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Sanjay Mehrotra

Publications and source records attributed to Sanjay Mehrotra.

At least 19 recordsLinked to original sources
arXiv2026

Robust Chance-Constrained Optimization using a Continuous Parameter Space Wasserstein-2 Ambiguity Set of Gaussian Mixtures

We study distributionally robust linear chance-constrained problems in which uncertainty is modeled by a Gaussian mixture model (GMM). Finite-support distributionally robust (FDR) formulations, widely used in data-driven robust optimization, robustify over empirical mixture support points and therefore primarily stress-test the fitted nominal mixture. This can be insufficient when service reliability depends on structural misspecification of the nominal mixture-support parameters. To address this limitation, we describe the ambiguity set of distributions by developing a novel formulation of a Wasserstein-2 metric that uses the Bures-Wasserstein (BW) metric over probability measures with finite second moments. Unlike FDR, which generally sets finitely many empirical support points a priori, the proposed ambiguity set allows the worst-case distribution to endogenously determine both how many mixture components receive mass and where their means and covariances lie within a continuous support. For the resulting ambiguity set, under mild regularity conditions, we prove strong duality for the inner worst-case chance-constraint problem and derive its semi-infinite reformulation. We then develop an adaptive cutting-surface algorithm, which endogenously determines the locations of mixture components receiving mass, and the mean and covariances of the Gaussian distributions at these locations. The algorithm attains any prescribed optimality gap in finitely many iterations, while a block-alternating local search identifies new components. A case study using the electric-vehicle charging-station energy-allocation problem demonstrates the framework's practical value in achieving any reliability targets. CDR also induces structural changes in energy allocations, unlike FDR, whose allocations remain close to the nominal solution.

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arXiv2026

Nonparametric Multi Change Point Detection for Markov Chains via Adaptive Clustering

Offline change point detection tries to detect time points of distribution change in a given data sequence; and is now routinely used in signal processing, speech processing, climatology etc. Despite this broad applicability across economics, computer science, and planetary sciences, rigorous, nonparametric techniques for change point detection with non-independent and identically distributed (i.i.d.) datasets has remained elusive. This paper establishes such guarantees by proposing a non-parametric clustering algorithm which can accurately obtain the change points from a given Markovian dataset of length $n$. It does so by bridging together two different components of mathematical statistics; Rademacher complexities of Markov chains, and adaptive clustering via penalisation. Our first result uses recent advances in Rademacher complexities of regenerating Markov chains to derive a Dvoretzky Kiefer Wolfowitz (DKW) type inequality for the empirical distribution of the Markov chain. We then use this to show that an adaptive clustering algorithm recovers the correct change points for a Markovian sequence. We establish the tightness of our rates by showing that they essentially coincide with the best known rates for i.i.d. data. We end the paper by discussing the computational considerations of the problem.

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arXiv2025

On Solving Chance-Constrained Models with Gaussian Mixture Distribution

We study linear chance-constrained problems where the coefficients follow a Gaussian mixture distribution. We provide mixed-binary quadratic programs that give inner and outer approximations of the chance constraint based on piecewise linear approximations of the standard normal cumulative density function. We show that $O\left(\sqrt{\ln(1/\tau)/\tau} \right)$ pieces are sufficient to attain $\tau$-accuracy in the chance constraint. We also show that any desired optimality gap can be achieved under a constraint qualification condition by controlling the approximation accuracy. Extensive computations using a commercial solver show that problems with up to one thousand random coefficients specified with up to fifteen Gaussian mixture components, generated under diverse settings, can be solved to near optimality within 18 hours, while satisfying chance constraint satisfaction probabilities of up to $0.999$. The solution times are significantly lower for problems with fewer random coefficients and mixture terms. For example, problems with one hundred random coefficients, ten mixture terms, and a constraint satisfaction probability of $0.999$ can be solved in a minute or less. Sample average approximations fail to provide meaningful solutions even for the smaller problems.

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arXiv2025

A Framework for Stochastic Fairness in Dominant Resource Allocation with Cloud Computing Applications

Allocation of limited resources under uncertain requirements often necessitates fairness considerations, with applications in computer systems, health systems, and humanitarian logistics. This paper introduces a distributionally robust (DR) stochastic fairness framework for multi-resource allocation, leveraging rough estimates of the mean and variance of resource requirement distributions. The framework employs a sampled approximation DR (SA-DR) model to develop the concept of stochastic fairness, satisfying key properties such as stochastic Pareto efficiency, stochastic sharing incentive, and stochastic envy-freeness under suitable conditions. We show the convergence of the SA-DR model to the DR model and propose a finitely convergent algorithm to solve the SA-DR model. We empirically evaluate the performance of our moment-based SA-DR model -- which uses only rough estimates of the mean and variance of the resource requirement distribution -- against alternative resource allocation models under varying levels of information availability. We demonstrate that our moment-based partial-information SA-DR model can achieve performance closer to the full-information model than the worst-case information model. Convergence of the sampled approximation model and comparisons across models are illustrated using data from cloud computing applications.

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arXiv2024

On Approximation of Robust Max-Cut and Related Problems using Randomized Rounding Algorithms

Goemans and Williamson proposed a randomized rounding algorithm for the MAX-CUT problem with a 0.878 approximation bound in expectation. The 0.878 approximation bound remains the best-known approximation bound for this APX-hard problem. Their approach was subsequently applied to other related problems such as Max-DiCut, MAX-SAT, and Max-2SAT, etc. We show that the randomized rounding algorithm can also be used to achieve a 0.878 approximation bound for the robust and distributionally robust counterparts of the max-cut problem. We also show that the approximation bounds for the other problems are maintained for their robust and distributionally robust counterparts if the randomization projection framework is used.

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arXiv2023

Elevating Industries with Unmanned Aerial Vehicles: Integrating Sustainability and Operational Innovation

Unmanned aerial vehicles, commonly known as drones, have emerged as a disruptive technology with the potential to revolutionize operations across various industries. Drones are the fast-growing internet-of-things technology and are estimated to have a $100 billion market value in the next decade. Exploring drone operations through research has the potential to yield innovative academic insights and create significant practical effects in diverse industries, offering a competitive edge. Drawing insights from both academic and industry literature, this article describes how technological advancements in UAVs may disrupt traditional operational practices in different industries (e.g., commercial last-mile delivery, commercial pickup and delivery, telecommunication, insurance, healthcare, humanitarian, environmental, urban planning, homeland security), identifies the value of this evolving disruptive technology from sustainability and operational innovation perspectives, argues the significance of this area for operations management by conceptualizing a research agenda. The current state of the art focuses on the computing aspect of analytical models to tackle a variety of synthetic drone-related problems, with mixed integer optimization being the primary tool. There is a very significant research gap that should focus on drone operations management with industry know-how by partnering with actual stakeholders and using a variety of tools (i.e., econometrics, field experiments, game theory, optimal control, utility functions). This article aims to promote research on UAVs from operations management and industry-specific point of view.

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arXiv2023

Optimization Modeling for Pandemic Vaccine Supply Chain Management: A Review and Future Research Opportunities

During various stages of the COVID-19 pandemic, countries implemented diverse vaccine management approaches, influenced by variations in infrastructure and socio-economic conditions. This article provides a comprehensive overview of optimization models developed by the research community throughout the COVID-19 era, aimed at enhancing vaccine distribution and establishing a standardized framework for future pandemic preparedness. These models address critical issues such as site selection, inventory management, allocation strategies, distribution logistics, and route optimization encountered during the COVID-19 crisis. A unified framework is employed to describe the models, emphasizing their integration with epidemiological models to facilitate a holistic understanding. This article also summarizes evolving nature of literature, relevant research gaps, and authors' perspectives for model selection. Finally, future research scopes are detailed both in the context of modeling and solutions approaches.

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arXiv2023

Optimizing Equitable Resource Allocation in Parallel Any-Scale Queues with Service Abandonment and its Application to Liver Transplant

We study the problem of equitably and efficiently allocating an arriving resource to multiple queues with customer abandonment. The problem is motivated by the cadaveric liver allocation system of the United States, which includes a large number of small-scale (in terms of yearly arrival intensities) patient waitlists with the possibility of patients abandoning (due to death) until the required service is completed (matched donor liver arrives). We model each waitlist as a GI/MI/1+GI queue, in which a virtual server receives a donor liver for the patient at the top of the waitlist, and patients may abandon while waiting or during service. To evaluate the performance of each queue, we develop a finite approximation technique as an alternative to fluid or diffusion approximations, which are inaccurate unless the queue's arrival intensity is large. This finite approximation for hundreds of queues is used within an optimization model to optimally allocate donor livers to each waitlist. A piecewise linear approximation of the optimization model is shown to provide the desired accuracy. Computational results show that solutions obtained in this way provide greater flexibility, and improve system performance when compared to solutions from the fluid models. Importantly, we find that appropriately increasing the proportion of livers allocated to waitlists with small scales or high mortality risks improves the allocation equity. This suggests a proportionately greater allocation of organs to smaller transplant centers and/or those with more vulnerable populations in an allocation policy. While our motivation is from liver allocation, the solution approach developed in this paper is applicable in other operational contexts with similar modeling frameworks.

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arXiv2023

Service Center Location with Decision Dependent Utilities with an Application to Early Stage Testing and Vaccination in Epidemic Planning

We study a service center location problem with ambiguous utility gains upon receiving service. The model is motivated by the problem of deciding medical clinic/service centers, possibly in rural communities, where residents need to visit the clinics to receive health services. A resident gains his utility based on travel distance, waiting time, and service features of the facility that depend on the clinic location. The elicited location-dependent utilities are assumed to be ambiguously described by an expected value and variance constraint. We show that despite a non-convex nonlinearity, given by a constraint specified by a maximum of two second-order conic functions, the model admits a mixed 0-1 second-order cone (MISOCP) formulation. We study the non-convex substructure of the problem, and present methods for developing its strengthened formulations by using valid tangent inequalities. Computational study shows the effectiveness of solving the strengthened formulations. Examples are used to illustrate the importance of including decision dependent ambiguity. An illustrative example to identify locations for Covid-19 testing and vaccination is used to further illustrate the model and its properties.

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arXiv2022

Diversity Subsampling: Custom Subsamples from Large Data Sets

Subsampling from a large data set is useful in many supervised learning contexts to provide a global view of the data based on only a fraction of the observations. Diverse (or space-filling) subsampling is an appealing subsampling approach when no prior knowledge of the data is available. In this paper, we propose a diversity subsampling approach that selects a subsample from the original data such that the subsample is independently and uniformly distributed over the support of distribution from which the data are drawn, to the maximum extent possible. We give an asymptotic performance guarantee of the proposed method and provide experimental results to show that the proposed method performs well for typical finite-size data. We also compare the proposed method with competing diversity subsampling algorithms and demonstrate numerically that subsamples selected by the proposed method are closer to a uniform sample than subsamples selected by other methods. The proposed DS algorithm is shown to be more efficient than known methods and takes only a few minutes to select tens of thousands of subsample points from a data set of size one million. Our DS algorithm easily generalizes to select subsamples following distributions other than uniform. We provide the FADS Python package to implement the proposed methods.

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arXiv2022

A New Method for Computing Stationary Distribution and Steady-State Performance Measures of a Continuous-State Markov Chain with a Queuing Application

Applications of stochastic models often involve the evaluation of steady-state performance, which requires solving a set of balance equations. In most cases of interest, the number of equations is infinite or even uncountable. As a result, numerical or analytical solutions are unavailable. This is true even when the system state is one-dimensional. This paper develops a general method for computing stationary distributions and steady-state performance measures of stochastic systems that can be described as continuous-state Markov chains supported on R. The balance equations are numerically solved by properly constructing a proxy Markov chain with finite states. We show the consistency of the approximate solution and provide deterministic non-asymptotic error bounds under the supremum norm. Our finite approximation method is near-optimal among all approximation methods using discrete distributions, including the empirical distributions generated by a simulation approach. We apply the developed method to compute the stationary distribution of virtual waiting time in a G/G/1+G queue and associated performance measures under certain mild but general differentiability and boundedness assumptions on the inter-arrival, service, and patience time distributions. Numerical experiments validate the accuracy and efficiency of our method, and show it outperforms a standard Markov chain Monte Carlo method by several orders of magnitude. The developed method is also significantly more accurate than the available fluid approximations for this queue.

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arXiv2021

A Cutting-plane and Benders' Decomposition Algorithm for Two-Stage Distributionally Robust Convex programs

We present a finitely convergent cutting-plane algorithm for solving a general mixed-integer convex program given an oracle for solving a general convex program. This method is extended to solve a family of two-stage mixed-integer convex programs using cutting planes, with applications to solving distributionally-robust two-stage stochastic mixed-integer convex programs. Analysis is also given for the case where convex programming oracle provides an $epsilon$-optimal solution. We combine the cut generation with a branch-and-union scheme to develop a more practical algorithm. Computational results on generated test problems show the practicality of our algorithm. Specifically, results show that in the tested problems our algorithm achieves < 5% optimality gap in 12 hours. This gap is >17% with a commercial solver.

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arXiv2020

A Model of Supply-Chain Decisions for Resource Sharing with an Application to Ventilator Allocation to Combat COVID-19

This paper presents a stochastic optimization model for allocating and sharing a critical resource in the case of a pandemic. The demand for different entities peaks at different times, and an initial inventory from a central agency is to be allocated. The entities (states) may share the critical resource with a different state under a risk-averse condition. The model is applied to study the allocation of ventilator inventory in the COVID-19 pandemic by the Federal Emergency Management Agency of the US Department of Homeland Security (FEMA) to different states in the US. Findings suggest that if less than 60% of the ventilator inventory is available for non-COVID-19 patients, FEMA's stockpile of 20,000 ventilators (as of 03/23/2020) would be nearly adequate to meet the projected needs. However, when more than 75% of the available ventilator inventory must be reserved for non-COVID-19 patients, various degrees of shortfall are expected. In an extreme case, where the demand is assumed to be concentrated in the top-most quartile of the forecast confidence interval, the total shortfall over the planning horizon (till 05/31/20) is about 28,500 ventilator days, with a peak shortfall of 2,700 ventilators on 04/12/20. The results also suggest that in the worse-than-average to severe demand scenario cases, NY requires between 7,600-9,200 additional ventilators for COVID-19 patients during its peak demand. However, between 400 to 2,000 of these ventilators can be given to a different state after the peak demand in NY has subsided.

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arXiv2019

A Decomposition Method for Distributionally-Robust Two-stage Stochastic Mixed-integer Cone Programs

We develop a decomposition algorithm for distributionally-robust two-stage stochastic mixed-integer convex cone programs, and its important special case of distributionally-robust two-stage stochastic mixed-integer second order cone programs. This generalizes the algorithm proposed by Sen and Sherali~[Mathematical Programming 106(2): 203-223, 2006]. We show that the proposed algorithm is finitely convergent if the second-stage problems are solved to optimality at incumbent first stage solutions, and solution to an optimization problem to identify worst-case probability distribution is available. The second stage problems can be solved using a branch-and-cut algorithm. The decomposition algorithm is illustrated with an example. Computational results on a stochastic programming generalization of a facility location problem show significant solution time improvements from the proposed approach. Solutions for many models that are intractable for an extensive form formulation become possible. Computational results suggest that solution time requirement does not increase significantly when considering distributional robust counterparts to the stochastic programming models.

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arXiv2019

A Geometric Branch and Bound Method for a Class of Robust Maximization Problems of Convex Functions

We investigate robust optimization problems defined for maximizing convex functions. For finite uncertainty set, we develop a geometric branch-and-bound algorithmic approach to solve this problem. The geometric branch-and-bound algorithm performs sequential piecewise-linear approximations of the convex objective, and solves linear programs to determine lower and upper bounds of nodes specified by the active linear pieces. Finite convergence of the algorithm to an $ε-$optimal solution is proved. Numerical results are used to discuss the performance of the developed algorithm. The algorithm developed in this paper can be used as an oracle in the cutting surface method for solving robust optimization problems with compact ambiguity sets.

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arXiv2019

Distributionally Robust Optimization: A Review

The concepts of risk-aversion, chance-constrained optimization, and robust optimization have developed significantly over the last decade. Statistical learning community has also witnessed a rapid theoretical and applied growth by relying on these concepts. A modeling framework, called distributionally robust optimization (DRO), has recently received significant attention in both the operations research and statistical learning communities. This paper surveys main concepts and contributions to DRO, and its relationships with robust optimization, risk-aversion, chance-constrained optimization, and function regularization.

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arXiv2018

Distributionally Robust Optimization with Decision Dependent Ambiguity Sets

We study decision dependent distributionally robust optimization models, where the ambiguity sets of probability distributions can depend on the decision variables. These models arise in situations with endogenous uncertainty. The developed framework includes two-stage decision dependent distributionally robust stochastic programming as a special case. Decision dependent generalizations of five types of ambiguity sets are considered. These sets are based on bounds on moments, Wasserstein metric, $ϕ$-divergence and Kolmogorov-Smirnov test. For the finite support case, we use linear, conic or Lagrangian duality to give reformulations of the models with a finite number of constraints. These reformulations allow solutions of such problems using global optimization techniques. Certain reformulations give rise to non-convex semi-infinite programs. Techniques from global optimization and semi-infinite programming can be used to solve these reformulations.

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