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 someone up for lost
stop expecting that a missing person will be found alive.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.