Dual-space optimization

In order to reduce the size of the PW basis set pseudo potentials (PP) of the dual-space type [12,13] are used. The latest implementation of the GPW method [34] has been done within the CP2K program and the corresponding module is called Quickstep [32]. In this implementation the linear scaling calculation of the GPW KS matrix elements is combined with an optimizer based on orbital transformations [33]. This optimization algorithm scales linearly in the number of basis functions for a given system size and, in combination with parallel computers, it can be used for systems with several thousands of basis functions [33,34]. 

The method for converting dual-energy data into a color-coded image is a task that embodies many subtleties [64]. Conceptually, the X-ray image can be broken down according to an optimized attenuation, which sets a pixel s intensity, and Zgfp, which sets the hue. The optimized attenuation is a combination of HI and LO attenuations. Each pixel s attenuation is a composite of the attenuation produced by aU objects in that line s path from the X-ray source and to the corresponding detector. This projection X-ray view superimposes a 3-D collection of objects into a 2-D data-space representation. 

In the high-dimensional space, the optimized super-planes were computed by substituting dot product operation for linear kernel function and by solving the dual problem using quadratic programming. The discrimination functions of quartz, feldspar and biotite in the high-dimensional space were obtained and represented as... 

The variables n x,t) are the dual prices associated with the nonoverlapping constraints (for the position (x, t) in space). With fixed r(x, t), the inner optimization problem is trivial. We use a version of subgradient algorithm, toown as the volume algorithm in the literature (cf Barahona and Anbil (2000)), to update the dual prices and to solve the above problem. To make the problem manageable so that the lower bound can be obtained within reasonable time, the space-time network is also discretized to moderate sizes. Note that the discretized version of the problem will still provide us with a lower bound for the static berth plaiming problem. 

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