New Bounds for the Stochastic Knapsack Problem
Keyword(s):
The knapsack problem is a problem in combinatorial optimization that seeks to maximize the objective function \(\sum_{i = 1}^{n} v_ix_i\) subject to the constraints \(\sum_{i = 1}^{n} w_ix_i \leq W\) and \(x_i \in \{0, 1\}\), where \(\mathbf{x}, \mathbf{v} \in \mathbb{R}^{n}\) and \(W\) are provided. We consider the stochastic variant of this problem in which \(\mathbf{v}\) remains deterministic, but \(\mathbf{x}\)is an \(n\)-dimensional vector drawn uniformly at random from \([0, 1]^{n}\). We establish a sufficient condition under which the summation-bound condition is almost surely satisfied. Furthermore, we discuss the implications of this result on the deterministic problem.
Keyword(s):
2013 ◽
Vol 10
(2)
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pp. 147-154
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Keyword(s):
1997 ◽
Vol 103
(3)
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pp. 584-594
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Keyword(s):
2009 ◽
Vol 176
(1)
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pp. 77-93
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2014 ◽
Vol 24
(3)
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pp. 1485-1506
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2008 ◽
Vol 46
(3)
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pp. 427-450
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