  Semi-online algorithms that are allowed to perform a bounded amount of
repacking achieve guaranteed good worst-case behaviour in a more realistic
setting. Most of the previous works focused on minimization problems that aim
to minimize some costs. In this work, we study maximization problems that aim
to maximize their profit.
  We mostly focus on a class of problems that we call choosing problems, where
a maximum profit subset of a set objects has to be maintained. Many known
problems, such as Knapsack, MaximumIndependentSet and variations of these, are
part of this class. We present a framework for choosing problems that allows us
to transfer offline $\alpha$-approximation algorithms into
$(\alpha-epsilon)$-competitive semi-online algorithms with amortized migration
$O(1/\epsilon)$. Moreover we complement these positive results with lower
bounds that show that our results are tight in the sense that no amortized
migration of $o(1/\epsilon)$ is possible.
