DOcplex:对偶的对偶和原始的不一样吗

Question

我有一个定义如下的非常简单的原始问题：

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)

我还写下了它的双重问题，比如这个

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)

原始代码运行完美。然而，当我处理双重问题时，CPLEX告诉我双重目标和原始目标是不同的。那么对偶的对偶问题和原始问题不一样吗？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。假设你用原始方法求解原始和对偶两个模型

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