Thesis (M.S.)--West Virginia University, 2010. / Title from document title page. Document formatted into pages; contains ix, 128 p. : ill. (some col.). Includes abstract. Includes bibliographical references (p. 93-102).

In this thesis, we address a multi-objective personnel scheduling problem where personnel’s workload is uncertain and propose a two-stage robust modelling approach with demand uncertainty. In the first stage, we model a multi-objective personnel scheduling problem without incorporating demand coverage and, in the second stage, we minimize over or under-staffing after the realization of the demand and the assignments from the first stage. Two solution approaches are introduced for this model. The first approach solves the proposed model through a cutting plane strategy known as Benders dual cutting plane method, and the second approach reformulates the problem based on the strong duality theory. As a case study, the proposed model and the first solution approach are applied to an existing scheduling problem in the pathology department at The Ottawa Hospital. It is shown that the proposed model is successful at reducing the unmet demand while maintaining the performance with respect to other metrics when compared against the deterministic alternative.

Cross-docking is an important transportation logistics strategy in supply chain management which reduces transportation costs, inventory holding costs, order-picking costs and response time. Careful planning is needed for successful cross-dock operations. Uncertainty in cross-dock problems is inevitable and needs to be addressed. Almost all research in the cross-dock area assumes determinism. This dissertation considers uncertainty in cross-dock problems and optimizes these problems under uncertainty. We consider uncertainty in processing times, using scenario-based and protection-based robust approaches. Using a heuristic method, we find a lower and upper bound and combine that with a meta-heuristic method to solve the problem. Also, we consider problems in two different industries (Goodwill and H-E-B) and address the uncertainties that happen frequently in their operations. The scenario-based robust optimization model for the unloading problem using a min max objective is presented with examples. A surrogate heuristic procedure is used to find a robust solution. Next, a two-space genetic algorithm, a meta-heuristic procedure, is applied to the unloading problem using the bounds obtained by the heuristic procedure. The results are closer to the optimal solution than those obtained by the two-space genetic algorithm without bounds. When compared with the regular genetic algorithm with bounds, the two-space algorithm performs well. The protection-based approach considers a limit on the number of coefficients allowed to change with data uncertainty, protecting against the degree of conservatism. The management of trucks and reduction of overtime pay in the cross-dock operations of Goodwill is addressed through two models and uncertainty is applied to those models. A combined cross-dock operations model together with demand is formulated and the uncertainties are discussed for H-E-B operations. This dissertation does not address the recycling operation within the cross-dock of Goodwill, or the uncertainty in H-E-B data. / text

The use of a stochastic model to predict the likelihood of future outcomes forms an integral part of decision optimization under uncertainty. In classical stochastic modeling uncertain parameters are often assumed to be driven by a particular form of probability distribution. In practice however, the distributional form is often difficult to infer from the observed data, and the incorrect choice of distribution can lead to significant quality deterioration of resultant decisions and unexpected losses. In this thesis, we present new approaches for evaluating expected future performance that do not rely on an exact distributional specification and can be robust against the errors related to committing to a particular specification. The notion of comprehensive robustness is promoted, where various degrees of model misspecification are studied. This includes fundamental one such as unknown distributional form and more involved ones such as stochastic moments and moment outliers. The approaches are developed based on the techniques of moment-based optimization, where bounds on the expected performance are sought based solely on partial moment information. They can be integrated into decision optimization and generate decisions that are robust against model misspecification in a comprehensive manner. In the first part of the thesis, we extend the applicability of moment-based optimization to incorporate new objective functions such as convex risk measures and richer moment information such as higher-order multivariate moments. In the second part, new tractable optimization frameworks are developed that account for various forms of moment uncertainty in the context of decision analysis and optimization. Financial applications such as portfolio selection and option pricing are studied.

The use of a stochastic model to predict the likelihood of future outcomes forms an integral part of decision optimization under uncertainty. In classical stochastic modeling uncertain parameters are often assumed to be driven by a particular form of probability distribution. In practice however, the distributional form is often difficult to infer from the observed data, and the incorrect choice of distribution can lead to significant quality deterioration of resultant decisions and unexpected losses. In this thesis, we present new approaches for evaluating expected future performance that do not rely on an exact distributional specification and can be robust against the errors related to committing to a particular specification. The notion of comprehensive robustness is promoted, where various degrees of model misspecification are studied. This includes fundamental one such as unknown distributional form and more involved ones such as stochastic moments and moment outliers. The approaches are developed based on the techniques of moment-based optimization, where bounds on the expected performance are sought based solely on partial moment information. They can be integrated into decision optimization and generate decisions that are robust against model misspecification in a comprehensive manner. In the first part of the thesis, we extend the applicability of moment-based optimization to incorporate new objective functions such as convex risk measures and richer moment information such as higher-order multivariate moments. In the second part, new tractable optimization frameworks are developed that account for various forms of moment uncertainty in the context of decision analysis and optimization. Financial applications such as portfolio selection and option pricing are studied.

Thesis (Ph.D.)--City University of Hong Kong, 2009. / "Submitted to Department of Management Sciences in partial fulfillment of the requirements for the degree of Doctor of Philosophy." Includes bibliographical references (leaves 114-124)

Thesis (Ph. D.)--Ohio State University, 2007. / Title from first page of PDF file. Includes bibliographical references (p. 178-183).

Today we are living in the era of big data, there is a pressing need for efficient, scalable and robust optimization methods to analyze the data we create and collect. Although Convex methods offer tractable solutions with global optimality, heuristic nonconvex methods are often more attractive in practice due to their superior efficiency and scalability. Moreover, for better representations of the data, the mathematical model we are building today are much more complicated, which often results in highly nonlinear and nonconvex optimizations problems. Both of these challenges require us to go beyond convex optimization. While nonconvex optimization is extraordinarily successful in practice, unlike convex optimization, guaranteeing the correctness of nonconvex methods is notoriously difficult. In theory, even finding a local minimum of a general nonconvex function is NP-hard – nevermind the global minimum. This thesis aims to bridge the gap between practice and theory of nonconvex optimization, by developing global optimality guarantees for nonconvex problems arising in real-world engineering applications, and provable, efficient nonconvex optimization algorithms. First, this thesis reveals that for certain nonconvex problems we can construct a model specialized initialization that is close to the optimal solution, so that simple and efficient methods provably converge to the global solution with linear rate. These problem include sparse basis learning and convolutional phase retrieval. In addition, the work has led to the discovery of a broader class of nonconvex problems – the so-called ridable saddle functions. Those problems possess characteristic structures, in which (i) all local minima are global, (ii) the energy landscape does not have any ''flat'' saddle points. More interestingly, when data are large and random, this thesis reveals that many problems in the real world are indeed ridable saddle, those problems include complete dictionary learning and generalized phase retrieval. For each of the aforementioned problems, the benign geometric structure allows us to obtain global recovery guarantees by using efficient optimization methods with arbitrary initialization.

The goal of Distributionally Robust Optimization (DRO) is to minimize the cost of running a stochastic system, under the assumption that an adversary can replace the underlying baseline stochastic model by another model within a family known as the distributional uncertainty region. This dissertation focuses on a class of DRO problems which are data-driven, which generally speaking means that the baseline stochastic model corresponds to the empirical distribution of a given sample. One of the main contributions of this dissertation is to show that the class of data-driven DRO problems that we study unify many successful machine learning algorithms, including square root Lasso, support vector machines, and generalized logistic regression, among others. A key distinctive feature of the class of DRO problems that we consider here is that our distributional uncertainty region is based on optimal transport costs. In contrast, most of the DRO formulations that exist to date take advantage of a likelihood based formulation (such as Kullback-Leibler divergence, among others). Optimal transport costs include as a special case the so-called Wasserstein distance, which is popular in various statistical applications. The use of optimal transport costs is advantageous relative to the use of divergence-based formulations because the region of distributional uncertainty contains distributions which explore samples outside of the support of the empirical measure, therefore explaining why many machine learning algorithms have the ability to improve generalization. Moreover, the DRO representations that we use to unify the previously mentioned machine learning algorithms, provide a clear interpretation of the so-called regularization parameter, which is known to play a crucial role in controlling generalization error. As we establish, the regularization parameter corresponds exactly to the size of the distributional uncertainty region. Another contribution of this dissertation is the development of statistical methodology to study data-driven DRO formulations based on optimal transport costs. Using this theory, for example, we provide a sharp characterization of the optimal selection of regularization parameters in machine learning settings such as square-root Lasso and regularized logistic regression. Our statistical methodology relies on the construction of a key object which we call the robust Wasserstein profile function (RWP function). The RWP function similar in spirit to the empirical likelihood profile function in the context of empirical likelihood (EL). But the asymptotic analysis of the RWP function is different because of a certain lack of smoothness which arises in a suitable Lagrangian formulation. Optimal transport costs have many advantages in terms of statistical modeling. For example, we show how to define a class of novel semi-supervised learning estimators which are natural companions of the standard supervised counterparts (such as square root Lasso, support vector machines, and logistic regression). We also show how to define the distributional uncertainty region in a purely data-driven way. Precisely, the optimal transport formulation allows us to inform the shape of the distributional uncertainty, not only its center (which given by the empirical distribution). This shape is informed by establishing connections to the metric learning literature. We develop a class of metric learning algorithms which are based on robust optimization. We use the robust-optimization-based metric learning algorithms to inform the distributional uncertainty region in our data-driven DRO problem. This means that we endow the adversary with additional which force him to spend effort on regions of importance to further improve generalization properties of machine learning algorithms. In summary, we explain how the use of optimal transport costs allow constructing what we call double-robust statistical procedures. We test all of the procedures proposed in this paper in various data sets, showing significant improvement in generalization ability over a wide range of state-of-the-art procedures. Finally, we also discuss a class of stochastic optimization algorithms of independent interest which are particularly useful to solve DRO problems, especially those which arise when the distributional uncertainty region is based on optimal transport costs.

Carbon dioxide is a main greenhouse gas that is responsible for global warming and climate change. The reduction in greenhouse gas emission is required to comply with the Kyoto Protocol. Looking at CO2 emissions distribution in Canada, the electricity and heat generation sub-sectors are among the largest sources of CO2 emissions. In this study, the focus is to reduce CO2 emissions from electricity generation through capacity expansion planning for utility companies. In order to reduce emissions, different mitigation options are considered including structural changes and non structural changes. A drawback of existing capacity planning models is that they do not consider uncertainties in parameters such as demand and fuel prices. Stochastic planning of power production overcomes the drawback of deterministic models by accounting for uncertainties in the parameters. Such planning accounts for demand uncertainties by using scenario sets and probability distributions. However, in past literature different scenarios were developed by either assigning arbitrary values or by assuming certain percentages above or below a deterministic demand. Using forecasting techniques, reliable demand data can be obtained and can be inputted to the scenario set. The first part of this thesis focuses on long term forecasting of electricity demand using autoregressive, simple linear, and multiple linear regression models. The resulting models using different forecasting techniques are compared through a number of statistical measures and the most accurate model was selected. Using Ontario electricity demand as a case study, the annual energy, peak load, and base load demand were forecasted, up to year 2025. In order to generate different scenarios, different ranges in economic, demographic and climatic variables were used. The second part of this thesis proposes a robust optimization capacity expansion planning model that yields a less sensitive solution due to the variation in the above parameters. By adjusting the penalty parameters, the model can accommodate the decision maker’s risk aversion and yield a solution based upon it. The proposed model is then applied to Ontario Power Generation, the largest power utility company in Ontario, Canada. Using forecasted data for the year 2025 with a 40% CO2 reduction from the 2005 levels, the model suggested to close most of the coal power plants and to build new natural gas combined cycle turbines and nuclear power plants to meet the demand and CO2 constraints. The model robustness was illustrated on a case study and, as expected, the model was found to be less sensitive than the deterministic model.