DOcplex:对偶的对偶和原始的不一样吗？

Question

我有一个非常简单的原始问题定义如下：

mdl = Model(name='tubs_primal')

aqua = mdl.continuous_var(name='aqua')
hydro = mdl.continuous_var(name='hydro')
typhoon = mdl.continuous_var(name='typhoon')

pump = mdl.add_constraint(aqua + hydro + typhoon <= 200, 'pump')
hour = mdl.add_constraint(9*aqua + 6*hydro + 8*typhoon<= 1566, 'hour')
tubing = mdl.add_constraint(12*aqua+16*hydro + 13*typhoon <= 2880, 'tubing')

profit = 350*aqua + 300*hydro + 320*typhoon

mdl.maximize(profit)

我还写下了这样的双重问题

mdl = Model(name='tubs_dual')

pump = mdl.continuous_var(name='pump', lb=None)    
hour = mdl.continuous_var(name='hour', lb=None)    
tubing = mdl.continuous_var(name='tubing', lb=None)

aqua = mdl.add_constraint(pump + 9*hour + 12*tubing >= 350, 'aqua')
hydro = mdl.add_constraint(pump + 6*hour + 16*tubing >= 300, 'hydro')
typhoon = mdl.add_constraint(pump + 8*hour + 13*tubing >= 320, 'typhoon')

cost = 200*pump + 1566*hour + 2880*tubing

mdl.minimize(cost)

原始代码工作得很完美。然而，当我运行对偶问题时，CPLEX告诉我，对偶目标和原始目标是不同的。那么对偶的对偶问题与原始问题是不一样的？我很困惑。有什么帮助吗？谢谢。

Iteration log . . .
Iteration:     1   Dual objective     =         60900.000000
objective: 66100.000
  pump=200.000
  hour=16.667

Answer 1

原因是日志显示不完整。假设用原始方法求解两个模型:原始模型和对偶模型。

from docplex.mp.model import Model

mdl = Model(name='tubs_primal')

aqua = mdl.continuous_var(name='aqua')
hydro = mdl.continuous_var(name='hydro')
typhoon = mdl.continuous_var(name='typhoon')

pump = mdl.add_constraint(aqua + hydro + typhoon <= 200, 'pump')
hour = mdl.add_constraint(9*aqua + 6*hydro + 8*typhoon<= 1566, 'hour')
tubing = mdl.add_constraint(12*aqua+16*hydro + 13*typhoon <= 2880, 'tubing')

profit = 350*aqua + 300*hydro + 320*typhoon
mdl.maximize(profit)

mdl.parameters.lpmethod=1

mdl.solve(log_output=True)

print("obj=",mdl.objective_value)

mdl2 = Model(name='tubs_dual')

pump = mdl2.continuous_var(name='pump', lb=None)    
hour = mdl2.continuous_var(name='hour', lb=None)    
tubing = mdl2.continuous_var(name='tubing', lb=None)

aqua = mdl2.add_constraint(pump + 9*hour + 12*tubing >= 350, 'aqua')
hydro = mdl2.add_constraint(pump + 6*hour + 16*tubing >= 300, 'hydro')
typhoon = mdl2.add_constraint(pump + 8*hour + 13*tubing >= 320, 'typhoon')

cost = 200*pump + 1566*hour + 2880*tubing

mdl2.minimize(cost)
mdl2.parameters.lpmethod=1

mdl2.solve(log_output=True)

print("obj=",mdl2.objective_value)

然后你会得到

Iteration log . . .
Iteration:     1    Objective     =         60900.000000
obj= 66100.0


Iteration log . . .
Iteration:     1    Scaled infeas =             0.000000
Switched to devex.
Iteration:     2    Objective     =         66100.000000
obj= 66100.0