Iterative optimization approach

However, there can be limits to this approach because it is not always clear whether the chosen CG mapping scheme can converge to an optimal fit. For liquid mixtures or solutions, the situation is more complex because several RDFs that mutually affect each other need to be simultaneously reproduced. In additicni, for dilute solutions, where we have a low concentration of solute, the solute-solute RDFs converge very slowly in the CG simulations. In this case, the PMF between the solute molecules can be obtained using free-energy calculation methods such as umbrella sampling or constraint dynamics. Recently, these methods have been used in an iterative optimization approach to study self-assembling dipeptides at the CG scale [75, 76]. The PMF between solute molecules in a solvent box, Fpi (r), is calculated by all-atom simulation from n distance constraint simulations ... 

Figure 1. Schematic of the iterative optimization approach to designing a tissue engineering scaffold with the addition of a biologically active peptide.

Wang, J. and Cohen, M. (2005) An iterative optimization approach for unified image segmentation and matting. Proceedings of Computer Vision and Pattern Recognition (CVPR), 2 936-943. 

A novel optimization approach based on the Newton-Kantorovich iterative scheme applied to the Riccati equation describing the reflection from the inhomogeneous half-space was proposed recently [7]. The method works well with complicated highly contrasted dielectric profiles and retains stability with respect to the noise in the input data. However, this algorithm like others needs the measurement data to be given in a broad frequency band. In this work, the method is improved to be valid for the input data obtained in an essentially restricted frequency band, i.e. when both low and high frequency data are not available. This... 

In our simulations of histone modifying enzymes, the computational approaches centered on the pseudobond ab initio quantum mechanical/molecular mechanical (QM/MM) approach. This approach consists of three major components [20,26-29] a pseudobond method for the treatment of the QM/MM boundary across covalent bonds, an efficient iterative optimization procedure which allows for the use of the ab initio QM/MM method to determine the reaction paths with a realistic enzyme environment, and a free energy perturbation method to take account... 

A more subjective approach to the multiresponse optimization of conventional experimental designs was outlined by Derringer and Suich (22). This sequential generation technique weights the responses by means of desirability factors to reduce the multivariate problem to a univariate one which could then be solved by iterative optimization techniques. The use of desirability factors permits the formulator to input the range of property values considered acceptable for each response. The optimization procedure then attempts to determine an optimal point within the acceptable limits of all responses. 

The classic methods use an ODE solver in combination with an optimization algorithm and solve the problem sequentially. This solution strategy is referred to as a sequential solution and optimization approach, since for each iteration the optimization variables are set and then the differential equation constraints are integrated. Though straightforward, this approach is generally inefficient because it requires the accurate solution of the model equations at each iteration within the optimization, even when iterates are far from the final optimal solution. 

A feasible path optimization approach can be very expensive because an iterative calculation is required to solve the undetermined model. A more efficient way is to use an unfeasible path approach to solve the NLP problem however, many of these large-scale NLP methods are only efficient in solving problems with few degrees of freedom. A decoupled SQP method was proposed by Tjoa and Biegler (1991) that is based on a globally convergent SQP method. 

Basically, two types of approaches are developed here iterative (optimization-based) approaches like the one by Sippl et al. [101] and direct approaches like the one by Kabsch [102, 103], based on Lagrange multipliers. Unfortunately, the much expedient direct methods may fail to produce a sufficiently accurate solution on some degenerate cases. Redington [104] suggested a hybrid method with an improved version of the iterative approach, which requires the computation of only two 3x3 matrix multiplications in the inner loop of the optimization. 

Although it is beyond the scope of this chapter, more sophisticated interpretive methods can be employed to compensate for the anomalous retention behavior. That is, the complex retention surfaces are broken down into smaller parts in an iterative fashion the smaller retention surfaces are more accurately modeled by simple functions. This iterative interpretive approach has recently been applied in micellar LC (MLC) for the optimization of organic modifier, surfactant concentration, and pH (57,58). 

In contrast to the explicit analytical solution of least-squares fit used in linear regression, our present treatment of data analysis relies on an iterative optimization, which is a completely different approach as a result of the operations discussed in the previous section, theoretical data are calculated, dependent on the model and choice of parameters, which can be compared with the experimental results. The deviation between theoretical and experimental data is usually expressed as the sum of the errors squared for all the data points, alternatively called the sum of squared deviations (SSD) ... 

The computational approach described here, based on the combination of the Kalman filter algorithm and iterative optimization by the simulated annealing method, was able to find the optimal alignment of the pure component peaks with respect to the shifted components in the overlapped spectra, and hence, to correctly estimate the contributions of each component in the mixture. The simulated annealing demonstrated superior ability over the other optimization methods, simplex and steepest descent, in yielding more reliable convergences at the expense of not much more computer time, at least for resolving ternary shifted overlapped spectra. 

Fig. 1 Systematic approach for iterative optimization of catalyst preparation procedure and preparation-performance model.

The first iterations w -w are needed to collect the information that is required to estimate the empirical gradients. Thereafter, the iterative improvement starts. It can be seen that despite a significant error in the chromatograms, the iterative optimization converges to the true optimum and establishes the desired purities and recoveries of the components. In recent work, this approach has been applied to continuous annular electrochromatography (Behrens and Engell, 2011). 

Orcun S., Joglekar G. and Clark S. 2002. An iterative optimization-simulation approach to account for yield variability and decentralized risk parameterization, AIChE Annual Meeting, Indianapolis, Indiana, paper 266b. 

Simulated annealing and taboo search are two techniques that can be viewed as generalizations of the iterative improvement approach to combinatorial optimization problems. 

The Tophss Tree is an enduring tool for the iterative optimization of a chemical series where it is not possible to synthesize aU close analogs (i.e., for all medicinal chemistry projects) Figure 8.2 illustrates the practical application of the method. The continued popularity of the Tophss Tree (although, in the authors view, the tool is shU underused by medicinal chemists) is perhaps explained by the simplicity and visual applicatiou of the approach. However, the method is without a strong mathematical framework, which limits its apphcabihty to more complex problems, such as multiparameter optimization (MPO). 

The team adonpted to synthesize the full 25(X) set of compounds—620 could not be made with a reasonable time investment, a sobering statistic. AU compounds were tested against MMP-12 to derive the fuU dataset. Independently, a series of iterative experiments were made, guided by a purely computational optimization approach, in this case a genetic algorithm (GA). At each iteration, 14 molecules were made and tested. The results were added to the known data, and the GA was... 

The third step is again preceded by a decision is the mix design to be performed with or without application of an optimization approach In both cases, iterative procedure is necessary. However, it is believed that the left-hand path in Figure 12.1 is shorter and leads to better results, because objective indications from the optimization approach are used. The tentative design may in many cases be started with a very distant solution from the final one. The end of the design is presented in the form of a Ust of components with their volume or mass fractions, computed for one cubic meter or for one batch. 

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