混合整数编程:每个条件的变量赋值(如果不是) [英] Mixed integer programming: variable assignment per condition (if then else)

问题描述

我(混合)整数编程相对较新，并且对约束的制定感到困惑.

I am relatively new to (mixed) integer programming and got stuck with the formulation of a constraint.

在我的简化模型中，我有一个参数和两个变量，它们是正实数，上限为321.我要表达的逻辑在这里:

In my simplified model I have one Parameter and two Variables that are positive Reals having the value 321 as upper bound. The logic I want to express is here:

if Parameter > Variable1:
    Variable2 = Variable1
else:
    Variable2 = Parameter

**edit** (while Variable1 is always >= Variable2)

实际上可以用线性输入(等式)来描述吗?

Is it actually possible to describe this using linear in(equalities)?

如果有帮助:对于实现，我正在使用Python，Pyomo和最新的gurobi求解器.

If it helps: For the implementation I am using Python, Pyomo and the newest gurobi solver.

感谢您的帮助！

推荐答案

将Variable2设置为等于Variable1和Parameter的最小值或最大值.

Setting Variable2 equal to min or max of Variable1 and Parameter.

min(Parameter,Variable1):

min(Parameter,Variable1):

如果确定目标函数中的Variable2希望"很小，那么您只需要Variable2小于或等于Parameter和Variable1:

If you are sure that Variable2 "wants" to be small in the objective function, then you just need to require Variable2 to be less than or equal to both Parameter and Variable1:

Variable2 <= Variable1
Variable2 <= Parameter

max(Parameter,Variable1):

max(Parameter,Variable1):

如果您确定目标函数中的Variable2希望"很大，那么您只需要求Variable2大于或等于Parameter和Variable1:

If you are sure that Variable2 "wants" to be large in the objective function, then you just need to require Variable2 to be greater than or equal to both Parameter and Variable1:

Variable2 >= Variable1
Variable2 >= Parameter

无论哪种情况:

如果最好将Variable2设置为严格小于min(Parameter,Variable1)/严格大于max(Parameter,Variable1)，那么(除了上述约束之外)您还需要引入一个新的如果Parameter > Variable1:

If there's a chance that it will be optimal to set Variable2 to something strictly less than min(Parameter,Variable1) / strictly greater than max(Parameter,Variable1), then you will also (in addition to the constraints above) need to introduce a new binary variable that equals 1 if Parameter > Variable1:

Parameter - Variable1 <= M * NewVar
Variable1 - Parameter <= M * (1 - NewVar)

其中M是一个大数字.因此，如果Parameter > Variable1则NewVar 必须等于1，而如果Parameter < Variable1则NewVar 必须等于0.

where M is a large number. So, if Parameter > Variable1 then NewVar must equal 1, while if Parameter < Variable1 then NewVar must equal 0.

min(Parameter,Variable1):

min(Parameter,Variable1):

介绍确保Variable2 >= min(Parameter,Variable1)的约束:

Variable2 >= Parameter - M * NewVar
Variable2 >= Variable1 - M * (1 - NewVar)

因此，如果Parameter > Variable1然后NewVar = 1，则第一个约束无效，而第二个约束表示Variable2 >= Variable1.如果Parameter < Variable1然后NewVar = 0，则第一个约束表示Variable2 >= Parameter，第二个约束无效.

So, if Parameter > Variable1 then NewVar = 1, the first constraint has no effect, and the second says Variable2 >= Variable1. If Parameter < Variable1 then NewVar = 0, the first constraint says Variable2 >= Parameter, and the second constraint has no effect.

max(Parameter,Variable1):

max(Parameter,Variable1):

介绍确保Variable2 <= max(Parameter,Variable1)的约束:

Variable2 <= NewVar * Parameter + M * (1 - NewVar)
Variable2 <= Variable1 + M * NewVar

因此，如果Parameter > Variable1然后NewVar = 1，则第一个约束条件表示为Variable2 <= Parameter，而第二个约束条件则无效.如果Parameter < Variable1然后NewVar = 0，则第一个约束无效，第二个约束表示Variable2 <= Variable1.

So, if Parameter > Variable1 then NewVar = 1, the first constraint says Variable2 <= Parameter, and the second constraint has no effect. If Parameter < Variable1 then NewVar = 0, the first constraint has no effect, and the second says Variable2 <= Variable1.

无论哪种情况:

请注意，M应该尽可能小，同时仍要确保在约束中触发M会使约束不具有约束力.我认为将其设置为|Parameter - Variable1|可能获得的最大值就足够了.通常，这些大质谱"会削弱配方，导致求解时间更长，因此您始终希望它们尽可能小.

Note that M should be as small as possible while still ensuring that triggering the M in the constraint makes the constraint non-binding. I think it's sufficient to set it equal to the largest value that |Parameter - Variable1| can possibly get. In general these "big-Ms" weaken the formulation and result in longer solve times, so you always want them as small as possible.

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