define appropriate big-M in linearize KKT Complementary slackness conditions 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 $s,\lambda$).
