Hi, i'm a bit in struggle with the optimization below and I'd be grateful for any help or hint. The problem deals with the "relaxation" of the nonnegativity constraint, i.e. my decision variables x can be negative. The cost function c_i(x) shall stay positive. Assume that x is bounded in between a lower and upper bound. Let
So the problem needs to be minimized:
min z(x) = sum(c(x)) s.t. where c(x) = n, m are constant. Do you know any to convert this function (e.g. by using binary variables?) such that I am able to solve this problem by a linear solver? Thank you in advance. asked 13 Jul '16, 05:00 Thomas 
Under the following conditions, you can model it without any binary variables:
The reason for the second condition is that for this approach to work, there cannot be any "pressure" for \(c(x)\) to be larger than the conditional definition you gave. Under those conditions (and the objective function you named), you just need the following two constraints (elementwise if \(x\) is a vector, which seems likely but is a bit unclear given your notation): \(c(x) \ge m * x,\) \(c(x) \ge n * x.\) For any nonzero \(x\), the rightside of one inequality will be negative, and the rightside of the other will be the correct value of \(c(x)\). answered 13 Jul '16, 15:34 Paul Rubin ♦♦ Sorry for the misunderstanding notation. Your approach worked fine for my problem. Thank you very much!
(15 Jul '16, 04:37)
Thomas
