All Poly-APX-complete problems I know are maximization problems, e.g. Max Clique, Max Independent Set, Max One for some set of contraints, and even choosing the attributes of a product to maximize (Max again!) the number of customers preferring it, provided that preferences are additive (http://www.sciencedirect.com/science/article/pii/S1568494617300510). Max, max, max... Is there any known Poly-APX-complete Min problem?
I suspect a problem I'm dealing with might be Poly-APX-complete, but it's a minimization problem. If there were some known Poly-APX-complete Min problem, an AP-reduction from it into my target problem would map Min into Min, so proving the Poly-APX-hardness of my target problem would probably be more natural and easier.