How to model linearly this complex piecewise objective function?

Question

I want to model linearly the following situation : my objective function $f(q)$ should be like this :

$f(q)=A\cdot q+B\cdot x$ if $x=1, \gamma=0, \delta=0$

$f(q)=A' \cdot q+B'\cdot x$ if $x=1, \gamma=1, \delta=0$

$f(q)=A'' \cdot q+B''\cdot x$ if $x=1, \gamma=0, \delta=1$

$x,\delta, \gamma$ are binary variables and when $x=0$, $\delta=0,\gamma=0$ but the reverse is not true.

$A,A',A", B,B',B"$ are constants.

Thanks for help

Answer 1

Your exact requirements aren't clear to me, but it seems like what you want is to model union of polyhedra using mixed-integer linear programming. A good reference for this is this review paper by Vielma. Section 2 of the above paper motivates modeling piecewise linear functions using mixed-integer linear programming, and Section 5 of this paper provides a formal mathematical treatment. Imposing any other logical conditions on the binary variables should be doable using standard mixed-integer linear programming techniques.