Simultaneous optimization method

These methods, which probably deserve more attention than they have received to date, simultaneously optimize the positions of a number of points along the reaction path. The method of Elber and Karpins [91] was developed to find transition states. It fiimishes, however, an approximation to the reaction path. In this method, a number (typically 10-20) equidistant points are chosen along an approximate reaction path coimecting two stationary points a and b, and the average of their energies is minimized under the constraint that their spacing remains equal. This is obviously a numerical quadrature of the integral s f ( (.v)where... 

Werner H-J and Meyer W 1981 A quadratically convergent MCSCF method for the simultaneous optimization of several states J. Chem. Phys 74 5794... 

As stated earlier, a commercially useful computational method must be fast, accurate, transferable, and interpretable. Logical steps one can take towards this goal are now presented. It is important to point out that under current resources, these four criteria cannot be simultaneously optimized. With current computational capabilities, the most complete theoretical description of protein-ligand interactions (which may involve many-body terms) cannot... 

The first ternary metal oxide catalyst of Ca0-Mn0/Ce02 was prepared by simultaneous impregnation method, while the second ternary metal oxide of Ca0/Mn0-Ce02 catalyst was prepared by combination of co-precipitation and impregnation method. The catalysts composition used in this paper were based on multi-responses optimization result [3]. H2-TPR was carried out using Micromeritics 2900 TPD/TPR equipped by TCD. A catalyst amount of... 

The procedure and methods for the MEP determination by the NEB and parallel path optimizer methods have been explained in detail elsewhere [25, 27], Briefly, these methods are types of chain of states methods [20, 21, 25, 26, 30, 31]. In these methods the path is represented by a discrete number of images which are optimized to the MEP simultaneously. This parallel optimization is possible since any point on the MEP is a minimum in all directions except for the reaction coordinate, and thus the energy gradient for any point is parallel to the local tangent of the reaction path. 

No single stopping criterion will suffice for Newton s method or any of the optimization methods described in this chapter. The following simultaneous criteria are recommended to avoid scaling problems ... 

An example of a multireference technique is the multiconfigurational SCF (MCSCF) approach, where the wave function is obtained by simultaneously optimizing both the molecular orbitals and the configuration coefficients, thereby blending the different resonance structures together. [28] Historically, the MCSCF approach has been used extensively to provide qualitatively accurate representations of surfaces however, this method still suffers two primary drawbacks (1) the ambiguous choice of configurations and (2) the lack of dynamical correlation. 

Wolters, R., and Kateman, G. (1990), The Constmction of Simultaneous Optimal Experimental Designs for Several Polynomials in the Calibration of Analytical Methods, J. Chemometrics, 4, 171-185. 

Several other approaches with the goal of simultaneous optimization of several criteria have been reported. One such approach is the generation of a library that is both focused and diverse via the dual fingerprint metric described by Bajorath [94], In this method, individual compounds are randomly generated and their similarity to a known inhibitor is evaluated by comparison of their minifingerprints [95] using the Tanimoto coefficient. Those molecules that are above a similarity threshold are then... 

The nature of combinatorial chemistry can present a considerable challenge because these libraries are generally produced as arrays of compounds and it is often inconvenient to synthesize individual compounds in order to achieve an optimal design. Two methods have been described that attempt to select optimal subset of reagents from a virtual library that has been partitioned into favorable and unfavorable compounds by some method of filtering. The PLUMS algorithm [97] was designed to simultaneously optimize the size of the library based on effectiveness and efEciency . 

The application of simultaneous optimization to reactor-based flowsheets leads us to consider the more general problem of differentiable/algebraic optimization problems. Again, the optimization problem needs to be reconsidered and reformulated to allow the application of efficient nonlinear programming algorithms. As with flowsheet optimization, older conventional approaches require the repeated execution of the differential/algebraic equation (DAE) model. Instead, we briefly describe these conventional methods and then consider the application and advantages of a simultaneous approach. Here, similar benefits are realized with these problems as with flowsheet optimization. 

Cuthrell, J. E., and Biegler, L. T., Simultaneous optimization and solution methods for batch reactor control profiles, Comp, and Chem. Eng. 13(1/2), 49-62 (1989). 

In principle, the analytical results obtained by the GPC spin column/HPLC ESI-MS methodology described in this chapter should be similar to the results obtained using the tandem chromatographic method of GPC/reversed-phase HPLC ESI-MS described in Chapter 3. There are practical advantages for each method. Since each of the chromatographic and mass spectral steps are done serially for the GPC spin column/HPLC ESI-MS methodology, each of the steps can be performed and optimized individually. In the event of mass spectrometer failure, the production of spin column eluate samples can proceed and samples can be stored for future analysis. In contrast, the parallel methodology of tandem GPC/ reversed-phase HPLC ESI-MS requires the simultaneous optimization of multi-... 

In practice often more than one quality criterion is relevant. In the case of the need to build in robustness, at least two criteria are already needed the quality criterion itself and its associated robustness criterion. Hence, optimization has to be done on more than one criterion simultaneously. If a simultaneous optimization technique is used then there are procedures to deal with multiple optimization criteria. Several methods for multi-criteria optimization have been proposed and recently a tutorial/review has appeared [22]. 

The choice for a sequential or a simultaneous method depends on the expected difficulty to obtain the optimum and the effort needed to perform one experiment. In this paper the problem is further complicated by the requirement of robustness. The optimization method will be determined after a discussion of robustness and the Taguchi method. 

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