Formulation c[i]=min(a[i],b[i])

> How to formulate c[i]=min(a[i],b[i]) for use with Cplex. A and b are integer.

There is not much we can exploit knowing that a, b are integer. In this formulation I use an extra binary variable bv[i] that implements the OR-functionality. The AMPL/Cplex formulation can look like:

var bv{I} binary;
e1{i in I}: c[i] <= a[i];
e2{i in I}: c[i] <= b[i];
e3{i in I}: bv[i]=0 ==> c[i]=a[i] else c[i]=b[i];

e3 implements c[i] = a[i] or c[i] = b[i] and e1,e2 make sure the minimum of the two is chosen.