When we transform our maximization problem to its equivalent KKT, It becomes non-linear because of Complementary slackness. $$lambda\times S=0,$$ We can easily linearize above condition with replace it with to two inequalities: $$lambda \le M\times Y$$ $$S\le M\times (1-Y)$$ Where $M$ is a very large positive number and $Y$ is a binary variable in which when $Y= 0$, $lambda=0$ and $S$ can take any value and when $Y= 1$, $S=0$ and $lambda$ can take any value.
To improve the quality of solving our problem whit aforementioned method with optimization solvers, it is necessary to define a appropriate $M$.
Does anybody Know how can define a good $M$?
Is there any method to linearize Complementary slackness whitout the presence of big-M?
This is to provide more detail on the comment about the use of SOS1 variables to implement the complementarity condition $\lambda \cdot s = 0$ with $\lambda, s\ge 0$.
Let $y_1,y_2\ge 0$ form a SOS1 set (i.e. only one of them can be nonzero). Then
$$ \begin{align} \lambda \le y_1\\ s \le y_2 \end{align} $$
will do what the poster wants. (Of course, more directly, we can also form a SOS1 set consisting of $s,\lambda$).