arXiv subjects

Omar El Housni

Publications and source records attributed to Omar El Housni.

18 recordsLinked to original sources

arXiv2026

Adaptive Two-Sided Assortment Optimization: Revenue Maximization

We study adaptive two-sided assortment optimization for revenue maximization in choice-based matching platforms. The platform has two sides of agents: an initiating side and a responding side. The decision-maker sequentially selects agents from the initiating side, shows each an assortment of agents from the responding side, and observes their choices. After processing all initiating agents, the responding agents are shown assortments and make their selections. A match occurs when two agents mutually select each other, generating pair-dependent revenue. Choices follow Multinomial Logit (MNL) models. This setting generalizes prior work focused on maximizing the number of matches under submodular demand assumptions, which do not hold in our revenue-maximization context. We show that the revenue maximization problem is APX-hard, so constant-factor loss is unavoidable. Our main contribution is the design of polynomial-time approximation algorithms with constant-factor guarantees. In particular, for general pairwise revenues, we develop a randomized algorithm that achieves a $(\frac{1}{2}-ε)$-approximation in expectation for any $ε> 0$. The algorithm is static and provides guarantees under various agent arrival settings, including fixed order, simultaneous processing, and adaptive selection. When revenues are uniform across all pairs involving any given responding-side agent, the guarantee improves to $(1 - \frac{1}{e} - ε)$. In structural settings where responding-side agents share a common revenue-based ranking, we design a simpler adaptive deterministic algorithm achieving a $\frac{1}{2}$-approximation. Our approach leverages novel linear programming relaxations, correlation-gap arguments, and structural properties of the revenue functions.

cs.GT↗

arXiv2026

Joint Inventory Placement, Assortment Personalization, and Order Fulfillment for Substitutable Products

Modern online retailers leverage networks of distributed warehouses to rapidly fulfill customer orders. We study a problem of jointly deciding (i) how to allocate inventories across warehouses in the network subject to warehouse capacity and product supply constraints, (ii) how to dynamically select personalized product assortments based on customer preferences and location, as well as real-time stock levels, and (iii) which warehouse to use to fulfill the product chosen by the customer to maximize expected profit. In our model, the firm chooses an inventory placement across warehouses at the start of the selling horizon. Customers of different types then arrive at discrete time periods over a finite time horizon according to a known distribution. When offered an assortment, each customer type chooses at most one product according to a discrete choice model. The firm then chooses a feasible warehouse from which to fulfill the chosen product, depleting its stock and earning a profit that depends on the product, customer type, and warehouse choice. We develop approximation algorithms and policies with provable theoretical guarantees and strong empirical performance. Under general probabilistic choice models, we establish asymptotically optimal approximation algorithms as warehouse capacities and product supplies scale. Under the multinomial logit (MNL) choice model, we obtain constant-factor approximation algorithms, with approximation factors ranging from 0.080 in the most general setting to 0.199 under stationarity and unlimited product supplies, which means that only warehouse capacity constraints are present. To the best of our knowledge, these results provide the first provably efficient algorithms for the joint inventory placement, assortment personalization, and order fulfillment problem.

math.OC↗

arXiv2026

Price and Assortment Optimization under the Multinomial Logit Model with Opaque Products

An opaque product is a product for which only partial information is disclosed to the buyer at the time of purchase. Opaque products are common in sectors such as travel and online retail, where the brand or product color is hidden in the opaque product. Opaque products enable sellers to target customers who prefer a price discount in exchange for being flexible about the product they receive. In this paper, we integrate opaque products and traditional products together into the multinomial logit (MNL) choice model and study the associated price and assortment optimization problems. For the price optimization problem, we show a surprising result that uniform pricing is optimal which implies it has the same optimal pricing solution and value as the traditional MNL model. Although adding an opaque product may enhance revenue given arbitrary traditional product prices, our result shows that this advantage disappears when all prices are optimized jointly. For the assortment problem, we first study the setting where the seller chooses both the offered assortment and the support of the opaque product, and prove that there exists an optimal assortment that is nested by revenue. We then study the case where the opaque support is required to coincide with the offered assortment which is common in practice. Here, we show that the optimal assortment is nested-by-valuation for uniformly priced products. For arbitrary prices, we propose a nested-by-revenue-and-valuation heuristic with a theoretical approximation guarantee that performs extremely well in our numerical study.

math.OC↗

arXiv2026

Two-sided Assortment Optimization: Adaptivity Gaps and Approximation Algorithms

To address efficiency and design challenges in choice-based matching platforms, we introduce a two-sided assortment optimization framework under general choice preferences. The goal in this problem is to maximize the expected number of matches by deciding which assortments are displayed to the agents and the order in which they are shown. In this context, we identify several classes of policies that platforms can use in their design. Our goals are: (1) to measure the value that one class of policies has over another one, and (2) to approximately solve the optimization problem itself for a given class. For (1), we define the adaptivity gap as the worst-case ratio between the optimal values of two different policy classes. First, we show that the gap between the class of policies that statically show assortments to one-side first and the class of policies that adaptively show assortments to one-side first is exactly $e/(e-1)$. Second, we show that the gap between the latter class of policies and the fully adaptive class of policies that show assortments to agents one by one is exactly $2$. We also note that the worst policies are those who simultaneously show assortments to all the agents. For (2), we first design a polynomial time algorithm that achieves a $1/4$ approximation factor within the class of policies that adaptively show assortments to agents one by one. Furthermore, when agents' preferences are governed by multinomial-logit models, we show that a 0.067 approximation factor can be obtained within the class of policies that show assortments to all agents at once. We further generalize our results to constrained assortment settings, where we impose an upper bound on the size of the displayed assortments. Finally, we present a computational study to evaluate the empirical performance of our theoretical guarantees.

math.OC↗

arXiv2026

Optimal Selection with Balanced Market Share: Static and Dynamic Assortment Optimization

Assortment optimization is a critical tool for online retailers aiming to maximize revenue. However, optimizing purely for revenue can lead to unbalanced sales across products, potentially causing a long tail of low-selling products and products with excessively large market shares, both of which could be harmful to the seller. To address these issues, we introduce a market share balancing constraint that limits the disparity in expected sales between any two offered products to a factor of a given parameter $α$. We study both static and dynamic assortment optimization under the multinomial logit (MNL) model with this fairness constraint. In the static setting, the seller selects a distribution over assortments that satisfies the market share balancing constraint while maximizing expected revenue. We show that this problem can be solved in polynomial time, and we characterize the structure of the optimal solution: a product is included if and only if its revenue and preference weight exceed certain thresholds. We further extend our analysis to settings with additional feasibility constraints on the assortment and demonstrate that, given a $β$-approximation oracle for the constrained problem, we can construct a $β$-approximation algorithm under the fairness constraint. In the dynamic setting, each product has a finite initial inventory, and the seller implements a dynamic policy to maximize total expected revenue while respecting both inventory limits and the market share balancing constraint in expectation. We design a policy that is asymptotically optimal, with its approximation ratio converging to one as inventories grow large.

cs.GT↗

arXiv2025

When Location Shapes Choice: Placement Optimization of Substitutable Products

Strategic product placement can have a strong influence on customer purchase behavior in physical stores as well as online platforms. Motivated by this, we consider the problem of optimizing the placement of substitutable products in designated display locations to maximize the expected revenue of the seller. We model the customer behavior as a two-stage process: first, the customer visits a subset of display locations according to a browsing distribution; second, the customer chooses at most one product from the displayed products at those locations according to a choice model. Our goal is to design a general algorithm that can select and place the products optimally for any browsing distribution and choice model, and we call this the Placement problem. We give a randomized algorithm that utilizes an $α$-approximate algorithm for cardinality constrained assortment optimization and outputs a $\frac{Θ(α)}{\log m}$-approximate solution (in expectation) for Placement with $m$ display locations, i.e., our algorithm outputs a solution with value at least $\frac{Ω(α)}{\log m}$ factor of the optimal and this is tight in the worst case. We also give algorithms with stronger guarantees in some special cases. In particular, we give a deterministic $\frac{Ω(1)}{\log m}$-approximation algorithm for the Markov choice model, and a tight $(1-1/e)$-approximation algorithm for the problem when products have identical prices.

cs.DS↗

arXiv2025

Assortment Optimization under the Multinomial Logit Model with Covering Constraints

We consider an assortment optimization problem under the multinomial logit choice model with general covering constraints. In this problem, the seller offers an assortment that should contain a minimum number of products from multiple categories. We refer to these constraints as covering constraints. Such constraints are common in practice due to service level agreements with suppliers or diversity considerations within the assortment. We consider both the deterministic version, where the seller decides on a single assortment, and the randomized version, where they choose a distribution over assortments. In the deterministic case, we provide a $1/(\log K+2)$-approximation algorithm, where $K$ is the number of product categories, matching the problem's hardness up to a constant factor. For the randomized setting, we show that the problem is solvable in polynomial time via an equivalent linear program. We also extend our analysis to multi-segment assortment optimization with covering constraints, where there are $m$ customer segments, and an assortment is offered to each. In the randomized setting, the problem remains polynomially solvable. In the deterministic setting, we design a $(1 - ε) / (\log K + 2)$-approximation algorithm for constant $m$ and a $1 / (m (\log K + 2))$-approximation for general $m$, which matches the hardness up to a logarithmic factor. Finally, we conduct a numerical experiment using real data from an online electronics store, categorizing products by price range and brand. Our findings demonstrate that, in practice, it is feasible to enforce a minimum number of representatives from each category while incurring a relatively small revenue loss. Moreover, we observe that the optimal expected revenue in both deterministic and randomized settings is often comparable, and the optimal solution in the randomized setting typically involves only a few assortments.

math.OC↗

arXiv2025

A Markovian Approach for Cross-Category Complementarity in Choice Modeling

While single-purchase choice models have been widely studied in assortment optimization, customers in modern retail and e-commerce environments often purchase multiple items across distinct product categories, exhibiting both substitution and complementarity. We consider the cross-category assortment optimization problem where retailers jointly determine assortments across categories to maximize expected revenue. Most prior work on the topic either overlooks complementarity or proposes models that lead to intractable optimization problems, despite being based on the multinomial logit (MNL) choice model. We propose a sequential multi-purchase choice model for cross-category choice that incorporates complementarity through a Markovian transition structure across categories, while allowing general Random Utility Maximization (RUM)-based choice models to capture the within-category substitution. We develop an Expectation-Maximization algorithm for estimation, and a polynomial-time algorithm for unconstrained assortment optimization that yields the optimal solution when the within-category substitution follows a Markov chain choice model. Furthermore, we introduce an empirical metric to quantify complementarity strength across product categories and conduct extensive numerical experiments on both synthetic data and a large-scale transaction-level dataset from a major US grocery store. Our model yields improvements in predictive accuracy, model fit, and expected revenue in setting with complementarity, and it reveals intuitive market structures such as brand-loyal cross-category purchasing. Overall, we believe that our model provides a theoretically-grounded and practical framework for modeling complementarity and making better cross-category assortment decisions.

math.OC↗

arXiv2025

Matching Drivers to Riders: A Two-stage Robust Approach

Matching demand (riders) to supply (drivers) efficiently is a fundamental problem for ride-sharing platforms who need to match the riders (almost) as soon as the request arrives with only partial knowledge about future ride requests. A myopic approach that computes an optimal matching for current requests ignoring future uncertainty can be highly sub-optimal. In this paper, we consider a two-stage robust optimization framework for this matching problem where future demand uncertainty is modeled using a set of demand scenarios (specified explicitly or implicitly). The goal is to match the current request to drivers (in the first stage) so that the cost of first-stage matching and the worst-case cost over all scenarios for the second-stage matching is minimized. We show that the two-stage robust matching is NP-hard under various cost functions and present constant approximation algorithms for different settings of our two-stage problem. Furthermore, we test our algorithms on real-life taxi data from the city of Shenzhen and show that they substantially improve upon myopic solutions and reduce the maximum wait time of the second-stage riders.

math.OC↗

arXiv2025

Assortment Optimization with Visibility Constraints

Motivated by applications in e-retail and online advertising, we study the problem of assortment optimization under visibility constraints, that we refer to as APV. Here, we are given a universe of substitutable products and a stream of customers. The objective is to determine the optimal assortment of products to offer to each customer in order to maximize the total expected revenue, subject to exogenously-given visibility constraints, stating that each product should be shown to a minimum number of customers. We assume that customer choices follow a Multinomial Logit model (MNL). We provide a structural characterization of optimal assortments and present a linear time algorithm for solving APV. To this end, we introduce a novel function called the ``expanded revenue" of an assortment and establish its supermodularity; our algorithm takes advantage of this structural property. Additionally, we prove that APV can be formulated as a compact linear program. Next, we consider APV with cardinality constraints, which limit the maximum number of products that can be included in an assortment. We prove this problem to be strongly NP-hard and not admitting a Fully Polynomial Time Approximation Scheme (FPTAS), even when all products have identical prices. Subsequently, we devise a Polynomial Time Approximation Scheme (PTAS) for APV under cardinality constraints with identical prices. We also examine the revenue loss resulting from the enforcement of visibility constraints, comparing it to the unconstrained problem. To offset this loss, we propose a novel strategy to distribute the loss incurred among the products subject to visibility constraints, charging each vendor an amount proportional to their product's contribution to the revenue loss.

math.OC↗

arXiv2025

Maximum Load Assortment Optimization: Approximation Algorithms and Adaptivity Gaps

Motivated by modern-day applications such as Attended Home Delivery and Preference-based Group Scheduling, where decision makers wish to steer a large number of customers toward choosing the exact same alternative, we introduce a novel class of assortment optimization problems, referred to as Maximum Load Assortment Optimization. In such settings, given a universe of substitutable products, we are facing a stream of customers, each choosing between either selecting a product out of an offered assortment or opting to leave without making a selection. Assuming that these decisions are governed by the Multinomial Logit choice model, we define the random load of any underlying product as the total number of customers who select it. Our objective is to offer an assortment of products to each customer so that the expected maximum load across all products is maximized. We consider both static and dynamic formulations. In the static setting, a single offer set is carried throughout the entire process of customer arrivals, whereas in the dynamic setting, the decision maker offers a personalized assortment to each customer, based on the entire information available at that time. The main contribution of this paper resides in proposing efficient algorithmic approaches for computing near-optimal static and dynamic assortment policies. In particular, we develop a polynomial-time approximation scheme (PTAS) for the static formulation. Additionally, we demonstrate that an elegant policy utilizing weight-ordered assortments yields a 1/2- approximation. Concurrently, we prove that such policies are sufficiently strong to provide a 1/4-approximation with respect to the dynamic formulation, establishing a constant-factor bound on its adaptivity gap. Finally, we design an adaptive policy whose expected maximum load is within factor 1-\eps of optimal, admitting a quasi-polynomial time implementation.

math.OC↗

arXiv2023

LP-based Approximations for Disjoint Bilinear and Two-Stage Adjustable Robust Optimization

We consider the class of disjoint bilinear programs $ \max \, \{ \mathbf{x}^T\mathbf{y} \mid \mathbf{x} \in \mathcal{X}, \;\mathbf{y} \in \mathcal{Y}\}$ where $\mathcal{X}$ and $\mathcal{Y}$ are packing polytopes. We present an $O(\frac{\log \log m_1}{\log m_1} \frac{\log \log m_2}{\log m_2})$-approximation algorithm for this problem where $m_1$ and $m_2$ are the number of packing constraints in $\mathcal{X}$ and $\mathcal{Y}$ respectively. In particular, we show that there exists a near-optimal solution $(\tilde{\mathbf{x}}, \tilde{\mathbf{y}})$ such that $\tilde{\mathbf{x}}$ and $\tilde{\mathbf{y}}$ are ``near-integral". We give an LP relaxation of the problem from which we obtain the near-optimal near-integral solution via randomized rounding. We show that our relaxation is tightly related to the widely used reformulation linearization technique (RLT). As an application of our techniques, we present a tight approximation for the two-stage adjustable robust optimization problem with covering constraints and right-hand side uncertainty where the separation problem is a bilinear optimization problem. In particular, based on the ideas above, we give an LP restriction of the two-stage problem that is an $O(\frac{\log n}{\log \log n} \frac{\log L}{\log \log L})$-approximation where $L$ is the number of constraints in the uncertainty set. This significantly improves over state-of-the-art approximation bounds known for this problem. Furthermore, we show that our LP restriction gives a feasible affine policy for the two-stage robust problem with the same (or better) objective value. As a consequence, affine policies give an $O(\frac{\log n}{\log \log n} \frac{\log L}{\log \log L})$-approximation of the two-stage problem, significantly generalizing the previously known bounds on their performance.

math.OC↗

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

Future Evolution of COVID-19 Pandemic in North Carolina: Can We Flatten the Curve?

On June 24th, Governor Cooper announced that North Carolina will not be moving into Phase 3 of its reopening process at least until July 17th. Given the recent increases in daily positive cases and hospitalizations, this decision was not surprising. However, given the political and economic pressures which are forcing the state to reopen, it is not clear what actions will help North Carolina to avoid the worst. We use a compartmentalized model to study the effects of social distancing measures and testing capacity combined with contact tracing on the evolution of the pandemic in North Carolina until the end of the year. We find that going back to restrictions that were in place during Phase 1 will slow down the spread but if the state wants to continue to reopen or at least remain in Phase 2 or Phase 3 it needs to significantly expand its testing and contact tracing capacity. Even under our best-case scenario of high contact tracing effectiveness, the number of contact tracers the state currently employs is inadequate.

q-bio.PE↗

arXiv2020

Can Testing Ease Social Distancing Measures? Future Evolution of COVID-19 in NYC

The "New York State on Pause" executive order came into effect on March 22 with the goal of ensuring adequate social distancing to alleviate the spread of COVID-19. Pause will remain effective in New York City in some form until early June. We use a compartmentalized model to study the effects of testing capacity and social distancing measures on the evolution of the pandemic in the "post-Pause" period in the City. We find that testing capacity must increase dramatically if it is to counterbalance even relatively small relaxations in social distancing measures in the immediate post-Pause period. In particular, if the City performs 20,000 tests per day and relaxes the social distancing measures to the pre-Pause norms, then the total number of deaths by the end of September can reach 250,000. By keeping the social distancing measures to somewhere halfway between the pre- and in-Pause norms and performing 100,000 tests per day, the total number of deaths by the end of September can be kept at around 27,000. Going back to the pre-Pause social distancing norms quickly must be accompanied by an exorbitant testing capacity, if one is to suppress excessive deaths. If the City is to go back to the "pre-Pause" social distancing norms in the immediate post-Pause period and keep the total number of deaths by the end of September at around 35,000, then it should be performing 500,000 tests per day. Our findings have important implications on the magnitude of the testing capacity the City needs as it relaxes the social distancing measures to reopen its economy.

q-bio.PE↗

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↗

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↗