Logarithmic regret in the dynamic and stochastic knapsack problem with equal rewards

We study a dynamic and stochastic knapsack problem in which a decision maker is sequentially presented with items arriving according to a Bernoulli process over $n$ discrete time periods. Items have equal rewards and independent weights that are drawn from a known non-negative continuous distributio...

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Hauptverfasser: Arlotto, Alessandro, Xie, Xinchang
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Sprache:eng
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Zusammenfassung:We study a dynamic and stochastic knapsack problem in which a decision maker is sequentially presented with items arriving according to a Bernoulli process over $n$ discrete time periods. Items have equal rewards and independent weights that are drawn from a known non-negative continuous distribution $F$. The decision maker seeks to maximize the expected total reward of the items that she includes in the knapsack while satisfying a capacity constraint and while making terminal decisions as soon as each item weight is revealed. Under mild regularity conditions on the weight distribution $F$, we prove that the regret---the expected difference between the performance of the best sequential algorithm and that of a prophet who sees all of the weights before making any decision---is, at most, logarithmic in $n$. Our proof is constructive. We devise a reoptimized heuristic that achieves this regret bound.
DOI:10.48550/arxiv.1809.02016