Inverse Optimization with Noisy Data

Inverse optimization refers to the inference of unknown parameters of an optimization problem based on knowledge of its optimal solutions. This paper considers inverse optimization in the setting where measurements of the optimal solutions of a convex optimization problem are corrupted by noise. We first provide a formulation for inverse optimization and prove it to be NP-hard. In contrast to existing methods, we show that the parameter estimates produced by our formulation are statistically consistent. Our approach involves combining a new duality-based reformulation for bilevel programs with a regularization scheme that smooths discontinuities in the formulation. Using epi-convergence theory, we show the regularization parameter can be adjusted to approximate the original inverse optimization problem to arbitrary accuracy, which we use to prove our consistency results. Next, we propose two solution algorithms based on our duality-based formulation. The first is an enumeration algorithm that is applicable to settings where the dimensionality of the parameter space is modest, and the second is a semiparametric approach that combines nonparametric statistics with a modified version of our formulation. These numerical algorithms are shown to maintain the statistical consistency of the underlying formulation. Lastly, using both synthetic and real data, we demonstrate that our approach performs competitively when compared with existing heuristics

Robust Facility Location under Demand Location Uncertainty

In this thesis, we generalize a set of facility location models within a two-stage robust optimization framework by assuming each demand is only known to lie within a continuous and bounded uncertainty region. Our approach involves discretizing each uncertainty region into a set of finite scenarios, each of which represents a potential location where the demand may be realized. We show that the gap between the optimal values of the theorized continuous uncertainty problem and our discretized model can be bounded by a function of the granularity of the discretization. We then propose a solution technique based on row-and-column generation, and compare its performance with existing solution methods. Lastly, we apply our robust location models to the problem of ambulance positioning using cardiac arrest location data from the City of Toronto, and show that hedging against demand location uncertainty may help decrease EMS response times to cardiac arrest emergencies.MAS

Inverse Optimization, Incentive Design and Healthcare Policy

This dissertation presents mathematical models and algorithms that draw from optimization and statistics and are motivated by practical problems in operations management. We discuss theoretical properties of the proposed models as well as their relevance to practice. In particular, we focus on the role these models can play in addressing challenges in healthcare operations and policy. In Chapter 2, we address the problem of building models of agent behavior from observational data regarding the agent's decisions. Concretely, we consider the inverse optimization problem, which refers to the estimation of unknown model parameters of a convex optimization problem from observations of its optimal solutions. First, we provide a new formulation for inverse optimization, which takes the form of a bi-level program where the optimality conditions of the lower level program are expressed using strong duality. In contrast to existing methods, we show that the parameter estimates produced by our formulation are statistically consistent under appropriate conditions. Second, we propose two solution algorithms based on our duality-based formulation: an enumeration algorithm that is applicable to settings where the dimensionality of the parameter space is modest, and a semiparametric approach that combines nonparametric statistics with a modified version of our formulation. These numerical algorithms are shown to maintain the statistical consistency of the underlying formulation. Lastly, using both synthetic and real data, we demonstrate that our approach performs competitively when compared with existing heuristics.In Chapter 3, we employ an inverse optimization approach to redesign a class of Medicare contracts. We formulate the existing contract between Medicare and a provider as a principal-agent model. We then propose an alternate contract, which we show to dominate the status quo contract under reasonable conditions by producing a strictly higher expected payoff for both Medicare and the provider. We then propose an estimator based on inverse optimization for estimating a model of provider behavior, using a dataset containing the financial performance of a group of Medicare providers that account for 7 million beneficiaries and over $70 billion in Medicare spending. We estimate a performance improvement -- in terms of savings accrued by Medicare -- of 40% under the alternate contract, which suggests significant room for improvement in the status quo.In Chapter 4, we propose a data-driven modeling approach to facility location in a setting where the location of demand points is subject to uncertainty. The model is motivated by the problem of placing automated external defibrillators in public locations in anticipation of sudden cardiac arrest. We propose a distributionally robust optimization approach where the demand distribution is continuous in the plane and uncertain. We propose a solution technique based on row-and-column generation that exploits the structure of the uncertainty set and allows us to solve practical-sized instances of the defibrillator deployment problem. Using real cardiac arrest data, we conduct an extensive numerical study and find that hedging against cardiac arrest location uncertainty can produce defibrillator deployments that outperform a intuitive sample average approximation by 9 to 15%. Our findings suggest that accounting for cardiac arrest location uncertainty can lead to improved accessibility of defibrillators during cardiac arrest emergencies and the potential for improved survival outcomes

Inverse Optimization, Incentive Design and Healthcare Policy

This dissertation presents mathematical models and algorithms that draw from optimization and statistics and are motivated by practical problems in operations management. We discuss theoretical properties of the proposed models as well as their relevance to practice. In particular, we focus on the role these models can play in addressing challenges in healthcare operations and policy. In Chapter 2, we address the problem of building models of agent behavior from observational data regarding the agent's decisions. Concretely, we consider the inverse optimization problem, which refers to the estimation of unknown model parameters of a convex optimization problem from observations of its optimal solutions. First, we provide a new formulation for inverse optimization, which takes the form of a bi-level program where the optimality conditions of the lower level program are expressed using strong duality. In contrast to existing methods, we show that the parameter estimates produced by our formulation are statistically consistent under appropriate conditions. Second, we propose two solution algorithms based on our duality-based formulation: an enumeration algorithm that is applicable to settings where the dimensionality of the parameter space is modest, and a semiparametric approach that combines nonparametric statistics with a modified version of our formulation. These numerical algorithms are shown to maintain the statistical consistency of the underlying formulation. Lastly, using both synthetic and real data, we demonstrate that our approach performs competitively when compared with existing heuristics.In Chapter 3, we employ an inverse optimization approach to redesign a class of Medicare contracts. We formulate the existing contract between Medicare and a provider as a principal-agent model. We then propose an alternate contract, which we show to dominate the status quo contract under reasonable conditions by producing a strictly higher expected payoff for both Medicare and the provider. We then propose an estimator based on inverse optimization for estimating a model of provider behavior, using a dataset containing the financial performance of a group of Medicare providers that account for 7 million beneficiaries and over $70 billion in Medicare spending. We estimate a performance improvement -- in terms of savings accrued by Medicare -- of 40% under the alternate contract, which suggests significant room for improvement in the status quo.In Chapter 4, we propose a data-driven modeling approach to facility location in a setting where the location of demand points is subject to uncertainty. The model is motivated by the problem of placing automated external defibrillators in public locations in anticipation of sudden cardiac arrest. We propose a distributionally robust optimization approach where the demand distribution is continuous in the plane and uncertain. We propose a solution technique based on row-and-column generation that exploits the structure of the uncertainty set and allows us to solve practical-sized instances of the defibrillator deployment problem. Using real cardiac arrest data, we conduct an extensive numerical study and find that hedging against cardiac arrest location uncertainty can produce defibrillator deployments that outperform a intuitive sample average approximation by 9 to 15%. Our findings suggest that accounting for cardiac arrest location uncertainty can lead to improved accessibility of defibrillators during cardiac arrest emergencies and the potential for improved survival outcomes

Modeling the impact of public access defibrillator range on public location cardiac arrest coverage

Background Public access defibrillation with automated external defibrillators (AEDs) can improve survival from out-of-hospital cardiac arrests (OHCA) occurring in public. Increasing the effective range of AEDs may improve coverage for public location OHCAs. Objective To quantify the relationship between AED effective range and public location cardiac arrest coverage. Methods This was a retrospective cohort study using the Resuscitation Outcomes Consortium Epistry database. We included all public-location, atraumatic, EMS-attended OHCAs in Toronto, Canada between December 16, 2005 and July 15, 2010. We ran a mathematical model for AED placement that maximizes coverage of historical public OHCAs given pre-specified values of AED effective range and the number of locations to place AEDs. Locations of all non-residential buildings were obtained from the City of Toronto and used as candidate sites for AED placement. Coverage was evaluated for range values from 10 to 300 m and number of AED locations from 10 to 200, both in increments of 10, for a total of 600 unique scenarios. Coverage from placing AEDs in all public buildings was also measured. Results There were 1310 public location OHCAs during the study period, with 25,851 non-residential buildings identified as candidate sites for AED placement. Cardiac arrest coverage increased with AED effective range, with improvements in coverage diminishing at higher ranges. For example, for a deployment of 200 AED locations, increasing effective range from 100 m to 200 m covered an additional 15% of cardiac arrests, whereas increasing range further from 200 m to 300 m covered an additional 10%. Placing an AED in each of the 25,851 public buildings resulted in coverage of 50% and 95% under assumed effective ranges of 50 m and 300 m, respectively. Conclusion Increasing AED effective range can improve cardiac arrest coverage. Mathematical models can help evaluate the potential impact of initiatives which increase AED range