具有不确定性的多种补充资源的开发
Exploitation of Multiple Replenishing Resources with Uncertainty
摘要：
我们考虑一个优化问题，其中一个（单个）蝙蝠的目的是利用花蜜在一组$n$仙人掌的目标是最大限度地提高预期的花蜜饮料总量。每个仙人掌$i \in [n]$的特征在于一个参数$r_{i} > 0$，该参数决定花蜜在$i$中积累的速率。在每一轮中，蝙蝠可以访问一个仙人掌，并喝下自上次访问以来积累的所有花蜜。此外，竞争与其他蝙蝠，也可以访问一些仙人掌和喝他们的花蜜，是仿照一个随机过程中，仙人掌$i$是空的每一轮（独立）的概率$0 < s_i < 1$。我们的注意力仅限于纯随机策略，其特征在于概率向量$(p_1, \ldots, p_n)$确定蝙蝠在每轮访问仙人掌的概率$p_i$ $i$。我们证明，对于每一个$ε> 0$，存在一个纯随机的策略，接近最佳的纯随机策略，在一个乘法因子的$1 + ε$，而利用只有一个小的核心仙人掌。具体而言，我们表明，它足以包括最多$\frac{2 (1 - σ)}{ε\cdot σ}$仙人掌的核心，其中$σ= \min_{i \in [n]} s_{i}$。我们还表明，这个上限的核心大小是渐近最优的一个显着较小的大小的核心不能提供一个$(1 + ε)$ -近似的最佳纯随机策略。这意味着当竞争更加激烈时（即，$σ$更大），基于利用更小内核的策略将是有利的。
英文摘要：
We consider an optimization problem in which a (single) bat aims to exploit the nectar in a set of $n$ cacti with the objective of maximizing the expected total amount of nectar it drinks. Each cactus $i \in [n]$ is characterized by a parameter $r_{i} > 0$ that determines the rate in which nectar accumulates in $i$. In every round, the bat can visit one cactus and drink all the nectar accumulated there since its previous visit. Furthermore, competition with other bats, that may also visit some cacti and drink their nectar, is modeled by means of a stochastic process in which cactus $i$ is emptied in each round (independently) with probability $0 < s_i < 1$. Our attention is restricted to purely-stochastic strategies that are characterized by a probability vector $(p_1, \ldots, p_n)$ determining the probability $p_i$ that the bat visits cactus $i$ in each round. We prove that for every $ε> 0$, there exists a purely-stochastic strategy that approximates the optimal purely-stochastic strategy to within a multiplicative factor of $1 + ε$, while exploiting only a small core of cacti. Specifically, we show that it suffices to include at most $\frac{2 (1 - σ)}{ε\cdot σ}$ cacti in the core, where $σ= \min_{i \in [n]} s_{i}$. We also show that this upper bound on core size is asymptotically optimal as a core of a significantly smaller size cannot provide a $(1 + ε)$-approximation of the optimal purely-stochastic strategy. This means that when the competition is more intense (i.e., $σ$ is larger), a strategy based on exploiting smaller cores will be favorable.
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