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, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} is the set of allowed decisions and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} is a decision in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} . Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u} bee a vector representing the set of parameters of interest in this problem. If the goal is to minimize some function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(u, x)} , and we want Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} to adhere to a set of constraints Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(u, x) \leq 0} , then the problem may be formulated as:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align}\text{minimize, choosing x: } f&(u, x)\\ \text{under constraints: } g&(u, x) \leq 0\end{align}}

Or more simply:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align}\text{min}(x) \; &f(u, x)\\ \text{s.t. } \; &g(u, x) \leq 0\end{align}}

In this formulation, we call Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} the objective function and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g} the constraint function. If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u} is known, then the problem can be solved using methods such as branch and cut or KKT conditions. However, in many real world scenarios, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u} is not known. To address this uncertainty, the robust optimization approach generates the set of possible values of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u} , called the uncertainty set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U} , and solves for the decision Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} such that the constraint Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g} is satisfied in all cases and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is optimized for the worst case. The problem can be written as:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align}\text{min}(x)\text{ max}(u)\;&f(u, x)\\ \text{s.t.}\;&g(u, x) \leq 0 \end{align}}

Adaptive robust optimization expands this robust optimization framework by separating the decision Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} into multiple stages. For simplicity, assume there are two stages of decisions: in the first stage, only a subset of the decisions are made. After these decisions are made, the true values of the parameters Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u} are revealed, then the remaining set of decisions need to be made. The model is like a game of poker: the player needs to make initial bets based on incomplete information (the cards in his hand), then makes further bets as more and more cards are dealt. Mathematically, let the set of possible decisions in the first stage be Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_1} and the set of possible decisions in the second stage be Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_2} , so that the objective and constraint functions become functions of the parameters Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u} , the first stage decision Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_1} (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_1 in S_1} ), and the second stage decision Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_2} (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_2} in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_2} ). Then, we can formulate the problem as:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align}\text{min}(x_1)\text{ max}(u)\text{ x_2} \; &f(u, x_1, x_2)\\ \text{s.t.}\;&g(u, x_1, x_2) \leq 0 \; \text{for all} u \text{in} U\end{align}}