如何使用 Google 的 CP-SAT 求解器计算“AddAbsEquality”或“AddMultiplicationEqualit”进行非线性优化？

问题描述

我的目标是根据预测序列恢复数据序列。假设原始数据序列是 x_org = [10,20,30,40,50] 但我收到的随机数据为 x_ran = [50,10,30]。现在，我的目标是通过让它们最接近原始模式来恢复模式（最小化恢复损失）。

我使用了与 Google OR-tool 网站 [https://developers.google.com/optimization/assignment/assignment_teams] 上提供的“Assignment with Teams of Workers”和“Solving an Optimization Problem”几乎相似的方法，以及[https://developers.google.com/optimization/cp/integer_opt_cp]。

我可以最小化损失的总和（误差），但无法计算平方和/绝对和。

from ortools.sat.python import cp_model

x_org = [10,50]
x_ran = [50,30]
n = len(x_org)


model = cp_model.CpModel()

# Defidning recovered data
x_rec = [model.NewIntvar(0,10000,'x_rec_%i') for i in range(n)]

# Defidning recovery loss        
x_loss = [model.NewIntvar(0,'x_loss_%i' % i) for i in range(n)]

# Defining a (recovery) mapping matrix 
M = {}
for i in range(n):
    for j in range(n):
        M[i,j] = model.NewBoolVar('M[%i,%i]' % (i,j)) 
    
# -----------------Constraints---------------%
# Each sensor is assigned one unique measurement.
for i in range(n):
    model.Add(sum([M[i,j] for j in range(n)]) == 1)

# Each measurement is assigned one unique sensor.
for j in range(n):
    model.Add(sum([M[i,j] for i in range(n)]) == 1)


# Recovering the remapped data x_rec=M*x_ran (like,Ax =b)
for i in range(n):   
    model.Add(x_rec[i] == sum([M[i,j]*x_ran[j] for j in range(n)]))

# Loss = orginal data - recovered data
for i in range(n):
    x_loss[i] = x_org[i] - x_rec[i]

    
# minimizing recovery loss
model.Minimize(sum(x_loss))

#--------------- Calling solver -------------%

# Solves and prints out the solution.
solver = cp_model.cpsolver()
status = solver.solve(model)

print('Solve status: %s' % solver.StatusName(status))

if status == cp_model.OPTIMAL:
    print('Optimal objective value: %i' % solver.ObjectiveValue())
    for i in range(n):
        print('x_loss[%i] = %i' %(i,solver.Value(x_loss[i]))) 

那么没有绝对误差和的输出是：

Solve status: OPTIMAL
Optimal objective value: 0
x_loss[0] = -10
x_loss[1] = -30
x_loss[2] = 0
x_loss[3] = 30
x_loss[4] = 10

这表明即使损失总和为零，恢复也不正确。但是，当我尝试添加另一个int变量来存储损失的绝对值时[如下图]，编译器报错。

# Defidning abs recovery loss        
x_loss_abs = [model.NewIntvar(0,'x_loss_abs_%i' % i) for i in range(n)] 
# Loss = orginal data - recovered data
for i in range(n):
    model.AddAbsEquality(x_loss_abs[i],x_loss[i])
    #model.AddMultiplicationEquality(x_loss_abs[i],[x_loss[i],x_loss[i]])

回溯的错误是：

TypeError                                 Traceback (most recent call last)
<ipython-input-42-2a043a8fef8b> in <module>
      3 # Loss = orginal data - recovered data
      4 for i in range(n):
----> 5     model.AddAbsEquality(x_loss_abs[i],x_loss[i])

~/anaconda3/envs/tensorgpu/lib/python3.7/site-packages/ortools/sat/python/cp_model.py in AddAbsEquality(self,target,var)
   1217         ct = Constraint(self.__model.constraints)
   1218         model_ct = self.__model.constraints[ct.Index()]
-> 1219         index = self.GetorMakeIndex(var)
   1220         model_ct.int_max.vars.extend([index,-index - 1])
   1221         model_ct.int_max.target = self.GetorMakeIndex(target)

~/anaconda3/envs/tensorgpu/lib/python3.7/site-packages/ortools/sat/python/cp_model.py in GetorMakeIndex(self,arg)
   1397         else:
   1398             raise TypeError('NotSupported: model.GetorMakeIndex(' + str(arg) +
-> 1399                             ')')
   1400 
   1401     def GetorMakeBooleanIndex(self,arg):

TypeError: NotSupported: model.GetorMakeIndex((-x_rec_%i + 10))

您能否建议如何最小化恢复损失的绝对和/平方和？谢谢。

解决方法

AddAbsEquality 要求参数是变量（不是诸如 x_org[i] - x_rec[i] 之类的表达式。因此必须在使用它之前创建一个临时决策变量（此处为 v）。以下似乎是工作：

# ...
x_loss_abs = [model.NewIntVar(0,10000,'x_loss_abs_%i' % i) for i in range(n)]

# ...
for i in range(n):
   # x_loss[i] = x_org[i] - x_rec[i] # Original
   v = model.NewIntVar(-1000,1000,"v") # Temporary variable
   model.Add(v == x_org[i] - x_rec[i] )
   model.AddAbsEquality(x_loss_abs[i],v)

# ....
model.Minimize(sum(x_loss_abs))

解决方案是（我改变了输出）：

Optimal objective value: 0
x_org: [[10,20,30,40,50]]
x_rec: [10,50]
x_loss: [0,0]