(433c) Insights into Nonconvex Robust Optimization for Chemical Engineering Applications

In this work we implement and compare several nonconvex semi-infinite programming approaches and assess their performance across multiple robust optimization problems in chemical and process systems engineering. The case studies described are documented thoroughly. We provide description of key characteristics, implementations, and optimal solutions with the aim of providing them as future benchmarks for nonconvex robust optimization in chemical engineering applications. Based on the results we characterize which methods work best across specific settings, outlining recommendations for practitioners looking to solve nonconvex robust optimization problems. To the authors knowledge this is the first unified demonstration and comparison of the array of nonconvex SIP methods past previous qualitative reviews [1].

Robust optimization is a method for handling uncertainty within optimization problems. Considered the most conservative of all methods, it aims to immunise optimal solutions against all potential realizations of uncertain parameters. Robust optimization has been successfully applied across chemical and process systems engineering for some time. Examples include robust formulations and solutions to supply chain problems such as facility planning [2,3] and distribution [4], the pooling problem [5], and portfolio selection problems [6]. Accounting for the complete set of uncertain parameters results in a semi-infinite formulation whereby an infinite number of constraints must be satisfied for a given solution to ensure robustness. A considerable amount of literature is concerned with trading off conservatism with solution quality [7, 8, 9], in other words obtaining better quality solutions at the expense of making assumptions regarding what values uncertain parameters can take. This is achieved by deriving new uncertainty sets either from data or otherwise. In this work we focus on intractability derived from ‘hard’ constraints which are unable to be reformulated as a key problem, as opposed to the inherent issue of trading off conservatism. Traditionally, analytical reformulations are derived from linear and convex constraints, transforming a single semi-infinite constraint into a single, or multiple (importantly finite) robust counterparts making the problem tractable. However, when constraints appear that are nonconvex as is common in chemical engineering, alternative approaches to reformulation must be sought. The field of semi-infinite programming is closely related to robust optimization and has provided many developments that are applicable to robust optimization particularly in nonconvex scenarios.

Several approaches have been characterised within semi-infinite programming literature for handling nonconvex constraints. These include the potentially first algorithm for nonconvex SIP the Blankenship & Faulk method [10] which is shown in Fig 1. and relies on an iteratively improved relaxation of the feasible region, as well as interval-based restrictions [11], concave restrictions [12], and methods that combine restriction and relaxation of the semi-infinite feasible region [13]. In this work we implement all of the above methods, in order to investigate their performance across a number of case studies.

Methods that apply over-estimators to ensure robust feasibility, for example natural interval extensions, are shown to have inherent advantages when applied to robust optimization problems as they are guaranteed to locate a robust feasible solution regardless of resolution. Figure 2 demonstrates how providing an over estimation of a semi-infinite constraint using a single interval extension still provides a robust solution, with an increasingly refined number of interval extensions producing a better approximation to the feasible region. Increasing the fidelity of the overestimation results in a better quality robust feasible solution, at the cost of an increasing number of constraints. However, methods that apply relaxations of the feasible region that are iteratively improved, such as the original Blankenship and Faulk method, suffer as solutions only become feasible in a robust sense as the number of iterations approaches a limit. We conclude with recommendations to practitioners regarding approaches to solve nonconvex RO problems from across the spectrum of chemical engineering, classifying methods by convergence, performance across a number of dimensions, solution quality and solution time.