On the tightness of an LP relaxation for rational optimization and its applications

Avadhanula, Vashist; Bhandari, Jalaj; Goyal, Vineet; Zeevi, Assaf

doi:10.1016/j.orl.2016.07.001

Cited by 11 publications

(6 citation statements)

References 6 publications

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“…Our approach follows very closely the Primal Dual algorithm with knapsack constraints of [10]. However, we also show how to reduce the problem of finding the optimal bid vector to the problem of maximizing the ratio between a linear function of the UCB of the valuations and a linear function of the LCB of the costs, and this problem can be solved in polynomial time for a set of totally unimodular linear constraints [8]. In Section 4, we use the discretization idea introduced in [11], to discretize the continuous bid space in [p 0 , 1], with p 0 being the minimum reserve price across all platforms, by using an grid.…”

Section: Overview Of Contributionsmentioning

confidence: 98%

“…The number of different arms that is exponential can be reduced in the analysis by pruning out suboptimal arms. The optimal arm according to the U CB and LCB approximations can actually be computed in polynomial time since this is the problem of optimizing a rational function subject to a set of linear constraints described by a totally unimodular matrix [8]. After the feedback is received, the UCB estimation of the rewards and the LCB estimation of the costs are updated.…”

Section: Discrete Bid Spacesmentioning

confidence: 99%

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Stochastic Bandits for Multi-platform Budget Optimization in Online Advertising

Avadhanula¹,

Colini-Baldeschi²,

Leonardi³

et al. 2021

Preprint

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We study the problem of an online advertising system that wants to optimally spend an advertiser's given budget for a campaign across multiple platforms, without knowing the value for showing an ad to the users on those platforms. We model this challenging practical application as a Stochastic Bandits with Knapsacks problem over T rounds of bidding with the set of arms given by the set of distinct bidding m-tuples, where m is the number of platforms. We modify the algorithm proposed in Badanidiyuru et al., [11] to extend it to the case of multiple platforms to obtain an algorithm for both the discrete and continuous bid-spaces. Namely, for discrete bid spaces we give an algorithm with regret O OP T mn B + *

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Section: Overview Of Contributionsmentioning

confidence: 98%

Section: Discrete Bid Spacesmentioning

confidence: 99%

Stochastic Bandits for Multi-platform Budget Optimization in Online Advertising

Avadhanula¹,

Colini-Baldeschi²,

Leonardi³

et al. 2021

Preprint

Self Cite

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“…Proof. We use the result in Avadhanula et al (2016) which shows that the rational optimization problem max…”

Section: Identical Choice Model and Common Assortmentmentioning

confidence: 99%

“…The optimal solution of the linear program (21a)-(21i) can easily be transformed into an optimal solution of the original formulation similar to what is done in Avadhanula et al (2016).…”

Section: Identical Choice Model and Common Assortmentmentioning

confidence: 99%

Constrained multi‐location assortment optimization under the multinomial logit model

Bebitoğlu

Şen

Kaminsky

2023

Naval Research Logistics

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We study the assortment optimization problem in an online setting where a retailer uses multiple distribution centers (DC) to fulfill orders from multiple regions. Customer choice in each region follows a multinomial logit model. Each DC can carry up to a pre‐specified number of products. Outbound shipping cost to a region depends on the DC that ships the order. The problem is to determine which products to carry in each DC and which products to offer for sale in each region to maximize the expected profit. We first show that the problem is NP‐complete. We develop a conic quadratic mixed integer programming formulation and suggest a family of valid inequalities. We also show that a special case with identical choice models can be solved as a linear program. This LP solution approach can be used to develop heuristics for the general case. Numerical experiments show that our conic approach outperforms the mixed integer linear programming formulation and enables us to solve moderately sized instances optimally. The experiments also show that not allowing cross‐shipments or not considering them in assortment decisions may lead to substantial losses and LP‐based heuristics can be effective in practice.

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“…There are efficient polynomial time algorithms available to solve this optimization problem (e.g., refer to Davis et al [10], Avadhanula et al [7] and Rusmevichientong et al [18]). The details of our procedure are provided in Algorithm 1.…”

Section: A Ts Algorithm With Independent Conjugate Beta Priorsmentioning

confidence: 99%

Thompson Sampling for the MNL-Bandit

Agrawal¹,

Avadhanula²,

Goyal³

et al. 2017

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We consider a sequential subset selection problem under parameter uncertainty, where at each time step, the decision maker selects a subset of cardinality K from N possible items (arms), and observes a (bandit) feedback in the form of the index of one of the items in said subset, or none. Each item in the index set is ascribed a certain value (reward), and the feedback is governed by a Multinomial Logit (MNL) choice model whose parameters are a priori unknown. The objective of the decision maker is to maximize the expected cumulative rewards over a finite horizon T , or alternatively, minimize the regret relative to an oracle that knows the MNL parameters. We refer to this as the MNL-Bandit problem. This problem is representative of a larger family of exploration-exploitation problems that involve a combinatorial objective, and arise in several important application domains. We present an approach to adapt Thompson Sampling to this problem and show that it achieves near-optimal regret as well as attractive numerical performance.

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On the tightness of an LP relaxation for rational optimization and its applications

Cited by 11 publications

References 6 publications

Stochastic Bandits for Multi-platform Budget Optimization in Online Advertising

Stochastic Bandits for Multi-platform Budget Optimization in Online Advertising

Constrained multi‐location assortment optimization under the multinomial logit model

Thompson Sampling for the MNL-Bandit

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