Chapter 4 Two-Stage Nonsmooth Stochastic Optimization and Iterative Stochastic Quasigradient Procedure for Robust Estimation, Machine Learning and Decision Making

Uncertainties, risks, and disequilibrium are pervasive characteristics of modern socio-economic, technological, and environmental systems involving interactions between humans, economics, technology and nature. The systems are characterized by interdependencies, discontinuities, endogenous risks and thresholds, requiring nonsmooth quantile-based performance indicators, goals and constraints for their analysis and planning. The paper discusses the need for the two-stage stochastic optimization and the stochastic quasigradient (SQG) procedures to manage such systems. The two-stage optimization enables designing a robust portfolio of interdependent precautionary strategic and adaptive operational decisions making the systems robust with respect to potential uncertainty and risks. The SQG iterative algorithms define a “searching” process, which resembles a sequential adaptive learning and improvement of decisions from data and simulations, i.e. the so-called Adaptive Monte Carlo optimization. The SQG methods are applicable in cases when traditional stochastic approximation, gradient or stochastic gradient methods do not work, in particular, to general two-stage problems with implicitly defined goals and constraints functions, nonsmooth and possibly discontinuous performance indicators, risk and uncertainties shaped by decision of various agents. Stylized models from statistics, machine learning, robust decision making are presented to illustrate the two-stage (strategic-adaptive) modeling concept and the SQG procedures. The stylized models are parts of larger integrated assessment models developed at IIASA, e.g. Global Biosphere Management model (GLOBIOM) and Integrated Catastrophe Risk Management model (ICRIM)