(400a) Efficient Flexibility Analysis of Computationally Expensive Black-Box Simulators Using Quantile-Based Bayesian Optimization

Uncertainty, which is inevitably present in real-world problems, can come from a variety of sources such as noisy and incomplete datasets, unknown parameters, environmental disturbances (such as product demand and prices), and implementation errors. Flexibility analysis provides a quantitative framework for identifying feasibility of a process design over a range of uncertainty values while accounting for feedback/recourse from manipulatable control variables available in the system. The notion of flexibility was first introduced for process systems engineering problems in [1] and has been extended to account for probabilistic uncertainty descriptions [2] and dynamic problems [3]. The key assumption in the majority of these works is that the structure of the model is exactly known and can be exploited by existing optimization algorithms that assume derivatives and/or relaxations of the objective and constraint functions can be easily computed. In many emerging applications, however, this assumption is violated since the process model is defined in terms of an expensive simulator whose internal structure cannot be accessed by the modeler. Relevant examples include the simulation of: (i) multi-product plants that often requires proprietary (e.g., Aspen Plus) models of thermodynamics or kinetics, (ii) environmental models (e.g., CALPUFF, SEEMAC, iTree) that can be used to evaluate important sustainability metrics, and (iii) multiphysics phenomena (e.g., COMSOL) that is needed to represent important spatial variability and boundary effects. These cases are often referred to as “black-box” (in contrast to the “white-box” known equation case) and have been the focus of more recent work on flexibility analysis [4-7]. It is worth noting that most previous works on black-box flexibility analysis focus on the special case in which there are no recourse variables – this can be interpreted as a “feasibility test,” as one is trying to determine if the system maintains feasible operation despite perturbations from the uncertainty. Such a representation, however, can lead to a highly pessimistic view of flexibility since, in most engineering systems, there is often significant capacity to adapt to these perturbations as they occur.

In this work, we develop a novel black-box flexibility analysis method that addresses two of the key limitations of existing approaches. First, our method simultaneously accounts for the effect of uncertain parameters and resource variables, which is known to lead to a challenging tri-level “max-min-max” structure that is significantly harder to solve than traditional feasibility problems. Second, our method is specifically constructed to be data-efficient, meaning that it aims to minimize the number of calls to the black-box system model. The rationale for this second feature is that the simulation models mentioned above are computationally expensive (i.e., they require a long time and/or substantial resources to be evaluated), so that they are very likely to be the main bottleneck for the execution of the method. The proposed method builds upon the Bayesian optimization (BO) paradigm, which has become one of the most popular methods for data-efficient optimization of expensive black-box functions due to its practical success in several real-world applications [8-12]. The BO framework consists of two main components: (i) a predictive surrogate model of the black-box functions (equipped with uncertainty estimates) learned according to the Bayesian paradigm and (ii) an acquisition function that depends on the posterior distribution of the surrogate models, whose value at any point quantifies the benefit of evaluating the black-box functions at this point. However, standard BO methods only apply to single-level optimization problems. Therefore, in this work, we develop a novel extension of BO that is suitable for the types of tri-level optimization problems underpinning quantitative flexibility analysis. The basis for our proposed method is the construction of a quantile function that can be used to evaluate tight, high probability upper and lower confidence bounds for the flexibility metric. By defining an acquisition function in terms of this quantile function, we can efficiently identify the settings of the next expensive simulation that are most informative for classifying the system as flexible or not. The effectiveness of the proposed method is compared to several state-of-the-art alternatives (e.g., [13]) on a benchmark problem and a significantly more challenging heat exchanger network (HEN) problem. The results show that our new quantile-based BO method can provide more accurate upper and lower bounds on the true flexibility metric as well as identify better simulation settings to make sure those bounds converge at a faster rate than all tested alternative methods.

References:

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