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If you have a problem in convex optimization where all $N$ constraints ($N >> 0$) yield no possible solution but you are able to rank, or weight the constraint in terms of their importance are there methods I can look into for approximate solutions which violate constraints minimally (by some, say, squared metric) taking into account the priorities?

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As a first Idea I would suggest to combine the Penalty method with weights for each constraint? en.wikipedia.org/wiki/Penalty_method –  sonystarmap Jan 10 '13 at 20:12
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