The types of constraints encountered in black-box simulation-based optimization problems differ significantly from those addressed in nonlinear programming. We introduce a characterization of constraints to address this situation. We provide formal definitions for several constraint classes and present illustrative examples in the context of the resulting taxonomy. This taxonomy, denoted KARQ, is useful for modeling and problem formulation, as well as optimization software development and deployment. It can also be used as the basis for a dialog with practitioners in moving problems to increasingly solvable branches of optimization.

We are grateful for the many insightful comments provided by the discussants. One team politely pointed out oversights in our literature review and the subsequent omission of a formidable comparator. Another made an important clarification about when a more aggressive variation (the so-called NoMax) would perform poorly. A third team offered enhancements to the framework, including a derivation of closed-form expressions and a more aggressive updating scheme; these enhancements were supported by an empirical study comparing new alternatives with old. The last team suggested hybridizing the statistical augmented Lagrangian (AL) method with modern stochastic search. Here we present our responses to these contributions and detail some improvements made to our own implementations in light of them. We conclude with some thoughts on statistical optimization using surrogate modeling and open-source software.

Constrained blackbox optimization is a difficult problem, with most approaches coming from the mathematical programming literature. The statistical literature is sparse, especially in addressing problems with nontrivial constraints. This situation is unfortunate because statistical methods have many attractive properties: global scope, handling noisy objectives, sensitivity analysis, and so forth. To narrow that gap, we propose a combination of response surface modeling, expected improvement, and the augmented Lagrangian numerical optimization framework. This hybrid approach allows the statistical model to think globally and the augmented Lagrangian to act locally. We focus on problems where the constraints are the primary bottleneck, requiring expensive simulation to evaluate and substantial modeling effort to map out. In that context, our hybridization presents a simple yet effective solution that allows existing objective-oriented statistical approaches, like those based on Gaussian process surrogates and expected improvement heuristics, to be applied to the constrained setting with minor modification. This work is motivated by a challenging, real-data benchmark problem from hydrology where, even with a simple linear objective function, learning a nontrivial valid region complicates the search for a global minimum. Supplementary materials for this article are available online.