What is the difference between "give-up" &"give-in" and when we consider one of them is positive or negative?

To "give up" means either to quit or to relinquish something. Here is a sentence that shows the former: "I am going to give up piano lessons because I don't enjoy them anymore". Here is a sentence that shows the latter: "He had to give up his prize money because he cheated at the contest." To "give in", on the other hand", means to surrender to something. Ex: "I tried to stay up, but I had to give in to my tiredness."

DIFFERENCE: Give-up is more physical, and give-in is more mental.

POSITIVE

1. Give-up:    Give-up to a stronger opponent in contact sports or martial arts;  give-up the dream if after looooong years, you get nowhere; give-up in an argumentation whether you realize you are dealing with a better equipped and experienced speaker, or you end up arguing with a mediocre one who talks nonsense, or trash yet thinks he is right. The last one is giving-up more on the action, not the talk.

2. Give-in:     Give-in to someone's better idea or plan. Concede. Again, this is more on the verbal, not the act (of giving-up).

                      Give-in to your good desire, or to someone's, like a child's need to have a new computer for research work. 

                      Give-in to the majority, when it is for the betterment of all concerned parties, like in a union or a football team.

NEGATIVE: 

                   Give-up and give-in become negative in meaning depending on the subject, plan, idea, move or course of an action. If you give-up learning English without trying your best first, then your'e doing wrong. If you give-in to your kids desire to have that new computer set when in fact you just bought one last month, then you are doing wrong. If you give-up not giving-in to the nagging of your wife to buy a more expensive and better looking car from what you have now because she wants to show-off; and you apply for a car loan in the company just to please her, you are obviously, absolutely, surely, undoubtedly wrong. The last makes you end up a loser, financially, and mentally, and all --ly's associated to the mistake you made. That goes on through the payment process, which will take up years!

Hope this helped!  Cheers!

GIVE UP means to stop trying something because you are having success doing it,

 GIVE IN is to letting something happen or give way to something.

Give-up

Man, these questions on this paper are so hard! "I give up" (The person stopped trying,hence gave up) 

Give in

(Police) He thinks he can hide in there forever,he will eventually give in (as in give up* and let the police arrest him)  

Or 

It's alreadyam! my teacher told me to "give in" this assignment yesterday,I hope I don't miss the deadline again.

I hope this makes it easier to understand