(282d) On the Relation Between Flexibility Analysis and Robust Optimization

Authors: 
Zhang, Q., Carnegie Mellon University
Lima, R. M., King Abdullah University of Science and Technology
Grossmann, I. E., Carnegie Mellon University

Optimization under uncertainty is a very active area of research, and many approaches have been proposed to systematically model uncertainty and make sound decisions using such models. One approach that has gained much popularity in recent years is robust optimization, which poses an optimization problem in a way such that it is feasible for any realization of the uncertainty within a given uncertainty set. It turns out that in the process systems engineering community, the same idea was considered in the work on process flexibility analysis (Swaney & Grossmann, 1985) a decade before the field of robust optimization emerged (Ben-Tal & Nemirovski, 1998). In this work, we establish a link between flexibility analysis and robust optimization, and show how the two approaches can potentially benefit from each other.

In the context of flexibility analysis, flexibility is defined as the capability of a system to ensure feasibility for all possible realizations of the uncertainty by proper adjustment of control variables, which correspond to the so-called recourse actions in two-stage stochastic programming. The traditional robust optimization approach does not consider recourse and therefore cannot be applied to address flexibility analysis problems. However, in recent works, researchers have proposed accounting for recourse in robust optimization by applying decision rules that are functions of the uncertain parameters (Ben-Tal, et al., 2004). The decision rule approach allows recourse; however, compared to flexibility analysis, it is conservative because we are typically restricted to affine decision rules in order to achieve tractable formulations.

Our analysis is limited to the case of linear inequality constraints and box uncertainty sets. Here, we consider the three classical problems in flexibility analysis: the flexibility test problem, the flexibility index problem, and design under flexibility constraints. For each problem, the flexibility analysis approach is compared with the robust optimization approach applying affine decision rules. Furthermore, for each problem, a new formulation is developed using both robust optimization techniques and insights from flexibility analysis. We show the connections between the three different approaches. In particular, we demonstrate that compared to flexibility analysis, the decision rule approach results in efficient formulations but is restrictive in terms of the allowed recourse actions. To overcome this limitation, the new proposed robust formulations allow full recourse by leveraging insights from flexibility analysis. Several examples are presented to illustrate the major concepts covered in this presentation.

References
Ben-Tal, A., Goryashko, A., Guslitzer, E. & Nemirovski, A., 2004. Adjustable robust solutions of uncertain linear programs. Mathematical Programming, 99(2), pp. 351-376.
Ben-Tal, A. & Nemirovski, A., 1998. Robust Convex Optimization. Mathematics of Operations Research, 23(4), pp. 769-805.
Swaney, R. E. & Grossmann, I. E., 1985. An Index for Operational Flexibility in Chemical Process Design - Part I: Formulation and Theory. AIChE Journal, 31(4), pp. 621-630.