Except in some rare/specific cases you are far better trying to eliminate algebraic loops entirely from your model (instead of trying to make the algebraic loop easier to numerically solve, which is what the suggestions you are receiving are trying to do).
A somewhat detailed description on ways to do this can be found in the doc at
http://www.mathworks.com/help/simulink/ug/simulating-dynamic-systems.html
For the specific case you give (which you hint is actually simpler than your real code), consider the following:
u = sine wave + y
So for the case u > 0.8 you have, if sine wave + y > 0.8 then sine wave + y = y.
The only solution for this is when sine wave = 0.
And for the case u <=0.8 you have, if sine wave + y <= 0.8 then y = 0.
So for the sine wave <= 0.8 there is a solution, but not otherwise.
So, what happens when sine wave > 0.8?
I'm not sure, and I suspect Simulink (quite rightly) is having problems too.
Phil.