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Statistical inference on the location of the optima (global maxima or minima) is one of the main goals in the area of Response Surface Methodology, with many applications in engineering and science. While there exist previous methods for computing confidence regions on the location of optima, these are for linear models based on a Normal distribution assumption, and do not address specifically the difficulties associated with guaranteeing global optimality. This paper describes distribution-free methods for the computation of confidence regions on the location of the global optima of response surface models. The methods are based on bootstrapping and Tukey's data depth, and therefore their performance does not rely on distributional assumptions about the errors affecting the response. An R language implementation, the package \code{OptimaRegion}, is described. Both parametric (quadratic and cubic polynomials in up to 5 covariates) and nonparametric models (thin plate splines in 2 covariates) are supported. A coverage analysis is presented demonstrating the quality of the regions found. The package also contains an R implementation of the Gloptipoly algorithm for the global optimization of polynomial responses subject to bounds. 

Dynamic traffic assignment models rely on a network performance module known as dynamic network loading (DNL), which expresses flow propagation, flow conservation, and travel delay at a network level. The DNL defines the so-called network delay operator , which maps a set of path departure rates to a set of path travel times (or costs). It is widely known that the delay operator is not available in closed form, and has undesirable properties that severely complicate DTA analysis and computation, such as discontinuity, nondifferentiability, nonmonotonicity, and computational inefficiency. This paper proposes a fresh take on this important and difficult issue, by providing a class of surrogate DNL models based on a statistical learning method known as Kriging . We present a metamodeling framework that systematically approximates DNL models and is flexible in the sense of allowing the modeler to make trade-offs among model granularity, complexity, and accuracy. It is shown that such surrogate DNL models yield highly accurate approximations (with errors below 8%) and superior computational efficiency (9 to 455 times faster than conventional DNL procedures such as those based on the link transmission model). Moreover, these approximate DNL models admit closed-form and analytical delay operators, which are Lipschitz continuous and infinitely differentiable, with closed-form Jacobians. We provide in-depth discussions on the implications of these properties to DTA research and model applications.