如何解决Gekko 错误地找不到整数解
以下 MINLP 问题返回 Warning: no more possible trial points and no integer solution
。我认为这是错误的,因为至少 b = [[1,1,1],[1,0]]
是一个可行的点。
当切换到 m.options.soLVER=3
时,我确实得到了一个合理的非整数解。当我使用更简单的目标函数(即 m.Obj(-sig[0][0])
)测试完全相同的脚本时,使用 m.options.soLVER=1
,Gekko 找到了整数解。
from gekko import GEKKO
import numpy as np
import pandas as pd
info_df = pd.DataFrame({'pos': ['qb','qb','rb','wr','wr'],'cost': [30,40,15,20,30]})
budget = 80
mu_post = np.array([15,16,8,9,10,14,15])
n = mu_post.shape[0]
sigma_post = np.identity(n)
for i in range(n):
sigma_post[i][i] = i+1
def get_football_position_range(pos,df):
return (df[df['pos'] == pos].index[0],df[df['pos'] == pos].index[-1])
qb_index_range = get_football_position_range('qb',info_df)
rb_index_range = get_football_position_range('rb',info_df)
wr_index_range = get_football_position_range('wr',info_df)
# Number of lineups
N = 2
pi = 3.14159
eps = 1.0E-6
def normal_cdf(x,m):
return 1/(1+m.exp(-1.65451*x))
def normal_pdf(x,m):
return (1/((2*pi)**(.5)))*m.exp((x**2)/2)
def theta(s,m):
return m.sqrt(s[0][0]+s[1][1] - 2*s[0][1])
#################################################
#Integer Optimization Program
m = GEKKO(remote=False)
b = m.Array(m.Var,(N,n),lb=0,ub=1,integer=True,value = 1e-3)
# CONSTRAINT: Each Lineup must be less than budget
z = np.array([None for i in range(N)])
for i in range(N):
z[i] = m.Intermediate(sum(b[i,:]*list(info_df['cost'])))
m.Equations([z[i] <= budget for i in range(N)])
# CONSTRAINT: Each Lineup has one QB
z_1 = np.array([None]*N)
for i in range(N):
z_1[i] = m.Intermediate(sum(b[i,qb_index_range[0]: qb_index_range[1]+1]))
m.Equations([z_1[i] == 1 for i in range(N)])
# CONSTRAINT: Each Lineup has one RB
z_2 = np.array([None for i in range(N)])
for i in range(N):
z_2[i] = m.Intermediate(sum(b[i,rb_index_range[0]: rb_index_range[1]+1]))
m.Equations([z_2[i] == 1 for i in range(N)])
# CONSTRAINT: Each Lineup has one WR
z_3 = np.array([None for i in range(N)])
for i in range(N):
z_3[i] = m.Intermediate(sum(b[i,wr_index_range[0]: wr_index_range[1]+1]))
m.Equations([z_3[i] == 1 for i in range(N)])
# OBJECTIVE: Maximize
mu = b@mu_post
sig = b@sigma_post@b.T
inter = m.if3(theta(sig,m)-eps,.5*mu[0]+.5*mu[1],(mu[0]*normal_cdf((mu[0]-mu[1])/theta(sig,m),m) + \
mu[1]*normal_cdf((mu[1]-mu[0])/theta(sig,m) + \
theta(sig,m)*normal_pdf((mu[0]-mu[1])/theta(sig,m)))
m.Obj(-inter)
m.options.soLVER = 1
m.solve(debug=0,disp=True)
请注意,这是对 Gekko returning incorrect successful solution 的后续问题,未完全指定。
解决方法
问题出在求解器以初始猜测 1e-3
开始时。以下是获得成功解决方案的两种方法。
- 使用您建议的初始猜测:
bv = np.array([[1,1,1],[1,0]])
for i in range(N):
for j in range(n):
b[i,j].value = bv[i,j]
然而,这通常不切实际,尤其是对于大问题。
- 先用
b
初始化value=1
并用IPOPT
求解,然后用APOPT
求解。
m.Maximize(inter)
m.options.SOLVER = 3
m.solve(debug=0,disp=True)
m.options.SOLVER = 1
m.solve(debug=0,disp=True)
print(b)
这给出了一个使用 IPOPT(非整数解)的成功解法,它改进了 IPOPT(整数解法)求解器寻找解法。
Number of state variables: 20
Number of total equations: - 11
Number of slack variables: - 4
---------------------------------------
Degrees of freedom : 5
----------------------------------------------
Steady State Optimization with APOPT Solver
----------------------------------------------
Iter: 1 I: 0 Tm: 0.06 NLPi: 21 Dpth: 0 Lvs: 3 Obj: -3.90E+01 Gap: NaN
Iter: 2 I: -9 Tm: 2.44 NLPi: 251 Dpth: 1 Lvs: 2 Obj: -3.90E+01 Gap: NaN
Iter: 3 I: 0 Tm: 0.01 NLPi: 7 Dpth: 1 Lvs: 3 Obj: -3.90E+01 Gap: NaN
Iter: 4 I: 0 Tm: 0.02 NLPi: 11 Dpth: 1 Lvs: 4 Obj: -3.90E+01 Gap: NaN
Iter: 5 I:-11 Tm: 0.00 NLPi: 1 Dpth: 2 Lvs: 3 Obj: -3.90E+01 Gap: NaN
--Integer Solution: -3.90E+01 Lowest Leaf: -3.90E+01 Gap: 0.00E+00
Iter: 6 I: 0 Tm: 0.00 NLPi: 1 Dpth: 2 Lvs: 3 Obj: -3.90E+01 Gap: 0.00E+00
Successful solution
---------------------------------------------------
Solver : APOPT (v1.0)
Solution time : 2.6365 sec
Objective : -39.000000008290634
Successful solution
---------------------------------------------------
[[[0.0] [1.0] [0.0] [1.0] [0.0] [1.0] [0.0]]
[[0.0] [1.0] [0.0] [1.0] [0.0] [1.0] [0.0]]]
完整代码如下:
from gekko import GEKKO
import numpy as np
import pandas as pd
info_df = pd.DataFrame({'pos': ['qb','qb','rb','wr','wr'],'cost': [30,40,15,20,30]})
budget = 80
mu_post = np.array([15,16,8,9,10,14,15])
n = mu_post.shape[0]
sigma_post = np.identity(n)
for i in range(n):
sigma_post[i][i] = i+1
def get_football_position_range(pos,df):
return (df[df['pos'] == pos].index[0],df[df['pos'] == pos].index[-1])
qb_index_range = get_football_position_range('qb',info_df)
rb_index_range = get_football_position_range('rb',info_df)
wr_index_range = get_football_position_range('wr',info_df)
# Number of lineups
N = 2
pi = 3.14159
eps = 1.0E-6
def normal_cdf(x,m):
return 1/(1+m.exp(-1.65451*x))
def normal_pdf(x,m):
return (1/((2*pi)**(.5)))*m.exp((x**2)/2)
def theta(s,m):
return m.sqrt(s[0][0]+s[1][1] - 2*s[0][1])
#################################################
#Integer Optimization Program
m = GEKKO(remote=False)
b = m.Array(m.Var,(N,n),lb=0,ub=1,integer=True,value=1)
#bv = np.array([[1,0]])
#for i in range(N):
# for j in range(n):
# b[i,j]
# CONSTRAINT: Each Lineup must be less than budget
z = np.array([None for i in range(N)])
for i in range(N):
z[i] = m.Intermediate(sum(b[i,:]*list(info_df['cost'])))
m.Equations([z[i] <= budget for i in range(N)])
# CONSTRAINT: Each Lineup has one QB
z_1 = np.array([None]*N)
for i in range(N):
z_1[i] = m.Intermediate(sum(b[i,qb_index_range[0]: qb_index_range[1]+1]))
m.Equations([z_1[i] == 1 for i in range(N)])
# CONSTRAINT: Each Lineup has one RB
z_2 = np.array([None for i in range(N)])
for i in range(N):
z_2[i] = m.Intermediate(sum(b[i,rb_index_range[0]: rb_index_range[1]+1]))
m.Equations([z_2[i] == 1 for i in range(N)])
# CONSTRAINT: Each Lineup has one WR
z_3 = np.array([None for i in range(N)])
for i in range(N):
z_3[i] = m.Intermediate(sum(b[i,wr_index_range[0]: wr_index_range[1]+1]))
m.Equations([z_3[i] == 1 for i in range(N)])
# OBJECTIVE: Maximize
mu = b@mu_post
sig = b@sigma_post@b.T
inter = m.if3(theta(sig,m)-eps,.5*mu[0]+.5*mu[1],(mu[0]*normal_cdf((mu[0]-mu[1])/theta(sig,m),m) + \
mu[1]*normal_cdf((mu[1]-mu[0])/theta(sig,m) + \
theta(sig,m)*normal_pdf((mu[0]-mu[1])/theta(sig,m)))
m.Maximize(inter)
m.options.SOLVER = 3
m.solve(debug=0,disp=True)
m.options.SOLVER = 1
m.solve(debug=0,disp=True)
print(b)
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