Friday, January 7, 2011

Max and Min Functions in a MIP

Previously I wrote about how to work the "positive part" function ($y^+=\max(y,0)$) into a mixed integer programming model. More general uses of the $\max$ and $\min$ functions are really just an extension of this, but the extension may not be entirely obvious. In this post I'll explain how to set $z=\max(x,y)$ in a MIP model, where $x$, $y$ and $z$ are all variables. If the model was in fact a linear program before introducing the $\max$ or $\min$ function, it may remain an LP, or it may become a MIP, depending on exactly what you are doing.  I'll use the $\max$ function for explanatory purposes; the necessary tweaks to deal with the $\min$ function are left to the reader as an exercise.

Let's start with the obvious part: if $z=\max(x,y)$ then $$z\ge x$$ and $$z \ge y.$$In fact, adding those two constraints will be sufficient if the nature of the model ensures that the smallest feasible value of $z$ will always be chosen  Typically this means that (a) $z$ has a positive coefficient in an objective function being minimized, or a negative coefficient in an objective function being maximized, and (b) $z$ does not connect to other variables, through constraints, in a way that might allow an inflated value of $z$ to cause other variables to change enough that their improved objective contribution would more than pay for the damage done by inflating $z$.

There are also occasional cases where inflation of $z$ is not actually a concern: you need $z$ to be at least as big as both $x$ and $y$ (which the constraints ensure), but a larger than necessary value of $z$ will not affect the objective function or the values of other variables, and can be corrected manually once a solution is found.

The (slightly) tricky case is when you cannot be sure that the "pressure" of the  objective function will prevent inflated values of $z$ from occurring, and you really do need the value of $z$ to be correct.  To handle this case, we will need finite bounds for $x$ and $y$, so from here on I will assume that $L_x \le x \le U_x$ and $L_y \le y \le U_y$.  We have to introduce a new binary variable $w$, which will switch values depending on whether $x$ or $y$ is the larger of the two. In addition to the two previous constraints, we add the following:$$z \le x + (U_y - L_x)w$$and$$z \le y + (U_x-L_y)(1-w).$$The combined effect of the constraints is that $w=0\implies z=x\ge y$ and $w=1\implies z=y\ge x$.


  1. Can you shed me a light on this subject regarding my problem? I need to find PoverBudget, which is PoverBudget = (summation_t=1:24) Pbattery - Pbudget . I want the PoverBudget to be zero, if the resulting value is negative or else keep the value if its positive. I am confused how to formulate it, as a constraint or as an objective function. Actually, I need to minimize (Cost_battery * PoverBudget) in the objective function, and I cannot find ways to model that in CPLEX.
    Any help will be greatly appreciated.

    1. Assuming that Cost_battery > 0, the solver will want to give it the smallest value possible (given the rest of the solution). That means you can declare PoverBudget to be a nonnegative variable and then constrain it to be >= the summation you gave. If the sum is negative, the solver will set it to zero, the smallest feasible solution (given the sign restriction).


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  4. Very useful !

    It seems to work for several variables :


    a,b,c have same bounds

    Isn't it ?


    1. Yes, you can "chain" max operations as you described, adding one binary variable for each $z_j$. You can equivalently start with one binary variable for each original variable ($w_a$ corresponding to $a$, $w_b$ corresponding to $b$, ...) with the constraint $w_a + w_b + ... = 1$ (exactly one variable is picked to be maximal), and similar upper bound constraints as before (where the original $w$ would now be $w_y$ and $1-w$ would be $w_x$). The lower bounds $L_a$ etc. would be as before, while the upper bounds would change so that, for example, $U_y$ in the first upper limit constraint would be replaced by the largest upper bound of any variable other than $x$, while $U_x$ would be replaced by the largest upper bound of any variable other than $y$. Either approach should work.


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