Data analytics and optimization theory are well-developed techniques to describe, predict and optimize real-world systems, and they have been widely used in engineering and science. This dissertation focuses on applications in biomedical systems, ranging from the scale of microbial communities to problems relating to human disease and health care. Starting from the microbial level, the first problem considered is to design metabolic division of labor in microbial communities. Given a number of microbial species living in a community, the starting point of the analysis is a list of all metabolic reactions present in the community, expressed in terms of the metabolite proportions involved in each reaction. Leveraging tools from Flux Balance Analysis (FBA), the problem is formulated as a Mixed Integer Program (MIP) and new methods are developed to solve large scale instances. The strategies found reveal a large space of nuanced and non-intuitive metabolic division of labor opportunities, including, for example, splitting the Tricarboxylic Acid Cycle (TCA) cycle into two separate halves. More broadly, the landscape of possible 1-, 2-, and 3-strain solutions is systematically mapped at increasingly tight constraints on the number of allowed reactions. The second problem addressed involves the prediction and prevention of short-term (30-day) hospital re-admissions. To develop predictive models, a variety of classification algorithms are adapted and coupled with robust (regularized) learning and heuristic feature selection approaches. Using real, large datasets, these methods are shown to reliably predict re-admissions of patients undergoing general surgery, within 30-days of discharge. Beyond predictions, a novel prescriptive method is developed that computes specific control actions with the effect of altering the outcome. This method, termed Prescriptive Support Vector Machines (PSVM), is based on an underlying SVM classifier. Applied to the hospital re-admission data, it is shown to reduce 30-day re-admissions after surgery through better control of the patient’s pre-operative condition. Specifically, using the new method the patient’s pre-operative hematocrit is regulated through limited blood transfusion. In the last problem in this dissertation, a framework for parameter estimation in Regularized Mixed Linear Regression (MLR) problems is developed. In the specific MLR setting considered, training data are generated from a mixture of distinct linear models (or clusters) and the task is to identify the corresponding coefficient vectors. The problem is formulated as a Mixed Integer Program (MIP) subject to regularization constraints on the coefficient vectors. A number of results on the convergence of parameter estimates for MLR are established. In addition, experimental prediction results are presented comparing the prediction algorithm with mean absolute error regression and random forest regression, in terms of both accuracy and interpretability.

<p>In this thesis, we study optimization problems which have a combinatorial aspect to them. Search space for such problems quickly grows large - exponentially - with respect to the problem dimension. Thus, exhaustive search becomes intractable and we need good relaxations to solve combinatorial problems efficiently. Another challenge arises due to the high dimensionality of such problems and lack of large number of samples. Our aim is to come up with innovative approaches that solve the problem in polynomial time and sample complexity. We discuss three combinatorial optimization problems and provide continuous relaxations for them. Our continuous relaxations involve both convex and nonconvex (invex) relaxations. Furthermore, we provide efficient first order algorithms to solve a general class of invex problems with provable convergence rate guarantees. The three combinatorial problems we study in this work are – learning the directed structure of a Bayesian network using blackbox data, fair sparse regression on a biased dataset where bias depends upon a hidden binary attribute and mixed linear regression. We propose convex relaxation for the first problem, while the other two are solved using invex relaxation. On the first problem, we come up with a novel notion of low rank representation of conditional probability tables for a Bayesian network and connect it to Fourier transformation of real valued set functions to recover the exact structure of the Bayesian networks. For the second problem, we propose a novel invex relaxation for the combinatorial version of sparse linear regression with fairness. For the final problem, we again use invex relaxation to learn a mixture of sparse linear regression models. We formally show correctness of our proposed methods and provide provable theoretical guarantees for efficient computational and sample complexity. We also develop efficient first order algorithms to solve invex problems. We provide convergence rate analysis for our proposed methods. Furthermore, we also discuss possible future research directions and the problems we want to tackle in future.</p>

<p dir="ltr">The National Flood Insurance Program (NFIP), which is run by the U.S. Federal Emergency Management Agency (FEMA), is presently under huge debt to the U.S. treasury. The debt is primarily caused by low flood insurance take-up rate, low willingness to pay for flood insurance, and large payouts after major disasters. Addressing this insolvency problem requires the NFIP to understand (1) what drives the demand for flood insurance so that it can be increased, (2) how risk factors contribute towards large flood insurance payouts so that effective risk reduction policies can be planned, and (3) how to predict the future flood insurance payouts so that the NFIP can be financially prepared. This research has answered these three fundamental questions by developing empirical models based on historical data. To answer the first question, this research has developed a propensity score-based causal model that analyzed one of the key components that influences the demand for flood insurance – the availability of post-disaster government assistance. It was found that the availability of the federal payout in a county in a year increased the number of flood insurance policies by 5.2% and the total insured value of the policies by 4.6% in the following year. Next, this research has developed Mixed Effects Regression model that quantified the causal relationships between the annual flood insurance payout in a county and flood related risk factors such as flood exposure, infrastructure vulnerability, social vulnerability, community resilience, and the number of mobile homes in the county. Based on the derived causal estimates, it was predicted that climate change, which is expected to increase flood exposure in coastal counties, will increase the annual NFIP payout in New Orleans, Louisiana by $2.04 billion in the next 30 years. Lastly, to make the NFIP financially prepared for future payouts, this research has developed a predictive model that can predict the annual NFIP payout in a county with adequate predictive accuracy. The predictive model was used to predict the NFIP payout for 2021 and it was able to predict that with a 9.8% prediction error. The outcomes of this research create new knowledge to inform policy decisions and strategies aimed at fortifying the NFIP. This includes strategies such as flood protection infrastructure, tailored disaster assistance, and other interventions that can bolster flood insurance uptake while mitigating the risk of substantial payouts. Ultimately, this research contributes to sustaining the NFIP's ability to provide vital flood insurance coverage to millions of Americans.</p>