Abstract
Optimization under uncertainty as an area of research has seen remarkable progress over the past few decades, under the headings of stochastic programming, robust and distributionally robust optimization. Duality in its various forms has been at the heart of these advances. This dissertation addresses two classes of problems at the interface of optimization and uncertainty which harness the power of linear and conic duality. Firstly, we focus on distributional uncertainty where the goal is to find the best possible bounds on tail probability and expected value functions of sums of Bernoulli random variables, attained by an extremal probability distribution from a set of distributions consistent with the given marginal probability and correlation information. We pri marily consider two types of correlation structure among the variables, i.e., pairwise independence (where the variables are uncorrelated) and extremal dependence (where the correlation between the variables is unknown). The tail probability function we consider is the right tail or the probability of occurrence of at least k out of n Bernoulli events, where k ? [0, n] is an integer while the expected value function is the expected value of a stop-loss type function. For pairwise independent variables, while some useful bounds on the tail proba bility function have been proposed in the literature, none of these bounds are tight in general. We provide several results towards finding tight probability bounds for this class of problems. When the individual and bivariate probabilities are known, even verifying if a joint distribution consistent with the given information exists, is known to be an NP-complete problem. Surprisingly however, we show that it is possible to capture the tightest upper bound on the probability of the union of n pairwise inde pendent events (k = 1) in a closed-form expression for any input marginal probability vector p ? [0, 1]n . The proof involves showing the existence of a positively correlated Bernoulli random vector for any p ? [0, 1]n , which is of independent interest in itself, since feasibility is typically not guaranteed for arbitrary correlation structures. Applica tions in correlation gap analysis, where pairwise independence provides better bounds (than extremal dependence) in some instances are discussed. For two random vari ables, we show that the correlation gap is upper bounded by 4/3 for any non-negative, non-decreasing, submodular function. We also prove that the Bonferroni lower bounds on the union of pairwise independent events are tight for small probabilities. Secondly, for k = 2 and any input marginal probability vector p ? [0, 1]n , new upper bounds are derived exploiting ordering of probabilities. Numerical examples are provided to illus trate when the bounds provide significant improvement over existing bounds. Thirdly, while the existing and new bounds are not always tight, we provide special instances when they are shown to be tight. Specifically, when the marginals are identical, we show that for any k ? [0, n], the bound derived in Boros and Prékopa (1989) is always tight and this result can be easily extended to identical t-wise independent variables. Further, we identify conditions under which this result provides useful small deviation bounds while other existing bounds are trivial. For the expected stop-loss function, in the special case when all the pairwise independent variables are identical, we provide an alternative proof to derive the tightest bound in a known closed-form expression, along with extremal distributions that attain the bound. With extremal dependence, we show that the tightest bounds on a weighted tail probability function can be computed as the optimal value of a compact linear program. Useful applications in a limited dependency system where only some of the variables are extremally dependent while the rest are mutually independent and these two sets of iv variables are independent of each other are explored. As a special case of the weighted tail probability function, we derive an earlier known closed-form bound (Rüger, 1978) on the probability that at least k out of n Bernoulli events occur. The usefulness of the closed-form bounds is subsequently demonstrated in solving star-shaped marginal sys tems. The results from the Bernoulli case are extended to derive useful upper bounds on the tail probability function of sums of random variables with discrete support. Nu merical illustrations show that these bounds are tight in many randomly generated in stances with identical and non-identical probabilities. For expected stop-loss functions, we prove that the comonotonic distribution attains the tightest upper bound while the Jensen (1906) bound is the tightest lower bound by similarly deriving a compact linear program. In the second part of this thesis, we shift attention to a recently proposed framework to deal with uncertain optimization problems known as robust satisficing (Long, Sim, and Zhou, 2021), where the uncertainty can be adversarially chosen from a pre-defined support set and which, for the same computational effort has the advantage of greater resistance to uncertainty as compared to the corresponding robust optimization model. We propose an alternative model inspired by the principle of satisficing but based on a constraint function that evaluates to the optimal objective value of a standard conic op timization problem, that can be used to model a wide range of constraint functions that are convex in the decision variables but can be either convex or concave in the uncer tain parameters. As a result, our model provides a unifying framework that generalizes and encompasses a wide variety of similar problems considered in recent papers. We derive an exact semidefinite optimization formulation when the constraint is biconvex quadratic with quadratic penalty and the support set is ellipsoidal. For more general conic uncertain problems with polyhedral support sets and penalty functions, we show the equivalence between the robust satisficing problems and the classical adaptive ro bust linear optimization models with conic uncertainty sets, where the latter can be solved approximately using affine recourse adaptation. More importantly, under the stated assumptions, we show that the exact reformulation and safe approximations do not lead to infeasible problems if the chosen target is above the optimum objective of the nominal problem. For the special case of a non-negative orthant cone, we prove that despite being simpler, the affine recourse approximation of the dual reformulation is closer to the original problem when compared to a specific non-affine recourse ap proximation of the original problem itself. Finally, we extend our framework to the data-driven setting and showcase the modeling and the computational benefits of the robust satisficing framework over classical robust optimization with three numerical examples: growth optimal portfolio selection, log-sum-exp optimization and adaptive lot-sizing problem.