give (someone or something) up for lost
To stop pursuing or searching for someone or something because it seems like a futile task. I haven't been able to find that watch for weeks, so I've given it up for lost at this point.
give up for lost
To consider lost or without hope. A noun or pronoun can be used after "give" or after "up." I gave the pizza up for lost when it didn't arrive in two hours. We had nearly given up the situation for lost when help arrived.
give somebody/something up for ˈlost
(formal) stop looking for somebody/something because you no longer expect to find them/it: The fishermen had been given up for lost in the storm but they have now arrived safely back.