
Constraint #  2  value:  1.7715790719554194
Constraint #  3  value:  8.137268636687622e-11
Constraint #  4  value:  -2.2235047136831554e-10
Constraint #  5  value:  0.27524657110337936
Constraint #  6  value:  2.0729351509689136
Constraint #  7  value:  0.2676534344356165
Constraint #  8  value:  0.09347837249122604
Constraint #  9  value:  0.5051967055706261
Constraint #  10  value:  0.6571754935710583
Constraint #  11  value:  1.5901376792721638
Constraint #  12  value:  2.1119945643862095
Constraint #  13  value:  0.8411451130595076
Constraint #  14  value:  0.6639056792092357
Constraint #  15  value:  0.23131403951409935
Constraint #  16  value:  1.6162662427554526
Constraint #  17  value:  1.0563610395273058
Constraint #  18  value:  0.43340178883510116
Constraint #  19  value:  1.5387662919992176