In practice, the presence of uncertain parameters in optimization problems introduces new challenges in modeling and solvability to operations research. There are three main paradigms proposed for optimization problems under uncertainty. These include stochastic programming, robust optimization, and sensitivity analysis. In this thesis, we examine, improve, and combine the latter two paradigms in several relevant models and applications. In the second chapter, we study a two-stage adjustable robust linear optimization problem in which the right-hand sides are uncertain and belong to a compact, convex, and tractable uncertainty set. Under standard and simple assumptions, we reformulate the two-stage problem as a copositive optimization program, which in turns leads to a class of tractable semidefinite-based approximations that are at least as strong as the affine policy, which is a well studied tractable approximation in the literature. We examine our approach over several examples from the literature and the results demonstrate that our tractable approximations significantly improve the affine policy. In particular, our approach recovers the optimal values of a class of instances of increasing size for which the affine policy admits an arbitrary large gap. In the third chapter, we leverage the concept of robust optimization to conduct sensitivity analysis of the optimal value of linear programming (LP). In particular, we propose a framework for sensitivity analysis of LP problems, allowing for simultaneous perturbations in the objective coefficients and right-hand sides, where the perturbations are modeled in a compact, convex, and tractable uncertainty set. This framework unifies and extends multiple approaches for LP sensitivity analysis in the literature and has close ties to worst-case LP and two-stage adjustable linear programming. We define the best-case and worst-case LP optimal values over the uncertainty set. As the concept aligns well with the general spirit of robust optimization, we denote our approach as robust sensitivity analysis. While the best-case and worst-case optimal values are difficult to compute in general, we prove that they equal the optimal values of two separate, but related, copositive programs. We then develop tight, tractable conic relaxations to provide bounds on the best-case and worst case optimal values, respectively. We also develop techniques to assess the quality of the bounds, and we validate our approach computationally on several examples from—and inspired by—the literature. We find that the bounds are very strong in practice and, in particular, are at least as strong as known results for specific cases from the literature. In the fourth chapter of this thesis, we study the expected optimal value of a mixed 0-1 programming problem with uncertain objective coefficients following a joint distribution. We assume that the true distribution is not known exactly, but a set of independent samples can be observed. Using the Wasserstein metric, we construct an ambiguity set centered at the empirical distribution from the observed samples and containing all distributions that could have generated the observed samples with a high confidence. The problem of interest is to investigate the bound on the expected optimal value over the Wasserstein ambiguity set. Under standard assumptions, we reformulate the problem into a copositive programming problem, which naturally leads to a tractable semidefinite-based approximation. We compare our approach with a moment-based approach from the literature for two applications. The numerical results illustrate the effectiveness of our approach.
Finally, we conclude the thesis with remarks on some interesting open questions in the field of optimization under uncertainty. In particular, we point out that some interesting topics that can be potentially studied by copositive programming techniques.