How to linearise constraint including summation and min in an optimisation problem?

Question

I have the following constraint to an optimisation problem:

Σ min(wj,0) ≥ −30 for j = 1,...,n

How can I linearise it introducing only n new non-binary decision variables?

Answer 1

sum(j, min(w(j),0)) >= -30

can be linearized with additional continuous variables y(j):

y(j) <= w(j)
y(j) <= 0
sum(j, y(j)) >= -30

Note that the interpretation of y(j) is a bit difficult. It is y(j) <= min(w(j),0) and not y(j) = min(w(j),0). As long as the limit of -30 is not reached it may be less than expected. So you probably should not report y(j) to the user.

I hope this was not a homework question.