Hi. Grammatically speaking, both are correct depending on how you look at it, but the past simple version is more common. Hamid
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AnonymousI have given up my idea.This portrays the situation as having some relevance for the present. It's almost like saying, I tried and tried, and my idea didn't work, so I am now finished with it. I'm not going to work on the basis of that idea anymore.
AnonymousI gave up my ideaThis portrays the situation as