LJ Optimization Procedure

One of the most reliable direct search methods is the LJ optimization procedure (Luus and Jaakola, 1973). This procedure uses random search points and systematic contraction of the search region. The method is easy to program and handles the problem of multiple optima with high reliability (Wang and Luus, 1977, 1978). A important advantage of the method is its ability to handle multiple nonlinear constraints. 

The adaptation of the original LJ optimization procedure to parameter estimation problems for algebraic equation models is given next. 

Given the fact that in parameter estimation we normally have a relatively smooth LS objective function, we do not need to be exceptionally concerned about local optima (although this may not be the case for ill-conditioned estimation problems). This is particularly true if we have a good idea of the range where the parameter values should be. As a result, it may be more efficient to consider using a value for NR which is a function of the number of unknown parameters. For example, we may consider 

Typical values would be NR=60 when p=l, NR=110 when p=5 and NR=I60 when p=10. 

At the same time we may wish to consider a slower contraction of the search region as the dimensionality of the parameter space increases. For example we could use a constant reduction of the volume (say 10%) of the search region rather than a constant reduction in the search region of each parameter. Namely we could use, 

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The Simplex algorithm and that of Powell s are examples of derivative-free methods (Edgar and Himmelblau, 1988 Seber and Wild, 1989, Powell, 1965). In this chapter only two algorithms will be presented (1) the LJ optimization procedure and (2) the simplex method. The well known golden section and Fibonacci methods for minimizing a function along a line will not be presented. Kowalik and Osborne (1968) and Press et al. (1992) among others discuss these methods in detail. 

Since we have a minimization problem, significant computational savings can be realized by noting in the implementation of the LJ optimization procedure that for each trial parameter vector, we do not need to complete the summation in Equation 5.23. Once the LS Objective function exceeds the smallest value found up to that point (S ), a new trial parameter vector can be selected. 

In this problem you are asked to determine the unknown parameters using the dominant zeros and poles of the original system as an initial guess. LJ optimization procedure can be used to obtain the best parameter estimates. 

Figure 8.4 Use of the LJ optimization procedure to bring the first parameter estimates inside the region of convergence of the Gauss-Newton method (denoted by the solid line). All test points are denoted by +. Actual path of some typical rims is shown by the dotted line.

In this section we briefly outline the parametrization protocol for determining the partial atomic charges, atomic polarizabilities, and the atom-based Thole damping factors, as well as the optimization procedure of the force field parameters not dependent of the Drude oscillator positions, namely the bonded and Lennard-Jones terms. While the overall parameter optimization is described linearly in the text, it is important to bear in mind that the bonded and nonbonded parameters are strongly interdependent, such that in practice, an iterative procedure is adopted, with the electrostatic and LJ nonbonded and bonded parameters optimized in turn until a self-consistent solution is reached, offering optimal agreement with all sets of target data. 

The pseudo parameters of the HSE theory are derived from an equation of state expanded in powers of 1/kT about a hard-sphere fluid, as is developed by the perturbation theory. Consequently, it is reasonable to expect that procedures for defining optimal diameters for the perturbation theory should work well with the HSE procedure. The first portion of this chapter shows that this is indeed correct. The Verlet-Weis (VW) (5) modification of the Weeks, Chandler, and Anderson (WCA) (6) procedure was used here to determine diameters in a mixture of Len-nard-Jones (LJ) (12-6) fluids. These diameters then were used in the HSE procedure to predict the mixture properties. 

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