Browsing by Author "Schaefer, Andrew J"

Now showing 1 - 8 of 8
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    A Machine Learning Approach for Quadratic and Linear Programming Value Functions
    (2023-04-21) Antley, Eric McSwain; Schaefer, Andrew J; Huchette, Joey
    Numerous modeling problems in optimization depend on how the optimal value of a linear or quadratic program changes as a function of its right-hand side. For example, in many stochastic programming formulations, the recourse function is explicitly a function of the second-stage right-hand sides. In bilevel programs, the follower's decisions are constrained to be optimal with respect to their own problem of interest which may be implicitly interpreted as a constraint contingent on how the value of follower optimization problem changes with respect to the leader's decisions. Computational interest in the wide range of applications stemming from the ability to predict how an optimization problem's optimal values vary with respect to changes in constraint values have motivated the study of the \textit{value function}. The value function of a linear or quadratic program parametrizes the optimal value of an optimization problem by the value of its right-hand side. Previous researchers have shown that value functions are highly structured objects that are amenable for various computational applications. This thesis develops and analyzes how machine learning techniques used for functional regression problems can be used to learn the value function of continuous linear and quadratic programs. We study and design models to verify the accuracy of neural network representations of the value function, and use the trained network value functions as constraints within various optimization problems.
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    Applications of Mixed Integer Programming to Cloud Computing: Modeling and Computation
    (2024-05-10) Alfant, Rachael M; Perez-Salazar, Sebastian; Schaefer, Andrew J
    Demand for computing capacity in the cloud is generally not easily forecast; however, sub-optimal pricing and mis-allocation of cloud computing resources both have negative consequences for users and providers of cloud computing. This thesis approaches pricing and capacity allocation in cloud computing through the lens of stochastic mixed integer programming (SMIP), which provides a particularly useful framework for solving large, complex decision-making problems under uncertainty. Often, the uncertainty inherent to SMIPs manifests in the right-hand side (demand) vector. Thus, it is important to have a framework by which to assess a mixed integer programming (MIP) model’s quality over unknown or stochastic right-hand sides. As such, this thesis explores both theoretical and practical applications of SMIPs and MIPs with unknown right-hand sides. In particular, this thesis develops theoretical evaluative metrics for MIPs over multiple right-hand sides via gap functions, presents several stochastic optimization approaches to optimal pricing in the cloud, and formulates waste-minimizing (revenue-maximizing) SMIP models that optimize capacity allocation in the cloud.
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    Evaluating Mixed-Integer Programming Models over Multiple Right-hand Sides
    (2022-04-14) Alfant, Rachael May; Schaefer, Andrew J
    A critical measure of the quality of a mixed-integer programming (MIP) model with fixed data is the difference, or gap, between the optimal objective value of the linear programming relaxation and that of the corresponding MIP. In many contexts, only an approximation of the right-hand sides may be available, or there may be multiple right-hand sides of interest. Yet, there is currently no consensus on appropriate measures for MIP model quality over a range of right-hand sides. In this thesis, we provide formulations of optimization problems that represent the expectation and extrema of both absolute and relative MIP gap functions over finite discrete sets. In addition, we provide the conditions under which absolute MIP gap functions are periodic. Thus, we provide a framework by which to determine a MIP model's quality over multiple right-hand sides.
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    Integer Programming Approaches to Cancer Treatment: Objective Selection in Intensity-Modulated Radiation Therapy and Chemotherapy Treatment Design
    (2020-08-03) Ajayi, Temitayo; Schaefer, Andrew J
    In this thesis, we present multiple problems that arise in cancer treatment decision-making, and we analyze and implement methods used to solve them. The first problem is the selection of objectives that reflect latent clinical preferences during radiation therapy treatment. By connecting an inverse optimization formulation with greedy and regularized solution approaches, we show that sparse sets of objectives can be retrieved effectively. The development of the greedy forward selection approaches for objective selection leads to an in-depth exploration of the greedy algorithm's performance when the optimized set function is approximately submodular. In addition to the greedy algorithm, we study approximate submodularity in other areas in discrete optimization. The second cancer treatment problem is combination chemotherapy optimization, which requires merging differential equations that model dynamics together with discrete decision variables for complex operational constraints. We formulate this problem as a mixed-integer linear program and solve two models, one that focuses on tumor shrinkage and another that minimizes toxicity.
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    Neural Network Verification as Piecewise Linear Optimization: Formulations for the Composition of Staircase Functions
    (2024-05-16) Nguyen, Tu Anh; Schaefer, Andrew J
    We present a technique for neural network verification using mixed-integer programming (MIP) formulations. We derive a \emph{strong formulation} for each neuron in a network using piecewise linear activation functions. Additionally, as in general, these formulations may require an exponential number of inequalities, we also derive a separation procedure that runs in super-linear time in the input dimension. We first introduce and develop our technique on the class of \emph{staircase} functions, which generalizes the ReLU, binarized, and quantized activation functions. We then use results for staircase activation functions to obtain a separation method for general piecewise linear activation functions. Empirically, using our strong formulation and separation technique, we can reduce the computational time in exact verification settings based on MIP and improve the false negative rate for inexact verifiers relying on the relaxation of the MIP formulation. While originally developed for neural network formulation, the MIP formulation and its technical results draw heavily on classical theory in linear optimization, and may be of independent interest to other applications.
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    Optimal Outcome-Based Regulation in Lung Transplantation: Modeling and ComputationOptimal Outcome-Based Regulation in Lung Transplantation: Modeling and Computation
    (2021-06-15) Mildebrath, David T. K.; Schaefer, Andrew J
    Organ transplantation is an increasingly common therapy for many types of end-stage organ failure, including heart disease, chronic obstructive pulmonary disease, and end-stage renal disease. Unfortunately, the supply of available organs has not kept pace with this increasing demand. In order to ensure the efficient utilization of scarce organs, the past 20 years has seen increased scrutiny of post-transplant outcomes in the United States. This scrutiny has come in the form of two sets of closely-related regulations, overseen by the Organ Procurement Transplantation Network (OPTN) and Centers for Medicare and Medicaid (CMS), which penalize transplant programs with worse-than-expected transplant outcomes. In spite of their stated goals of improving outcomes, these regulations have led to adverse unintended consequences. Most notably, there is evidence that these rules may have caused some programs to reject certain medically-suitable patients that are perceived to be ``high-risk,'' in order to avoid penalization. However, there remains debate in the literature over what drives this risk-averse behavior, and whether it is even a rational response by transplant programs seeking to avoid penalization. In this dissertation, we study the problem of regulatory-induced risk aversion in lung transplantation from the perspective of both the transplant program and regulators. To study program behavior, we present the first mathematical models of CMS and OPTN outcome-based regulations from the transplant program perspective. By calibrating our models with real data, we demonstrate that a rational program may reduce its risk of penalization by rejecting certain medically-suitable patients, thereby answering an open question in the clinical literature. We explore the incentives created by the regulations using a game-theoretic model, and find evidence that the large penalties associated with CMS penalization may be the primary driver of observed adverse patient selection. Motivated by this finding, we propose a new regulatory mechanism that is similar to current CMS regulations in other healthcare domains, and demonstrate that our proposed scheme may eliminate the incentive for programs to reject medically-suitable patients.
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    Optimizing Waitlist Composition from the Transplant Center's Perspective
    (2018-11-13) Mildebrath, David T.K.; Schaefer, Andrew J
    We present the first model to optimize patient selection for lung transplantation from the perspective of the transplant center. In 2007, the Centers for Medicare and Medicaid Services (CMS) introduced regulations designed to improve medical outcomes at transplant centers by financially penalizing transplant centers with high one-year post-transplant mortality rates. Since then, much work has focused on optimal allocation of organs within organ donation networks, but none has focused on optimizing the center-level patient mix in order to comply with CMS regulations. We introduce a chance-constrained non-linear mixed-integer programming model to maximize the number of patients who receive a transplant, while constraining the risk of penalization by the CMS. In addition, our model can be used to model dynamic pay-for-performance systems, which have seen growing popularity in the medical context. The present work may contribute to significant medical cost reduction and improve post-surgery survival rates for transplant recipients, thereby serving to keep more transplant centers open.
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    Single-Scenario Facet Preservation for Stochastic Mixed-Integer Programs
    (2023-12-04) Karagoz, Aysenur; Schaefer, Andrew J
    We consider improving the polyhedral representation of the extensive form of an SMIP. Given a facet-defining valid inequality for a single-scenario version of the SMIP, we provide conditions under which the same inequality remains facet-defining for the extensive form. Our main result gives necessary and sufficient conditions for a facet-defining inequality for a single-scenario version to be facet-defining for the extensive form. We then present several implications, which show that various recourse structures from the literature satisfy these conditions. For example, for an SMIP with simple recourse, any single-scenario facet is also a facet for the extensive form. More general recourse structures require additional mild assumptions for these conditions to hold.