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.
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
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.
We present a two-step approach to modeling the transmembrane spanning helical bundles of integral membrane proteins using only sparse distance constraints, such as those derived from chemical crosslinking, dipolar EPR and FRET experiments. In Step 1, using an algorithm, we developed, the conformational space of membrane protein folds matching a set of distance constraints is explored to provide initial structures for local conformational searches. In Step 2, these structures refined against a custom penalty function that incorporates both measures derived from statistical analysis of solved membrane protein structures and distance constraints obtained from experiments. We begin by describing the statistical analysis of the solved membrane protein structures from which the theoretical portion of the penalty function was derived. We then describe the penalty function, and, using a set of six test cases, demonstrate that it is capable of distinguishing helical bundles that are close to the native bundle from those that are far from the native bundle. Finally, using a set of only 27 distance constraints extracted from the literature, we show that our method successfully recovers the structure of dark-adapted rhodopsin to within 3.2 Å of the crystal structure.Keywords: helix packing; transmembrane helices; distance constraints; molecular refinement Integral membrane proteins are essential components of the cell membrane that participate in many important cellular processes such as energy transduction, cell signaling, mediation of senses such as vision, cell intoxication, and pathogenesis, and immune recognition. Their significance is emphasized by the fact that approximately one-third of the proteins encoded for by a typical genome are membrane proteins (Buchan et al. 2002). Furthermore, at least 70% of current pharmaceuticals are thought to act on membrane proteins (Wilson and Bergsma 2000). Despite their obvious importance, to date, the structures of fewer than 75 integral membrane proteins have been solved (see White 2003 and references therein), and this number includes redundant structures across species. This is a vast contrast to the over 25,000 soluble proteins whose structures have been solved using X-ray crystallography and NMR. Reasons for the slow progress in the structural analysis of membrane proteins include the instability of membrane proteins in environments lacking phospholipids, their tendency to aggregate and precipitate, and protein abundance, expression, and purification issues. These characteristics highlight why the application of standard structure determination methods to membrane proteins is nontrivial.Given the nature of the difficulties in generating highresolution structural data from methods such as X-ray crystallography and NMR, it is unlikely that these experimental techniques will yield a significant increase in the number of solved membrane protein structures in the near future. As an alternative approach, the focus here is on modeling transmembrane proteins using a set of s...
Black box functions, such as computer experiments, often have multiple optima over the input space of the objective function. While traditional optimization routines focus on finding a single best optimum, we sometimes want to consider the relative merits of multiple optima. First we need a search algorithm that can identify multiple local optima. Then we consider that blindly choosing the global optimum may not always be best. In some cases, the global optimum may not be robust to small deviations in the inputs, which could lead to output values far from the optimum. In those cases, it would be better to choose a slightly less extreme optimum that allows for input deviation with small change in the output; such an optimum would be considered more robust. We use a Bayesian decision theoretic approach to develop a utility function for selecting among multiple optima.
W e examine the problem of transmembrane protein structure determination. Like many questions that arise in biological research, this problem cannot be addressed generally by traditional laboratory experimentation alone. Instead, an approach that integrates experiment and computation is required. We formulate the transmembrane protein structure determination problem as a bound-constrained optimization problem using a special empirical scoring function, called Bundler, as the objective function. In this paper, we describe the optimization problem and its mathematical properties, and we examine results obtained using two different derivative-free optimization algorithms.
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