Adaptive robust optimization

Author: Ralph Wang (ChemE 6800 Fall 2020)

Steward: Allen Yang, Fengqi You

Introduction

Adaptive Robust Optimization (ARO), also known as adjustable robust optimization, models situations where decision makers make two types of decisions: here-and-now decisions that must be made immediately, and wait-and-see decisions that can be made at some point in the future. ARO improves on the robust optimization framework by accounting for any information the decision maker does not know now, but may learn before making future decisions. In the real-world, ARO is applicable whenever past decisions and new information together influence future decisions. Common applications include power systems control, inventory management, shift scheduling, and other resource allocation problems.

Compared to traditional robust optimization models, ARO gives less conservative and more realistic solutions, however, this improvement comes at the cost of computation time. Indeed, even the general linear ARO with linear uncertainty is proven computationally intractable. However, researchers have developed a wide variety of solution and approximation methods for specific industrial ARO problems over the past 15 years and the field continues to grow rapidly.

Problem Formulation

Suppose, for an optimization problem of interest, $S$ is the set of allowed decisions and $x$ is a decision in $S$. Let $u$ bee a vector representing the set of parameters of interest in this problem. If the goal is to minimize some function $f(u,x)$, and we want $x$ to adhere to a set of constraints $g(u,x)\leq 0$, then the problem may be formulated as:

${\begin{aligned}{\text{minimize, choosing x: }}f&(u,x)\\{\text{under constraints: }}g&(u,x)\leq 0\end{aligned}}$