Essays on distributionally robust optimization for decision-making under risk and ambiguity

This dissertation presents four research papers on the subject of decision under uncertainty, with a focus on decision-making in a risk-averse setting. The aim is to develop robust optimization and statistical techniques to obtain and analyze solutions for stochastic optimization problems, where the uncertain objective function is evaluated by a risk measure. Chapter 1 presents several robust optimization algorithms for solving risk-averse decision problems, where the risk measure is motivated by the theory of rank-dependent utility. Chapter 2 discusses the construction of uncertainty sets for applying robust optimization in risk management. Chapter 3 examines the statistical properties of sample average approximations of risk-averse stochastic optimization problems and provides finite-sample guarantees. Finally, Chapter 4 explores applications of risk measures in distributionally robust optimization for machine learning.