[AMPL 2627] obtaining minimum between two variables

[plozmar]

Hi,

I need to obtain the minimum between two variables. The idea is the
following:

--------------
set NODES;

var X {NODES};
var Y {NODES};
var Z {NODES};

minimize:

sum {n in NODES} Z[n];

s.t.
s.t.
....

subject to minimum {n in NODES}:

Z[n]  = min { X[n], Y[n] };

---------------------------------------------
I'm working with cplex and all the constraints must be linear and the
variables X and Y contains a cost, with different elements I have
calculated too with the other constraints...

Can I make it with CPLEX?

Thanks,

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[Paul A. Rubin]

On Jul 6, 2:24 pm, plozmar <lozano_pal...@...> wrote:

> Hi,
>
> I need to obtain the minimum between two variables. The idea is the
> following:
>
> --------------
> set NODES;
>
> var X {NODES};
> var Y {NODES};
> var Z {NODES};
>
> minimize:
>
> sum {n in NODES} Z[n];
>
> s.t.
> s.t.
> ....
>
> subject to minimum {n in NODES}:
>
> Z[n]  = min { X[n], Y[n] };
>
> ---------------------------------------------
> I'm working with cplex and all the constraints must be linear and the
> variables X and Y contains a cost, with different elements I have
> calculated too with the other constraints...
>
> Can I make it with CPLEX?
>

I assume X and Y (and hence Z) are nonnegative.  If you know upper
bounds for X and Y, and can cope with binary variables (making the
model a MILP), then yes.

var B {NODES} binary;
param MX {NODES};  # MX[n] is a valid a priori upper bound on X[n]
param MY {NODES};  # MY[n] is a valid a priori upper bound on Y[n]

...

s.t. Zdef1 {n in NODES}: Z[n] >= X[n] - MX[n]*B[n];
s.t. Zdef2 {n in NODES}: Z[n] >= Y[n] - MY[n]*(1 - B[n]);

If Z[n] is not used anywhere in the model other than the objective,
this is sufficient; the solver will set B[n] based on whichever value
produces the smaller value of Z[n].  If Z[n] is used in other
constraints, so that there is a danger the solver might "want" to pick
a value of Z[n] greater than the min of X[n] and Y[n], then you also
need the following:

s.t. Xsmaller {n in NODES}: X[n] <= Y[n] + MX[n]*B[n];        # X <= Y
if B = 0
s.t. Ysmaller {n in NODES}: Y[n] <= X[n] + MY[n]*(1 - B[n]);  # Y <= X
if B = 1
s.t. Zmax1 {n in NODES}: Z[n] <= X[n] + MX[n]*B[n];
s.t. Zmax2 {n in NODES}: Z[n] <= Y[n] + MY[n]*(1 - B[n]);

/Paul
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