Distributionally Robust Optimization

With its early roots in the development of calculus by Isaac Newton, Gottfried Wilhelm Leibniz, Pierre de Ferma t and others in the late 17th century, mathematical optimization has a rich his tory that involves contributions from numerous mathematicians, economists, eng ineers, and scientists. The birth of modern mathematical optimization is commonly c redited to George Dantzig, whose simplex algorithm developed in 1947 solves l inear optimization problems where ℓ is affine and X is a polyhedron ( Dantzig 1956). Subsequent milestones include the development of the rich theory of convex a nalysis ( Rockafellar 1970) as well as the discovery of polynomial-time solution metho ds for linear ( Khachiyan 1979, Karmarkar 1984) and broad classes of nonlinear convex optimization problems ( Nesterov and Nemirovskii 1994). Classical optimization problems are deterministic, that is, all problem data are assumed to be known with certainty. However, most decision pro blems encountered in practice depend on parameters that are corrupted by measu rement errors or that are revealed only after a decision must be determined and committed. A naïve approach to model uncertainty-affected decision problems a s deterministic optimization problems would be to replace all uncertain paramete rs with their expected values or with appropriate point predictions. However, it h as long been known and well-documented that decision-makers who replace an un certain parameter of an optimization problem with its mean value fall victim to th e'flaw of averages' ( Savage, Scholtes and Zweidler 2006, Savage 2012).