arXiv subjects

Vineet Goyal

Publications and source records attributed to Vineet Goyal.

26 records · Page 2Linked to original sources

arXiv2020

A Generalized Markov Chain Model to Capture Dynamic Preferences and Choice Overload

Assortment optimization is an important problem that arises in many industries such as retailing and online advertising where the goal is to find a subset of products from a universe of substitutable products which maximize seller's expected revenue. One of the key challenges in this problem is to model the customer substitution behavior. Many parametric random utility maximization (RUM) based choice models have been considered in the literature. However, in all these models, probability of purchase increases as we include more products to an assortment. This is not true in general and in many settings more choices hurt sales. This is commonly referred to as the choice overload. In this paper we attempt to address this limitation in RUM through a generalization of the Markov chain based choice model considered in Blanchet et al. (2016). As a special case, we show that our model reduces to a generalization of MNL with no-purchase attractions dependent on the assortment S and strictly increasing with the size of assortment S. While we show that the assortment optimization under this model is NP-hard, we present fully polynomial-time approximation scheme (FPTAS) under reasonable assumptions.

econ.TH↗

arXiv2020

On the Power of Static Assignment Policies for Robust Facility Location Problems

We consider a two-stage robust facility location problem on a metric under an uncertain demand. The decision-maker needs to decide on the (integral) units of supply for each facility in the first stage to satisfy an uncertain second-stage demand, such that the sum of first stage supply cost and the worst-case cost of satisfying the second-stage demand over all scenarios is minimized. The second-stage decisions are only assignment decisions without the possibility of adding recourse supply capacity. This makes our model different from existing work on two-stage robust facility location and set covering problems. We consider an implicit model of uncertainty with an exponential number of demand scenarios specified by an upper bound $k$ on the number of second-stage clients. In an optimal solution, the second-stage assignment decisions depend on the scenario; surprisingly, we show that restricting to a fixed (static) fractional assignment for each potential client irrespective of the scenario gives us an $O(\log k/\log \log k)$-approximation for the problem. Moreover, the best such static assignment can be computed efficiently giving us the desired guarantee.

math.OC↗

arXiv2020

Online Allocation of Reusable Resources via Algorithms Guided by Fluid Approximations

We consider the problem of online allocation (matching and assortments) of reusable resources where customers arrive sequentially in an adversarial fashion and allocated resources are used or rented for a stochastic duration that is drawn independently from known distributions. Focusing on the case of large inventory, we give an algorithm that is $(1-1/e)$ competitive for general usage distributions. At the heart of our result is the notion of a relaxed online algorithm that is only subjected to fluid approximations of the stochastic elements in the problem. The output of this algorithm serves as a guide for the final algorithm. This leads to a principled approach for seamlessly addressing stochastic elements (such as reusability, customer choice, and combinations thereof) in online resource allocation problems, that may be useful more broadly.

cs.DS↗

arXiv2019

Beyond Worst-case: A Probabilistic Analysis of Affine Policies in Dynamic Optimization

Affine policies (or control) are widely used as a solution approach in dynamic optimization where computing an optimal adjustable solution is usually intractable. While the worst case performance of affine policies can be significantly bad, the empirical performance is observed to be near-optimal for a large class of problem instances. For instance, in the two-stage dynamic robust optimization problem with linear covering constraints and uncertain right hand side, the worst-case approximation bound for affine policies is $O(\sqrt m)$ that is also tight (see Bertsimas and Goyal (2012)), whereas observed empirical performance is near-optimal. In this paper, we aim to address this stark-contrast between the worst-case and the empirical performance of affine policies. In particular, we show that with high probability affine policies give a good approximation for two-stage dynamic robust optimization problems on random instances generated from a large class of distributions; thereby, providing a theoretical justification of the observed empirical performance. The approximation bound depends on the distribution, but it is significantly better than the worst-case bound for a large class of distributions.

math.OC↗

arXiv2019

On the Optimality of Affine Policies for Budgeted Uncertainty Sets

In this paper, we study the performance of affine policies for two-stage adjustable robust optimization problem with fixed recourse and uncertain right hand side belonging to a budgeted uncertainty set. This is an important class of uncertainty sets widely used in practice where we can specify a budget on the adversarial deviations of the uncertain parameters from the nominal values to adjust the level of conservatism. The two-stage adjustable robust optimization problem is hard to approximate within a factor better than $Ω\left( \frac{\log n}{\log \log n} \right)$ even for budget of uncertainty sets and fixed non-negative recourse where $n$ is the number of decision variables. Affine policies, where the second-stage decisions are constrained to be an affine function of the uncertain parameters, provide a tractable approximation for the problem and have been observed to exhibit good empirical performance. We show that affine policies give an $O\left( \frac{\log n}{\log \log n} \right)$-approximation for the two-stage adjustable robust problem with fixed non-negative recourse for budgeted uncertainty sets. This matches the hardness of approximation and therefore, surprisingly affine policies provide an optimal approximation for the problem (up to a constant factor). We also show strong theoretical performance bounds for affine policy for significantly more general class of intersection of budgeted sets including disjoint constrained budgeted sets, permutation invariant sets and general intersection of budgeted sets. Our analysis relies on showing the existence of a near-optimal feasible affine policy that satisfies certain nice structural properties. Based on these structural properties, we also present an alternate algorithm to compute near-optimal affine solution that is significantly faster than computing the optimal affine policy by solving a large linear program.

math.OC↗

arXiv2019

Thompson Sampling for the MNL-Bandit

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.

cs.LG↗

arXiv2018

MNL-Bandit: A Dynamic Learning Approach to Assortment Selection

We consider a dynamic assortment selection problem, where in every round the retailer offers a subset (assortment) of $N$ substitutable products to a consumer, who selects one of these products according to a multinomial logit (MNL) choice model. The retailer observes this choice and the objective is to dynamically learn the model parameters, while optimizing cumulative revenues over a selling horizon of length $T$. We refer to this exploration-exploitation formulation as the MNL-Bandit problem. Existing methods for this problem follow an "explore-then-exploit" approach, which estimate parameters to a desired accuracy and then, treating these estimates as if they are the correct parameter values, offers the optimal assortment based on these estimates. These approaches require certain a priori knowledge of "separability", determined by the true parameters of the underlying MNL model, and this in turn is critical in determining the length of the exploration period. (Separability refers to the distinguishability of the true optimal assortment from the other sub-optimal alternatives.) In this paper, we give an efficient algorithm that simultaneously explores and exploits, achieving performance independent of the underlying parameters. The algorithm can be implemented in a fully online manner, without knowledge of the horizon length $T$. Furthermore, the algorithm is adaptive in the sense that its performance is near-optimal in both the "well separated" case, as well as the general parameter setting where this separation need not hold.

cs.LG↗

arXiv2018

A Tractable Approach for designing Piecewise Affine Policies in Two-stage Adjustable Robust Optimization

We consider the problem of designing piecewise affine policies for two-stage adjustable robust linear optimization problems under right-hand side uncertainty. It is well known that a piecewise affine policy is optimal although the number of pieces can be exponentially large. A significant challenge in designing a practical piecewise affine policy is constructing good pieces of the uncertainty set. Here we address this challenge by introducing a new framework in which the uncertainty set is "approximated" by a "dominating" simplex. The corresponding policy is then based on a mapping from the uncertainty set to the simplex. Although our piecewise affine policy has exponentially many pieces, it can be computed efficiently by solving a compact linear program given the dominating simplex. Furthermore, we can find the dominating simplex in a closed form if the uncertainty set satisfies some symmetries and can be computed using a MIP in general. The performance of our policy is significantly better than the affine policy for many important uncertainty sets, such as ellipsoids and norm-balls, both theoretically and numerically. For instance, for hypersphere uncertainty set, our piecewise affine policy can be computed by an LP and gives a $O(m^{1/4})$-approximation whereas the affine policy requires us to solve a second order cone program and has a worst-case performance bound of $O(\sqrt m)$.

math.OC↗