Target function minimization

Related Approaches Target Function Minimization, the Diffusion Equation Method, and the Ellipsoid Algorithm... 

To determine the optimal parameters, traditional methods, such as conjugate gradient and simplex are often not adequate, because they tend to get trapped in local minima. To overcome this difficulty, higher-order methods, such as the genetic algorithm (GA) can be employed [31,32]. The GA is a general purpose functional minimization procedure that requires as input an evaluation, or test function to express how well a particular laser pulse achieves the target. Tests have shown that several thousand evaluations of the test function may be required to determine the parameters of the optimal fields [17]. This presents no difficulty in the simple, pure-state model discussed above. 

When the structural information is available, then the components of the diffusion tensor can be derived from the minimization of the target function,... 

These models are nested the search starts with the simplest model and proceeds to the models of increasing degree of complexity (number of fitting parameters). An Ockham s razor principle is assumed here if more than one model is consistent with the data, the simplest model is preferred. For each model of motion, all parameters are determined from fitting, based on the simplex algorithm, to minimize the following target function ... 

Patterson Correlation Refinement. To select which of the orientations determined from the rotation search is the correct solution a Patterson correlation refinement of the peak list of the rotation function was performed. This was carried out by minimization against a target function defined by Brunger (1990) and as implemented in XPLOR. The search model, P2, was optimized for each of the selected peaks of the rotation function. 

The bond polarization formula can also be used to refine molecular structures with a molecular force field that uses NMR data as an additional target function by minimizing the differences between calculated and experimental chemical shift data. 

Rather than regarding the force field as a fixed part of the refinement procedure, it may be quite reasonable to adjust force field parameters to make barriers easier to surmount or to use force field parameters as variables that can be altered to implement a refinement method. In one sense, this principle of minimizing on a changing potential surface could be seen as the heart of the variable target function method discussed previously.8... 

The above discussion suggests that we cannot minimize the action directly to compute classical trajectories. We focus instead on the residuals (13). One boundary value formulation that we frequently use minimizes the squares of the residual vectors, i.e., we define a target function T that we wish to minimize as a function of all the intermediate coordinates Xj,j = 2,..., N — 1. 

There are important differences between the initial value formulation and the boundary value approach. Initial value solutions are based on interpolation forward in time one coordinate set after another. The boundary value approach is based on minimization of a target function of the whole trajectory. Minimization (and the study of a larger system) is more expensive in the boundary value formulation compared to initial value solver. However, the calculations of state to state trajectories and the abilities to use approximations (next section), make it a useful alternative for a large number of problems. 

We have started from expressions for the classical action, replaced them by a local differential equation, and by a local difference equation which we solve either as is using relaxation methods (a topic that was not discussed in this manuscript) or by minimizing a target function T T = at-i ) ... 

In RMC, random moves, i.e., changes in the configuration of the system, are performed as in the metropolis MC algorithm. Random moves are accepted or rejected so that the difference between the calculated and target S q) or g r) is minimized. If g(r) is used as the target function, the parameter to be minimized is ... 

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