I have intermediate knowledge of optimization and mathematical modeling I have this constraint. I know how to model it with integers (which leads to a mixed-integer linear program). However,I was wondering if there is a way to avoid integer variables.
let: $x_1, x_2$ have undefined signal, they can be positive or negative, but both of them are bounded by an upper and lower limit $$ x_1 = x_{1}^+ + x_1^-\\ \underline{x}_1 \lt x_1 \lt \bar{x}_1 \\ \underline{x}_1 \le x_1^- \le 0, \quad \quad 0 \le x_1^+ \le \bar{x}_1 $$ similarly $$ \\ x_2 = x_2^+ + x_2^- \\ \underline{x}_2 \lt x_2 \lt \bar{x}_2 \\ \underline{x}_2 \le x_2^- \le 0, \quad \quad 0 \le x_2^+ \le \bar{x}_2 $$
These above are just definitions. Now what I want to do is: $$ x_1 \times x_2 \ge 0 $$ which basically means: $x_1$ and $x_2$ must have the same sign or be zero Another way to describe this constraint is or: $$ x_1^+ > 0 \Rightarrow x_2^- = 0 \\ x_2^+ > 0 \Rightarrow x_1^- = 0 $$
I know how to do it with auxiliary integer variables; I was trying to avoid integers in my formulation to continue using a QP or an LP solver.
I tried a big-M constraints and failed. I mention it here to show my work, and save you the effort of exploring it: $$ x_1^- +M\cdot x_2^+ \le 0 $$ This is wrong because it forces $x_2^+$ to be always $0$, even when $x_1^-=0$
$$ \frac{x_1^-}{\underline{x}_1} -\frac{x_2^+}{\bar{x_2}} \le 0 $$ Normalize both variables to handle a percentage, or a surrogate between 0 and 1. Not effective either because it allows $x_2^+$ to be near its upper bound, while $x_1^-$ is slightly below zero.
It has just occurred to me that I can achieve my goal with a penalty on their product.
$$ Penalty = M\cdot(-x_2^- \times x_1^+) $$
Where $M$ is large enough. Doesn't this qualify as a quadratic cost term?