Optimization under Connected Uncertainty

TL;DR

This paper introduces a novel framework for optimization under uncertainty that explicitly models the dependence of future uncertainties on past realizations, improving decision-making in multi-period problems.

Contribution

It develops a connected uncertainty set framework that captures temporal dependence, extending robust optimization methods to more realistic, history-dependent scenarios.

Findings

Framework effectively models dependence of uncertainties over time

Reformulated robust constraints for common set structures

Numerical demonstrations on knapsack and portfolio problems

Abstract

Robust optimization methods have shown practical advantages in a wide range of decision-making applications under uncertainty. Recently, their efficacy has been extended to multi-period settings. Current approaches model uncertainty either independent of the past or in an implicit fashion by budgeting the aggregate uncertainty. In many applications, however, past realizations directly influence future uncertainties. For this class of problems, we develop a modeling framework that explicitly incorporates this dependence via connected uncertainty sets, whose parameters at each period depend on previous uncertainty realizations. To find optimal here-and-now solutions, we reformulate robust and distributionally robust constraints for popular set structures and demonstrate this modeling framework numerically on broadly applicable knapsack and portfolio-optimization problems.