sci.op-research

I have an optimization problem, where I want to minimize the objective
function. On the other hand I need to have a constraint which says
that optimal value of the objective function should be greater than or
equal to a specific value:

min z
s.t.
[some other constraints]
z_min >= k

where, k is a scalar, and z_min is the optimal value of the objective
function for the problem below:

min z
s.t.
[some other constraints]

Does anybody have an idea for formulating this problem?

Peter