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.

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APPSPACK is software for solving unconstrained and bound constrained optimization problems. It implements an asynchronous parallel pattern search method that has been specifically designed for problems characterized by expensive function evaluations. Using APPSPACK to solve optimization problems has several advantages: No derivative information is needed; the procedure for evaluating the objective function can be executed via a separate program or script; the code can be run in serial or parallel, regardless of whether or not the function evaluation itself is parallel; and the software is freely available. We describe the underlying algorithm, data structures, and features of APPSPACK version 4.0 as well as how to use and customize the software.3

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We propose a novel, variational inversion methodology for the electrical impedance tomography problem, where we seek electrical conductivity σ inside a bounded, simply connected domain Ω, given simultaneous measurements of electric currents I and potentials V at the boundary. Explicitly, we make use of natural, variational constraints on the space of admissible functions σ, to obtain efficient reconstruction methods which make best use of the data. We give a detailed analysis of the variational constraints, we propose a variety of reconstruction algorithms and we discuss their advantages and disadvantages. We also assess the performance of our algorithms through numerical simulations and comparisons with other, well established, numerical reconstruction methods.