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We consider a version of the knapsack problem in which an item size is random and revealed only when the decision maker attempts to insert it. After every successful insertion the decision maker can choose the next item dynamically based on the remaining capacity and available items, while an unsuccessful insertion terminates the process. We propose a new semi-infinite relaxation based on an affine value function approximation, and show that an existing pseudo-polynomial relaxation correspondsdoi:10.1137/15m1036233 fatcat:m76mygtz2vhutdoifwrfua2n6m