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.

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Lau, Fu Man. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2009. / Includes bibliographical references (leaves 94-97). / Abstract also in Chinese. / Abstract --- p.i / Acknowledgements --- p.iii / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Notations and Preliminaries --- p.4 / Chapter 2.1 --- Introduction --- p.4 / Chapter 2.2 --- Basic notations --- p.4 / Chapter 2.3 --- On the properties of subdifferentials --- p.8 / Chapter 2.4 --- On the properties of normal cones --- p.9 / Chapter 2.5 --- Some computation rules for conjugate functions --- p.13 / Chapter 2.6 --- On the properties of epigraphs --- p.15 / Chapter 2.7 --- Set-valued analysis --- p.19 / Chapter 2.8 --- Weakly* sum of sets in dual spaces --- p.21 / Chapter 3 --- Sum of Epigraph Constraint Qualification (SECQ) --- p.31 / Chapter 3.1 --- Introduction --- p.31 / Chapter 3.2 --- Definition of the SECQ and its basic properties --- p.33 / Chapter 3.3 --- Relationship between the SECQ and other constraint qualifications --- p.39 / Chapter 3.3.1 --- The SECQ and the strong CHIP --- p.39 / Chapter 3.3.2 --- The SECQ and the linear regularity --- p.46 / Chapter 3.4 --- Interior-point conditions for the SECQ --- p.58 / Chapter 3.4.1 --- I is finite --- p.59 / Chapter 3.4.2 --- I is infinite --- p.61 / Chapter 4 --- Duality theory of semi-infinite optimization via weakly* sum of epigraph of conjugate functions --- p.70 / Chapter 4.1 --- Introduction --- p.70 / Chapter 4.2 --- Fenchel duality in semi-infinite convex optimization --- p.73 / Chapter 4.3 --- Sufficient conditions for Fenchel duality in semi-infinite convex optimization --- p.79 / Chapter 4.3.1 --- Continuous real-valued functions --- p.80 / Chapter 4.3.2 --- Nonnegative-valued functions --- p.84 / Bibliography --- p.94

Sit, Loi Yuk. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (leaves 64-65). / Abstracts in English and Chinese. / Abstract --- p.i / Acknowledgement --- p.iii / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Introduction to Genetic Algorithm --- p.5 / Chapter 2.1 --- The Basic Genetic Algorithm --- p.5 / Chapter 2.1.1 --- Selection Mechanisms --- p.7 / Chapter 2.1.2 --- Variation Operators --- p.8 / Chapter 2.2 --- Implementation of Genetic Algorithm --- p.10 / Chapter 2.3 --- Examples of Combinatorial Optimization --- p.12 / Chapter 2.3.1 --- Max-Cut Problem --- p.12 / Chapter 2.3.2 --- Transportation Problem --- p.15 / Chapter 2.3.3 --- Travelling Salesman Problem --- p.23 / Chapter 3 --- Model Building --- p.27 / Chapter 3.1 --- Introduction --- p.27 / Chapter 3.2 --- Sampling Mechanism --- p.28 / Chapter 3.3 --- Sampling Algorithm --- p.34 / Chapter 3.4 --- Parameters Estimation --- p.35 / Chapter 3.4.1 --- Parameters α and β of f(y) --- p.36 / Chapter 3.4.2 --- "Parameters p of f(z\x1,x2)" --- p.38 / Chapter 4 --- Design and Results of the Simulation Study --- p.40 / Chapter 4.1 --- Introduction --- p.40 / Chapter 4.2 --- Selection Mechanism --- p.41 / Chapter 4.3 --- Choice of Parameters' Values --- p.42 / Chapter 4.4 --- Performance Index --- p.43 / Chapter 4.5 --- Results and Interpretation --- p.48 / Chapter 5 --- Empirical Checking of the Selection Rule --- p.54 / Chapter 5.1 --- Introduction --- p.54 / Chapter 5.2 --- Max-Cut Problem --- p.54 / Chapter 5.3 --- Transportation Problem --- p.56 / Chapter 5.4 --- Travelling Salesman Problem --- p.57 / Chapter 6 --- Conclusion and Discussion --- p.60 / Bibliography --- p.64

Digitized by Kansas Correctional Industries

Minimizing cost and optimization of nonlinear problems are important for industries in order to be competitive. The need of optimization strategies provides significant benefits for companies when providing quotes for products. Accurate and easily attained estimates allow for less waste, tighter tolerances, and better productivity. The Nelder-Mead Simplex method with exterior penalty functions was employed to solve optimum machining parameters. Two case studies were presented for optimizing cost and time for a multiple tools scenario. In this study, the optimum machining parameters for milling operations were investigated. Cutting speed and feed rate are considered as the most impactful design variables across each operation. Single tool process and scalable multiple tool milling operations were studied. Various optimization methods were discussed. The Nelder-Mead Simplex method showed to be simple and fast.

Bibliography: leaves 273-287. / xviii, 289 leaves : ill. ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / The aim of this thesis is to develop a new method for the conceptual design of heat exchanger networks. The initial designs can be optimized using conventional non-linear optimization techniques in the subset of the problem's initial dimensionality. / Thesis (Ph.D.)--University of Adelaide, Dept. of Chemical Engineering, 1994

Thesis (Ph. D.)--University of Hong Kong, 2007. / Title proper from title frame. Also available in printed format.