python - 在 Python 中使用 Pyomo 库的强对偶约束

Question

我正在实现一个逆优化问题，该问题不使用 KKT 条件，而是应用强对偶定理(原始最优目标和对偶最优目标相等)。为此，需要制定原始问题和对偶问题。我有 4 个发电机，每个发电机有 8 个能量 block (b)，并且有固定需求。这是每小时的市场清算。 主要问题是简单的市场清算，使用库 Pyomo 如下:

model = ConcreteModel()
model.g1=Var(b, within=NonNegativeReals)
model.g2=Var(b, within=NonNegativeReals)
model.g3=Var(b, within=NonNegativeReals)
model.g4=Var(b, within=NonNegativeReals)

model.obj = Objective(expr=
                      (sum(g1price[i]*model.g1[i] for i in b)+
                       sum(g2price[i]*model.g2[i] for i in b)+
                       sum(g3price[i]*model.g3[i] for i in b)+
                       sum(g4price[i]*model.g4[i] for i in b)))
model.con_power_balance=Constraint(
                      expr=
                      (sum(model.g1[i] for i in b)+
                       sum(model.g2[i] for i in b)+
                       sum(model.g3[i] for i in b)+
                       sum(model.g4[i] for i in b)- demand) == 0)


model.con_g1max=ConstraintList()
for i in b:
    model.con_g1max.add(model.g1[i] <= gsize[i])
model.con_g2max=ConstraintList()
for i in b:
    model.con_g2max.add(model.g2[i] <= gsize[i])
model.con_g3max=ConstraintList()
for i in b:
    model.con_g3max.add(model.g3[i]< = gsize[i])
model.con_g4max=ConstraintList()
for i in b:
    model.con_g4max.add(model.g4[i] <= gsize[i])

假设前面的方程被命名为 (1a-1f)。如果没有指定，则设置为默认值以最小化目标函数。由于原问题是一个最小化问题，所以它的对偶问题一定是一个最大化问题。但它等效于 max[F(x)] == min[-F(x)]，这就是我所应用的。这是对偶问题(方程 2a-2e):

model = ConcreteModel()
model.mu_g1max=Var(b, within=NonNegativeReals)
model.mu_g2max=Var(b, within=NonNegativeReals)
model.mu_g3max=Var(b, within=NonNegativeReals)
model.mu_g4max=Var(b, within=NonNegativeReals)
model.mu_g1min=Var(b, within=NonNegativeReals)
model.mu_g2min=Var(b, within=NonNegativeReals)
model.mu_g3min=Var(b, within=NonNegativeReals)
model.mu_g4min=Var(b, within=NonNegativeReals)
model.lambda=Var(b, within=NonNegativeReals)

model.obj = Objective(expr=
                      (sum(gsize[i]*model.mu_g1max[i] for i in b)+
                       sum(gsize[i]*model.mu_g2max[i] for i in b)+
                       sum(gsize[i]*model.mu_g3max[i] for i in b)+
                       sum(gsize[i]*model.mu_g4max[i] for i in b))

model.con_g1_dual=ConstraintList()
for i in b:
    model.con_g1_dual.add(model.lambda[i]+model.mu_g1min[i]-model.mu_g1max <= gsize[i])
model.con_g2_dual=ConstraintList()
for i in b:
    model.con_g2_dual.add(model.lambda[i]+model.mu_g2min[i]-model.mu_g2max <= gsize[i])
model.con_g3_dual=ConstraintList()
for i in b:
    model.con_g3_dual.add(model.lambda[i]+model.mu_g3min[i]-model.mu_g3max <= gsize[i])
model.con_g4_dual=ConstraintList()
for i in b:
    model.con_g4_dual.add(model.lambda[i]+model.mu_g4min[i]-model.mu_g4max <= gsize[i])

一旦提出原始问题和对偶问题，我必须解决的真正问题如下:

model = ConcreteModel()
model.lambda=Var(b, within=NonNegativeReals)
model.obj = Objective(expr=
                      np.absolute(sum(model.lambda[i]-lambda_ini[i] for i in b*4)))

在这里，我必须将之前编写的对偶问题 (2b-2e) 中的约束作为约束，并应用强对偶约束:

min(目标函数原问题)= max(目标函数对偶问题)

这是我的问题:最后一个约束如何编写？

Answer 1

嗯，我不太明白你的问题，我还不能发表评论，所以我发布一个答案，希望它能对你有所帮助。

首先在pyomo中，就像@Qi Chen提到的那样，您可以访问由求解器本身生成的对偶值。为此，您需要在模型中添加以下内容:

model.dual = Suffix(direction=Suffix.IMPORT)

然后您可以在其他地方调用模型的实例，并且可以通过以下方式获取模型内部的约束的对偶值:

dual_value = inst.dual[inst.constraint_name]

例如:

dual_con_g1max = inst.dual[inst.con_g1max]

但是，由于 con_g1max 是一个 Constraint_List，因此您需要将其索引提供给对偶获取算法。像:

inst.dual[inst.con_g1max[1]]