使用Benders分解法求解整数规划问题时遇CPLEX错误2002

尝试用Benders分解求解整数规划，将变量x₁、x₂归入主问题，y₁、y₂、y₃归入子问题，但运行代码时触发CPLEX Error 2002: Invalid Benders decomposition。原代码如下：

from docplex.mp.model import Model

# Creating model
my_bdrex=Model('My Benders Model',log_output=True)
 
# Defining variables
x_1=my_bdrex.integer_var(name='x_1', lb=0)
x_2=my_bdrex.integer_var(name='x_2', lb=0)
y_1=my_bdrex.integer_var(name='y_1', lb=0)
y_2=my_bdrex.integer_var(name='y_2', lb=0)
y_3=my_bdrex.integer_var(name='y_3', lb=0)

# Adding constraints
my_bdrex.add_constraint(2*x_1+4*x_2+4*y_1-2*y_2+3*y_3<=12)
my_bdrex.add_constraint(3*x_1+5*x_2+2*y_1+3*y_2-y_3<=10)
my_bdrex.add_constraint(x_1<=2)
my_bdrex.add_constraint(x_2<=2)

# Defining the objective function
objective_bdrex=4*x_1+7*x_2+2*y_1-3*y_2+y_3

# Solving the Model
my_bdrex.maximize(objective_bdrex)

my_bdrex.parameters.benders.strategy = 1

x_1.benders_annotation=0
x_2.benders_annotation=0
y_1.benders_annotation=1
y_2.benders_annotation=1
y_3.benders_annotation=1

my_bdrex_MP.print_information()
print(my_bdrex_MP.export_as_lp_string())
my_bdrex_MP.solve(clean_before_solve=True)
my_bdrex_MP.print_solution()

1. 子问题变量类型不兼容：默认Benders分解要求子问题变量为连续型，原代码中y₁、y₂、y₃被定义为整数变量，这会导致CPLEX无法生成有效的子问题对偶。若需整数子问题，需额外设置相关参数，但通常优先将子问题变量改为连续型验证分解逻辑。

2. 约束结构不符合Benders规则：Benders分解要求主问题约束仅包含主变量（x₁、x₂），子问题约束可包含主变量（作为参数）和子变量（y系列）。原代码中前两个约束同时混合主、子变量，且未标注约束的Benders归属，CPLEX无法自动识别分解结构。

3. 未定义变量引用错误：代码中错误使用了未初始化的my_bdrex_MP，实际应使用创建的模型实例my_bdrex。

from docplex.mp.model import Model

# 创建模型
my_bdrex = Model('My Benders Model', log_output=True)

# 定义变量：主变量为整数，子变量改为连续型
x_1 = my_bdrex.integer_var(name='x_1', lb=0)
x_2 = my_bdrex.integer_var(name='x_2', lb=0)
y_1 = my_bdrex.continuous_var(name='y_1', lb=0)
y_2 = my_bdrex.continuous_var(name='y_2', lb=0)
y_3 = my_bdrex.continuous_var(name='y_3', lb=0)

# 添加约束：分离主问题约束和子问题约束
# 主问题约束（仅含主变量）
my_bdrex.add_constraint(x_1 <= 2, name='main_c1')
my_bdrex.add_constraint(x_2 <= 2, name='main_c2')

# 子问题约束（含主变量作为参数，子变量）
sub_c1 = my_bdrex.add_constraint(4*y_1 - 2*y_2 + 3*y_3 <= 12 - 2*x_1 -4*x_2, name='sub_c1')
sub_c2 = my_bdrex.add_constraint(2*y_1 + 3*y_2 - y_3 <= 10 -3*x_1 -5*x_2, name='sub_c2')

# 标注约束的Benders归属：主约束归0，子约束归1
sub_c1.benders_annotation = 1
sub_c2.benders_annotation = 1

# 定义目标函数
objective_bdrex = 4*x_1 +7*x_2 +2*y_1 -3*y_2 + y_3
my_bdrex.maximize(objective_bdrex)

# 设置Benders策略：1表示自动生成割平面
my_bdrex.parameters.benders.strategy = 1

# 标注变量的Benders归属：主变量0，子变量1
x_1.benders_annotation = 0
x_2.benders_annotation = 0
y_1.benders_annotation = 1
y_2.benders_annotation = 1
y_3.benders_annotation = 1

# 执行求解
my_bdrex.print_information()
print(my_bdrex.export_as_lp_string())
solution = my_bdrex.solve(clean_before_solve=True)
my_bdrex.print_solution()

内容的提问来源于stack exchange，提问作者Mohammad Samiullah