arXiv subjects

David Bergman

Publications and source records attributed to David Bergman.

At least 19 recordsLinked to original sources

arXiv2024

The Sensitivity of the U.S. Presidential Election to Coordinated Voter Relocation

U.S. presidential elections are decided by the Electoral College, established in 1789, and designed to mitigate potential risks arising from the collusion of large groups of citizens. A statewide winner-take-all popular voting system for electors is implemented in all but two states, which has led to instances where narrow victories in key states were decisive in several recent elections. Small groups of voters can significantly impact the election, for example, through voter turnout. However, another dynamic can also influence this: a surprisingly small number of dedicated voters moving short distances across state lines. The extent to which the election's outcome is sensitive to small and well-coordinated movements of people has not been investigated in detail. Using a combination of forecasting, simulation, and optimization, we show that a candidate's probability of winning can be increased by 1% through the strategic relocation of approximately 10,000 people no farther than 100 miles from their current county of residence, less than 0.006% of the eligible voting population. Moreover, an 8% probability increase can be realized by a mere 50,000 voters relocating across state lines, or 0.03% of the voting population. Given the remarkably small number of people involved and the fact that establishing electoral residence in many states takes about a month, this coordinated relocation of voters is not nearly as challenging as previously thought. As it stands, U.S. presidential elections may be vulnerable to the exploitation of the aforementioned loophole. Therefore, we anticipate our findings will have direct consequences on policymaking and campaign strategy, as well as motivate new operations research methods within the political sciences.

physics.soc-ph↗

arXiv2024

Network Relaxations for Discrete Bilevel Optimization under Linear Interactions

We investigate relaxations for a class of discrete bilevel programs where the interaction constraints linking the leader and the follower are linear. Our approach reformulates the upper-level optimality constraints by projecting the leader's decisions onto vectors that map to distinct follower solution values, each referred to as a state. Based on such a state representation, we develop a network-flow linear program via a decision diagram that captures the convex hull of the follower's value function graph, leading to a new single-level reformulation of the bilevel problem. We also present a reduction procedure that exploits symmetry to identify the reformulation of minimal size. For large networks, we introduce parameterized relaxations that aggregate states by considering tractable hyperrectangles based on lower and upper bounds associated with the interaction constraints, and can be integrated into existing mixed-integer bilevel linear programming (MIBLP) solvers. Numerical experiments suggest that the new relaxations, whether used within a simple cutting-plane procedure or integrated into state-of-the-art MIBLP solvers, significantly reduce runtimes or solve additional benchmark instances. Our findings also highlight the correlation between the quality of relaxations and the properties of the interaction matrix, underscoring the potential of our approach in enhancing solution methods for structured bilevel optimization instances.

math.OC↗

arXiv2024

The Madness of Multiple Entries in March Madness

This paper explores multi-entry strategies for betting pools related to single-elimination tournaments. In such betting pools, participants select winners of games, and their respective score is a weighted sum of the number of correct selections. Most betting pools have a top-heavy payoff structure, so the paper focuses on strategies that maximize the expected score of the best-performing entry. There is no known closed-formula expression for the estimation of this metric, so the paper investigates the challenges associated with the estimation and the optimization of multi-entry solutions. We present an exact dynamic programming approach for calculating the maximum expected score of any given fixed solution, which is exponential in the number of entries. We explore the structural properties of the problem to develop several solution techniques. In particular, by extracting insights from the solutions produced by one of our algorithms, we design a simple yet effective problem-specific heuristic that was the best-performing technique in our experiments, which were based on real-world data extracted from recent March Madness tournaments. In particular, our results show that the best 100-entry solution identified by our heuristic had a 2.2% likelihood of winning a $1 million prize in a real-world betting pool.

cs.DS↗

arXiv2024

Heuristic Multiobjective Discrete Optimization using Restricted Decision Diagrams

Decision diagrams (DDs) have emerged as a state-of-the-art method for exact multiobjective integer linear programming. When the DD is too large to fit into memory or the decision-maker prefers a fast approximation to the Pareto frontier, the complete DD must be restricted to a subset of its states (or nodes). We introduce new node-selection heuristics for constructing restricted DDs that produce a high-quality approximation of the Pareto frontier. Depending on the structure of the problem, our heuristics are based on either simple rules, machine learning with feature engineering, or end-to-end deep learning. Experiments on multiobjective knapsack, set packing, and traveling salesperson problems show that our approach is highly effective, recovering over 85% of the Pareto frontier while achieving 2.5x speedups over exact DD enumeration on average, with very few non-Pareto solutions. The code is available at https://github.com/rahulptel/HMORDD.

cs.AI↗

arXiv2023

BDD for Complete Characterization of a Safety Violation in Linear Systems with Inputs

The control design tools for linear systems typically involves pole placement and computing Lyapunov functions which are useful for ensuring stability. But given higher requirements on control design, a designer is expected to satisfy other specification such as safety or temporal logic specification as well, and a naive control design might not satisfy such specification. A control designer can employ model checking as a tool for checking safety and obtain a counterexample in case of a safety violation. While several scalable techniques for verification have been developed for safety verification of linear dynamical systems, such tools merely act as decision procedures to evaluate system safety and, consequently, yield a counterexample as an evidence to safety violation. However these model checking methods are not geared towards discovering corner cases or re-using verification artifacts for another sub-optimal safety specification. In this paper, we describe a technique for obtaining complete characterization of counterexamples for a safety violation in linear systems. The proposed technique uses the reachable set computed during safety verification for a given temporal logic formula, performs constraint propagation, and represents all modalities of counterexamples using a binary decision diagram (BDD). We introduce an approach to dynamically determine isomorphic nodes for obtaining a considerably reduced (in size) decision diagram. A thorough experimental evaluation on various benchmarks exhibits that the reduction technique achieves up to $67\%$ reduction in the number of nodes and $75\%$ reduction in the width of the decision diagram.

eess.SY↗

arXiv2022

Recursive McCormick Linearization of Multilinear Programs

Linear programming (LP) relaxations are widely employed in exact solution methods for multilinear programs (MLP). One example is the family of Recursive McCormick Linearization (RML) strategies, where bilinear products are substituted for artificial variables, which deliver a relaxation of the original problem when introduced together with concave and convex envelopes. In this article, we introduce the first systematic approach for identifying RMLs, in which we focus on the identification of linear relaxation with a small number of artificial variables and with strong LP bounds. We present a novel mechanism for representing all the possible RMLs, which we use to design an exact mixed-integer programming (MIP) formulation for the identification of minimum-size RMLs; we show that this problem is NP-hard in general, whereas a special case is fixed-parameter tractable. Moreover, we explore structural properties of our formulation to derive an exact MIP model that identifies RMLs of a given size with the best possible relaxation bound is optimal. Our numerical results on a collection of benchmarks indicate that our algorithms outperform the RML strategy implemented in state-of-the-art global optimization solvers.

math.OC↗

arXiv2022

Constrained Shortest-Path Reformulations via Decision Diagrams for Structured Two-stage Optimization Problems

Many discrete optimization problems are amenable to constrained shortest-path reformulations in an extended network space, a technique that has been key in convexification, bound strengthening, and search. In this paper, we propose a constrained variant of these models for two challenging classes of discrete two-stage optimization problems, where traditional methods (e.g., dualize-and-combine) are not applicable compared to their continuous counterparts. Specifically, we propose a framework that models problems as decision diagrams and introduces side constraints either as linear inequalities in the underlying polyhedral representation, or as state variables in shortest-path dynamic programming models. For our first structured class, we investigate two-stage problems with interdiction constraints. We show that such constraints can be formulated as indicator functions in the arcs of the diagram, providing an alternative single-level reformulation of the problem via a network-flow representation. Our second structured class is classical robust optimization, where we leverage the decision diagram network to iteratively identify label variables, akin to an L-shaped method. We evaluate these strategies on a competitive project selection problem and the robust traveling salesperson with time windows, observing considerable improvements in computational efficiency as compared to general methods in the respective areas.

math.OC↗

arXiv2022

Constraint Learning to Define Trust Regions in Predictive-Model Embedded Optimization

There is a recent proliferation of research on the integration of machine learning and optimization. One expansive area within this research stream is predictive-model embedded optimization, which proposes the use of pre-trained predictive models as surrogates for uncertain or highly complex objective functions. In this setting, features of the predictive models become decision variables in the optimization problem. Despite a recent surge in publications in this area, only a few papers note the importance of incorporating trust region considerations in this decision-making pipeline, i.e., enforcing solutions to be similar to the data used to train the predictive models. Without such constraints, the evaluation of the predictive model at solutions obtained from optimization cannot be trusted and the practicality of the solutions may be unreasonable. In this paper, we provide an overview of the approaches appearing in the literature to construct a trust region, and propose three alternative approaches. Our numerical evaluation highlights that trust-region constraints learned through isolation forests, one of the newly proposed approaches, outperform all previously suggested approaches, both in terms of solution quality and computational time.

cs.LG↗

arXiv2021

Optimizing the expected maximum of two linear functions defined on a multivariate Gaussian distribution

We study stochastic optimization problems with objective function given by the expectation of the maximum of two linear functions defined on the component random variables of a multivariate Gaussian distribution. We consider random variables that are arbitrarily correlated, and we show that the problem is NP-hard even if the space of feasible solutions is unconstrained. We exploit a closed-form expression for the objective function from the literature to construct a cutting-plane algorithm that can be seen as an extension of the integer L-shaped method for a highly nonlinear function, which includes the evaluation of the c.d.f and p.d.f of a standard normal random variable with decision variables as part of the arguments. To exhibit the model's applicability, we consider two featured applications. The first is daily fantasy sports, where the algorithm identifies entries with positive returns during the 2018-2019 National Football League season. The second is a special case of makespan minimization for two parallel machines and jobs with uncertain processing times; for the special case where the jobs are uncorrelated, we prove the equivalence between its deterministic and stochastic versions and show that our algorithm can deliver a constant-factor approximation guarantee for the problem. The results of our computational evaluation involving synthetic and real-world data suggest that our discretization and upper bounding techniques lead to significant computational improvements and that the proposed algorithm outperforms sub-optimal solutions approaches.

math.OC↗

arXiv2021

Optimizing over an ensemble of neural networks

We study optimization problems where the objective function is modeled through feedforward neural networks with rectified linear unit (ReLU) activation. Recent literature has explored the use of a single neural network to model either uncertain or complex elements within an objective function. However, it is well known that ensembles of neural networks produce more stable predictions and have better generalizability than models with single neural networks, which motivates the investigation of ensembles of neural networks rather than single neural networks in decision-making pipelines. We study how to incorporate a neural network ensemble as the objective function of an optimization model and explore computational approaches for the ensuing problem. We present a mixed-integer linear program based on existing popular big-M formulations for optimizing over a single neural network. We develop a two-phase approach for our model that combines preprocessing procedures to tighten bounds for critical neurons in the neural networks with a Lagrangian relaxation-based branch-and-bound approach. Experimental evaluations of our solution methods suggest that using ensembles of neural networks yields more stable and higher quality solutions, compared to single neural networks, and that our optimization algorithm outperforms (the adaption of) a state-of-the-art approach in terms of computational time and optimality gaps.

cs.LG↗

arXiv2019

JANOS: An Integrated Predictive and Prescriptive Modeling Framework

Business research practice is witnessing a surge in the integration of predictive modeling and prescriptive analysis. We describe a modeling framework JANOS that seamlessly integrates the two streams of analytics, for the first time allowing researchers and practitioners to embed machine learning models in an optimization framework. JANOS allows for specifying a prescriptive model using standard optimization modeling elements such as constraints and variables. The key novelty lies in providing modeling constructs that allow for the specification of commonly used predictive models and their features as constraints and variables in the optimization model. The framework considers two sets of decision variables; regular and predicted. The relationship between the regular and the predicted variables are specified by the user as pre-trained predictive models. JANOS currently supports linear regression, logistic regression, and neural network with rectified linear activation functions, but we plan to expand on this set in the future. In this paper, we demonstrate the flexibility of the framework through an example on scholarship allocation in a student enrollment problem and provide a numeric performance evaluation.

cs.LG↗

arXiv2019

Template-based Minor Embedding for Adiabatic Quantum Optimization

Quantum Annealing (QA) can be used to quickly obtain near-optimal solutions for Quadratic Unconstrained Binary Optimization (QUBO) problems. In QA hardware, each decision variable of a QUBO should be mapped to one or more adjacent qubits in such a way that pairs of variables defining a quadratic term in the objective function are mapped to some pair of adjacent qubits. However, qubits have limited connectivity in existing QA hardware. This has spurred work on preprocessing algorithms for embedding the graph representing problem variables with quadratic terms into the hardware graph representing qubits adjacencies, such as the Chimera graph in hardware produced by D-Wave Systems. In this paper, we use integer linear programming to search for an embedding of the problem graph into certain classes of minors of the Chimera graph, which we call template embeddings. One of these classes corresponds to complete bipartite graphs, for which we show the limitation of the existing approach based on minimum Odd Cycle Transversals (OCTs). One of the formulations presented is exact, and thus can be used to certify the absence of a minor embedding using that template. On an extensive test set consisting of random graphs from five different classes of varying size and sparsity, we can embed more graphs than a state-of-the-art OCT-based approach, our approach scales better with the hardware size, and the runtime is generally orders of magnitude smaller.

cs.DS↗

arXiv2018

Binary Decision Diagrams for Bin Packing with Minimum Color Fragmentation

Bin Packing with Minimum Color Fragmentation (BPMCF) is an extension of the Bin Packing Problem in which each item has a size and a color and the goal is to minimize the sum of the number of bins containing items of each color. In this work, we introduce BPMCF and present a decomposition strategy to solve the problem, where the assignment of items to bins is formulated as a binary decision diagram and an optimal integrated solutions is identified through a mixed-integer linear programming model. Our computational experiments show that the proposed approach greatly outperforms a direct formulation of BPMCF and that its performance is suitable for large instances of the problem.

cs.DS↗

arXiv2018

Improving Optimization Bounds using Machine Learning: Decision Diagrams meet Deep Reinforcement Learning

Finding tight bounds on the optimal solution is a critical element of practical solution methods for discrete optimization problems. In the last decade, decision diagrams (DDs) have brought a new perspective on obtaining upper and lower bounds that can be significantly better than classical bounding mechanisms, such as linear relaxations. It is well known that the quality of the bounds achieved through this flexible bounding method is highly reliant on the ordering of variables chosen for building the diagram, and finding an ordering that optimizes standard metrics is an NP-hard problem. In this paper, we propose an innovative and generic approach based on deep reinforcement learning for obtaining an ordering for tightening the bounds obtained with relaxed and restricted DDs. We apply the approach to both the Maximum Independent Set Problem and the Maximum Cut Problem. Experimental results on synthetic instances show that the deep reinforcement learning approach, by achieving tighter objective function bounds, generally outperforms ordering methods commonly used in the literature when the distribution of instances is known. To the best knowledge of the authors, this is the first paper to apply machine learning to directly improve relaxation bounds obtained by general-purpose bounding mechanisms for combinatorial optimization problems.

cs.AI↗

arXiv2018

Seamless Multimodal Transportation Scheduling

Ride-hailing services have expanded the role of shared mobility in passenger transportation systems, creating new markets and creative planning solutions for major urban centers. In this paper, we consider their use for the first-mile or last-mile passenger transportation in coordination with a mass transit service to provide a seamless multimodal transportation experience for the user. A system that provides passengers with predictable information on travel and waiting times in their commutes is immensely valuable. We envision that the passengers will inform the system of their desired travel and arrival windows so that the system can jointly optimize the schedules of passengers. The problem we study balances minimizing travel time and the number of trips taken by the last-mile vehicles, so that long-term planning, maintenance, and environmental impact are all taken into account. We focus on the case where the last-mile service aggregates passengers by destination. We show that this problem is NP-hard, and propose a decision diagram-based branch-and-price decomposition model that can solve instances of real-world size (10,000 passengers spread over an hour, 50 last-mile destinations, 600 last-mile vehicles) in computational time (~1 minute) that is orders-of-magnitude faster than other methods appearing in the literature. Our experiments also indicate that aggregating passengers by destination on the last-mile service provides high-quality solutions to more general settings.

math.OC↗

arXiv2018

Predictive and Prescriptive Analytics for Location Selection of Add-on Retail Products

In this paper, we study an analytical approach to selecting expansion locations for retailers selling add-on products whose demand is derived from the demand of another base product. Demand for the add-on product is realized only as a supplement to the demand of the base product. In our context, either of the two products could be subject to spatial autocorrelation where demand at a given location is impacted by demand at other locations. Using data from an industrial partner selling add-on products, we build predictive models for understanding the derived demand of the add-on product and establish an optimization framework for automating expansion decisions to maximize expected sales. Interestingly, spatial autocorrelation and the complexity of the predictive model impact the complexity and the structure of the prescriptive optimization model. Our results indicate that the models formulated are highly effective in predicting add-on product sales, and that using the optimization framework built on the predictive model can result in substantial increases in expected sales over baseline policies.

stat.AP↗

arXiv2018

On Finding Stable and Efficient Solutions for the Team Formation Problem

The assignment of personnel to teams is a fundamental and ubiquitous managerial function, typically involving several objectives and a variety of idiosyncratic practical constraints. Despite the prevalence of this task in practice, the process is seldom approached as a precise optimization problem over the reported preferences of all agents. This is due in part to the underlying computational complexity that occurs when quadratic (i.e., intra-team interpersonal) interactions are taken into consideration, and also due to game-theoretic considerations, when those taking part in the process are self-interested agents. Variants of this fundamental decision problem arise in a number of settings, including, for example, human resources and project management, military platooning, sports-league management, ride sharing, data clustering, and in assigning students to group projects. In this paper, we study a mathematical-programming approach to "team formation" focused on the interplay between two of the most common objectives considered in the related literature: economic efficiency (i.e., the maximization of social welfare) and game-theoretic stability (e.g., finding a core solution when one exists). With a weighted objective across these two goals, the problem is modeled as a bi-level binary optimization problem, and transformed into a single-level, exponentially sized binary integer program. We then devise a branch-cut-and-price algorithms and demonstrate its efficacy through an extensive set of simulations, with favorable comparisons to other algorithms from the literature.

math.OC↗

arXiv2018

Network Models for Multiobjective Discrete Optimization

This paper provides a novel framework for solving multiobjective discrete optimization problems with an arbitrary number of objectives. Our framework formulates these problems as network models, in that enumerating the Pareto frontier amounts to solving a multicriteria shortest path problem in an auxiliary network. We design techniques for exploiting the network model in order to accelerate the identification of the Pareto frontier, most notably a number of operations to simplify the network by removing nodes and arcs while preserving the set of nondominated solutions. We show that the proposed framework yields orders-of-magnitude performance improvements over existing state-of-the-art algorithms on five problem classes containing both linear and nonlinear objective functions.

math.OC↗