Merit function

The mathematical procedure for a single merit function optimization for many design variables involves derivatives of the merit function with respect to each of the design variables (as a generalization to multiple... 

The goodness-of-fit between the experimental and theoretically calculated CBED rocking curves is described by a merit function, and in the present study we use the chi-square merit function defined as... 

Because of the occurrence of ill-conditioning, pure penalty methods have been replaced by more efficient algorithms. In SLP and SQP, a merit function is used within the line search phase of these algorithms. 

Numerous demonstrations in recent years have shown that the level of performance of present-day polymer electrolyte fuel cells can compete with current energy conversion technologies in power densities and energy efficiencies. However, for large-scale commercialization in automobile and portable applications, the merit function of fuel cell systems—namely, the ratio of power density to cost—must be improved by a factor of 10 or more. Clever engineering and empirical optimization of cells and stacks alone cannot achieve such ambitious performance and cost targets. 

This merit function ensures that the percentage error at each comparison point is given equal weight. 

The derivation of the elastic constant tensor to be used in RUS is an indirect iterative procedure. Provided the sample dimensions, density and elastic constants are known, the spectrum of resonant frequencies can be easily calculated. The inverse problem viz. calculating the elastic constants from a measured spectrum of mechanical resonances) has no known solution, however. For the indirect method, a starting resonant frequency spectrum, 7 (n = 1,2, ), can be calculated by elastic constants estimated from theory or from literature data for similar materials in addition to the known sample dimension and density. The difference between the calculated and measured resonance frequency spectrum, 7 (n = 1,2, ), is identified by a figure-of-merit function,... 

Comparisons between points are usually based on some form of merit function, P , e.g.. 

If line search criteria satisfied and merit function reduced then... 

This method has been found to reduce the overall computational effort in approaching the optimum by up to 30% compared to that obtained by accepting the first point satisfying an Armijo cone condition on the merit function while allowing step reductions of up to a factor of 10 at each line search iteration and using a quadratic interpolation formula to estimate the step reduction. 

In earlier literature, the term merit function has been used (cf. Ref. 15). But because an increased value corresponds to a worse force field, penalty function is more appropriate. 

The operation of a method based on this principle requires the possibility of evaluating the distance between the actual and the desired optical characteristic of the film system. This is defined by a merit function ... 

The nonlinear least squares fitting is to find a set of optimal parameters (P) that make the nonlinear model Y =/(X P) produce the results for the object (y) most close to a set of reference values. The overall performance for the parameters can be described with a merit function (F(P)), i.e., the sum of squares of all deviations between the reference values and the values from the model... 

To solve equation (18), both Gradient VF and Hessian H of the merit function F(P) should be derived first. For each of the systems for which the CHARMm force field is required to be reparameterized, only the two Lennard-Jones parameters (e , and of a specific atom pair a-b)... 

The contribution from the base charge is removed from the standard ESP, which is usually obtained from quantum mechanical calculations. Instead of fitting the total atomic charge directly, only the perturbation is fitted to the residual ESP. The figure-of-merit function is now written as... 

Although the solution of nonlinear systems results in the problem of finding a merit function that drives the search toward the solution of the problem (see Section 7.3), in the case of unconstrained minimization, this problem seems not to arise the unique function to be used is the obvious one it is the function to be minimized. In the previous sections, we saw how such a function can be used to accept or reject the iteration. Nevertheless, suppose we have a problem that can be... 

Convergence to the solution of a nonlinear equation system is often checked by means of an appropriate merit function, such as the Euclidean norm of the residuals. 

It is, therefore, reasonable to think that the problem of solving nonlinear systems can be tackled by a minimization program applied to a merit function of this kind. Unfortunately, multimodality issues arise if this is done and many additional minima, to which the real solution does not correspond, are introduced, transforming the original problem of the nonlinear system into a multidimensional multimodal optimization problem. 

The criterion represented by equations (7.11) and (7.12) should be avoided in a general purpose program when the f functions are unbalanced and have significantly different orders of magnitude. In such cases, the equations with lower orders of magnitude play no part in the merit function. 

The above-mentioned criterion can be improved by weighting each equation with a suitable weight wj. Consequently, the merit function becomes... 

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