TOWARDS TIGHTNESS WITH PAIRWISE INDEPENDENCE, EXTREMAL DEPENDENCE AND ROBUST SATISFICING USING LINEAR AND CONIC DUALITY

Abstract

This dissertation discusses two classes of problems at the interface of optimization and uncertainty that 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 expectation functions of sums of pairwise independent Bernoulli random variables, given their marginal probabilities. We show that the tightest bound can be captured in closed form for some special cases such as when the union probability is considered or when the variables are identical or extremally dependent. The second part provides a unified framework to address conic uncertain problems in the context of a recently proposed <em>robust satisficing</em> framework specified by a target objective, where nature adversarially chooses the uncertainty from a pre-defined support set. Conic duality is exploited to derive exact or approximate tractable robust counterparts, while numerical examples demonstrate the improved performance over classical robust optimization models.