Maximizing expected utility over a knapsack constraint

Yu, Jiajin; Ahmed, Shabbir

doi:10.1016/j.orl.2015.12.016

Cited by 7 publications

(7 citation statements)

References 14 publications

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“…Li and Deshpande [20] obtain a PTAS for the expected utility maximization (EUM) problem for several classes of utility functions (including for example increasing concave functions which typically indicate risk-averseness), and a large class of feasibility constraints (including cardinality constraint, s-t simple paths, matchings, and knapsacks). Similar results for other utility functions and feasibility constraints can be found in [27,21,4]. In the online problem, we can apply our algorithms, using their PTASs as the offline oracle.…”

Section: Applicationssupporting

confidence: 71%

“…We remark that the above independence assumption is also made for past studies on the offline EUM and K-MAX problems [27,20,21,4,13], so it is not an extra assumption for the online learning case. Assumption 2 (Bounded reward value).…”

Section: Setup and Notationmentioning

confidence: 99%

“…Expected utility maximization (EUM) encompasses a large class of stochastic optimization problems and has been well studied (e.g. [27,20,21,4]). To the best of our knowledge, we are the first to study the online learning version of these problems, and we provide a general solution to systematically address all these problems as long as there is an available offline (approximation) algorithm.…”

Section: Related Workmentioning

confidence: 99%

“…Beyond the K-MAX problem, many expected utility maximization (EUM) problems are studied in stochastic optimization literature [27,20,21,4]. The problem can be formulated as maximizing E[u( i∈S X i )] among all feasible sets S, where X i 's are independent random variables and u(•) is a utility function.…”

Section: Introductionmentioning

confidence: 99%

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Combinatorial Multi-Armed Bandit with General Reward Functions

Chen¹,

Hu²,

Li³

et al. 2016

Preprint

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In this paper, we study the stochastic combinatorial multi-armed bandit (CMAB) framework that allows a general nonlinear reward function, whose expected value may not depend only on the means of the input random variables but possibly on the entire distributions of these variables. Our framework enables a much larger class of reward functions such as the max() function and nonlinear utility functions. Existing techniques relying on accurate estimations of the means of random variables, such as the upper confidence bound (UCB) technique, do not work directly on these functions. We propose a new algorithm called stochastically dominant confidence bound (SDCB), which estimates the distributions of underlying random variables and their stochastically dominant confidence bounds. We prove that SDCB can achieve O(log T ) distribution-dependent regret and Õ( √ T ) distribution-independent regret, where T is the time horizon. We apply our results to the K-MAX problem and expected utility maximization problems. In particular, for K-MAX, we provide the first polynomial-time approximation scheme (PTAS) for its offline problem, and give the first Õ( √ T ) bound on the (1 − ǫ)approximation regret of its online problem, for any ǫ > 0.

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Section: Applicationssupporting

confidence: 71%

Section: Setup and Notationmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Combinatorial Multi-Armed Bandit with General Reward Functions

Chen¹,

Hu²,

Li³

et al. 2016

Preprint

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“…Together with his coauthors, Shabbir Ahmed made substantial contributions in advancing the state-of-the-art of mathematical programming methods for solving this challenging set of problems (Ahmed and Atamtürk [2], Yu and Ahmed [38,39]). In this article, we study optimization problems in which the expected value of the function f composed with a set union operator is maximized over a discrete feasible region.…”

Section: Introductionmentioning

confidence: 99%

Submodular maximization of concave utility functions composed with a set-union operator with applications to maximal covering location problems

2022

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We study a family of discrete optimization problems asking for the maximization of the expected value of a concave, strictly increasing, and differentiable function composed with a set-union operator. The expected value is computed with respect to a set of coefficients taking values from a discrete set of scenarios. The function models the utility function of the decision maker, while the set-union operator models a covering relationship between two ground sets, a set of items and a set of metaitems. This problem generalizes the problem introduced by Ahmed S, Atamtürk A (Mathematical programming 128(1-2):149–169, 2011), and it can be modeled as a mixed integer nonlinear program involving binary decision variables associated with the items and metaitems. Its goal is to find a subset of metaitems that maximizes the total utility corresponding to the items it covers. It has applications to, among others, maximal covering location, and influence maximization problems. In the paper, we propose a double-hypograph decomposition which allows for projecting out the variables associated with the items by separately exploiting the structural properties of the utility function and of the set-union operator. Thanks to it, the utility function is linearized via an exact outer-approximation technique, whereas the set-union operator is linearized in two ways: either (i) via a reformulation based on submodular cuts, or (ii) via a Benders decomposition. We analyze from a theoretical perspective the strength of the inequalities of the resulting reformulations, and embed them into two branch-and-cut algorithms. We also show how to extend our reformulations to the case where the utility function is not necessarily increasing. We then experimentally compare our algorithms inter se, to a standard reformulation based on submodular cuts, to a state-of-the-art global-optimization solver, and to the greedy algorithm for the maximization of a submodular function. The results reveal that, on our testbed, the method based on combining an outer approximation with Benders cuts significantly outperforms the other ones.

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Maximizing a class of submodular utility functions with constraints

Ahmed

2016

Math. Program.

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Maximizing expected utility over a knapsack constraint

Cited by 7 publications

References 14 publications

Combinatorial Multi-Armed Bandit with General Reward Functions

Combinatorial Multi-Armed Bandit with General Reward Functions

Submodular maximization of concave utility functions composed with a set-union operator with applications to maximal covering location problems

Maximizing a class of submodular utility functions with constraints

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