Functions penalty

A particularly important application of molecular dynamics, often in conjunction with the simulated annealing method, is in the refinement of X-ray and NMR data to determine the three-dimensional structures of large biological molecules such as proteins. The aim of such refinement is to determine the conformation (or conformations) that best explain the experimental data. A modified form of molecular dynamics called restrained moleculai dynarrdcs is usually used in which additional terms, called penalty functions, are added tc the potential energy function. These extra terms have the effect of penalising conformations... 

The additional penalty function that is added to the empirical potential energy function in restrained dynamics X-ray refinement has the form ... 

The generation of the penalty function requires selection from among many possible different functions as well as selection of many parameters of the penalty function. The contribution to the objective by the penalty function, relative to the contribution of the original objective function, changes the nature of the original objective function, which ia turn influences the ease or difficulty of finding the optimum. 

Penalty functions with augmented Lagrangian method (an enhancement of the classical Lagrange multiplier method)... 

There are two main methods for enforcing such constraints. One is the Penalty Function approach, the other the metlrod of Lagrange Undetermined Multipliers. 

The penalty function approach adds a tenn of tire type k r — ro) to the function to be optimized. The variable r is constrained to be near the target value ro, and the force constant k describes how important the constraint is compared with the unconstrained optimization. By making k arbitrary large, tire constraint may be fulfilled to any given... 

Equation (4.c) is discussed in Appendix A. For a symmetric molecule that does not possess a dipole moment and interacts with the electric field of the laser pulse through its polarizability, the choice of the penalty function for the... 

The penalty function for the laser fiuence, which gives rise to the term involving (3 in Eq. (10), acts to limit the magnitude of the electric field to within physically acceptable limits. In order to further limit the field strength the search direction is projected as follows [101] ... 

To improve the convergence of the gradient-type method, Tannor et al. [81, 93] suggested employing the Krotov iteration method [102]. In formulating their method, they utilize a penalty function of the form /[e(f)] = pe (f). In Tannor s Krotov method, the fcth iteration step of the solution process is given by... 

Reinhart and Rippin (1986) proposed two methods for design under uncertainty (1) introduction of a penalty function for the probability of exceeding the available production time, whereby the probability can be generated by standard error propagation techniques for technical or commercial uncertainties, and (2) the Here and Now method. 

One of the simplest and yet very effective penalty function that keeps the parameters in the interval (kmin i, km ,) is... 

The functions essentially place an equally weighted penalty for small or large-valued parameters on the overall objective function. If penalty functions for... 

The user supplied weighting constant, (>0), should have a large value during the early iterations of the Gauss-Newton method when the parameters are away from their optimal values. As the parameters approach the optimum, should be reduced so that the contribution of the penalty function is essentially negligible (so that no bias is introduced in the parameter estimates). 

With a few minor modifications, the Gauss-Newton method presented in Chapter 4 can be used to obtain the unknown parameters. If we consider Taylor series expansion of the penalty function around the current estimate of the parameter we have,... 

While prior information may be used to influence the parameter estimates towards realistic values, there is no guarantee that the final estimates will not reach extreme values particularly when the postulated grid cell model is incorrect and there is a large amount of data available. A simple way to impose inequality constraints on the parameters is through the incorporation of a penalty function as already discussed in Chapter 9 (Section 9.2.1.2). By this approach extra terms are added in the objective function that tend to explode when the parameters approach near the boundary and become negligible when the parameters are far. One can easily construct such penalty functions. For example a simple and yet very effective penalty function that keeps the parameters in the interval (kmjnkmaXil) is... 

In the case of the Porod-Kratky model, the polymer backbones have a constant curvature c0. Accounting for the polymer stiffness in generating the dense configuration of stream lines, the vector field used must have a homogeneous curvature field with a unique value cq in the entire simulation box T. In order to quantify the success in creating such a vector field, the deviation of the curvature from the ideal Porod-Kratky case, a volume integral has been used by Santos as a penalty function ... 

The Complex algorithm was modified slightly to include the two extra summation constraints as penalty functions. The response of leachability was minimized at a value of -3.53 with the following formulation ... 

Viswanathan, J. and Grossmann, I.E. (1990) A combined penalty function and outer approximation method for MINLP optimization. Comput. Chem. Eng., 14, 769-782. 

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