I remember we had something called got books in highschool which was 30 minutes where we were supposed to read but most everyone just did homework, I think if you set aside 30 minutes of the day for homework assignments that it could be beneficial, although on the other hand many students may not do the homework at home and then not have enough time to finish the homework. I also like the idea of an optional hour homework session with a teacher after school, but many kids in highschool dont care enough to go, so that may not work either but would help some.
The only way to get 100% HW turn in rates are by not giving HW, sad but true.
Also, I always used an agenda to organize hw assignments, try mandating that they log in their homework assignments into an agenda of some sort, many kids just forget.
Lastly, keep homework within a reasonable amount of problems. Most kids are not going to spend more than an hour on homework, and thats with all the classes combined, so coordinate with the other teachers to keep it in an achievable amount, you dont want kids getting a feeling of learned helplessness because they have to spend 4 hours a day on homework.