Diminish means "to make smaller, less, or less important," "to reduce, shrink, decrease, or contract": "Two aspirin tablets will diminish your pain." Minimize means "to reduce to the smallest possible amount, degree, size, or extent" and also means "to belittle": "It is always unwise to minimize the horrors of war." "Since I am doing the best I can, you should not try to minimize my efforts." Minimize is an absolute term and should not be accompanied by adverbs such as somewhat and greatly. Instead of writing greatly minimize, write diminish or reduce.
Dictionary of problem words and expressions. Harry Shaw. 1975.