Robust and Data-Driven Optimization: Modern Decision Making Under Uncertainty

Abstract

Traditional models of decision-making under uncertainty assume perfect information, i.e., ac-curate values for the system parameters and specific probability distributions for the random variables. However, such precise knowledge is rarely available in practice, and a strategy based on erroneous inputs might be infeasible or exhibit poor performance when implemented. The purpose of this tutorial is to present a mathematical framework that is well-suited to the limited information available in real-life problems and captures the decision-maker’s attitude towards uncertainty; the proposed approach builds upon recent developments in robust and data-driven optimization. In robust optimization, random variables are modeled as uncertain parameters be-longing to a convex uncertainty set and the decision-maker protects the system against the worst case within that set. Data-driven optimization uses observations of the random variables as direct inputs to the mathematical programming problems. The first part of the tutorial describes the robust optimization paradigm in detail in single-stage and multi-stage problems. In the second part, we address the issue of constructing uncertainty sets using historical realizations of the random variables and investigate the connection between convex sets, in particular polyhedra, and a specific class of risk measures.