Local minimization algorithms

Generating Inherent Structures of Liquids Comparison of Local Minimization Algorithms. 

Figure 8 Descent structure of local minimization algorithms.

Simple energy minimization methods generally find the nearest local minimum. There are three major families of local minimization algorithms ... 

A drop of water that is placed on a hillside will roll down the slope, following the surface curvature, until it ends up in the valley at the bottom of the hill. This is a natural minimization process by which the drop minimizes its potential energy until it reaches a local minimum. Minimization algorithms are the analogous computational procedures that find minima for a given function. Because these procedures are downhill methods that are unable to cross energy barriers, they end up in local minima close to the point from which the minimization process started (Fig. 3a). It is very rare that a direct minimization method... 

The goal of all minimization algorithms is to find a local minimum of a given function. They differ in how closely they try to mimic the way a drop of water or a small ball would roll down the slope, following the surface curvature, until it ends up at the bottom. Consider a Taylor expansion around a minimum point Xq of the general one-dimensional function F(X), which can be written as... 

Together, these general considerations of sampling and complexity have led to two broad classes of multivariate minimization algorithms local and global (see Figure 7). 

Figure 7 Structure of local and global minimization algorithms.

The restricted step method of the type discussed below were originally proposed by Levenberg and Marquardt [37,38] and extended to minimization algorithms by Goldfeld, Quandt and Trotter [39]. Recently Simons [13] discussed the restricted step method with respect to molecular energy hypersurfaces. The basic idea again is that the energy hypersurface E(x) can reasonably be approximated, at least locally, by the quadratic function... 

To improve the efficiency of adjusting side chain rotamers, Schaffer and Verkhiv-ker implemented a postdocking side chain optimization procedure using the deadend elimination (DEE) algorithm, follo ved by local minimizations and energy evaluations of all generated DEE solutions [66]. 

Hibbert examined several variants of the SGA for the problem of determining a set of kinetic rate constants for coupled reactions where the concentrations of all species are not known. This lack of full knowledge gives rise to multiple local minima. In particular, he compared SGA with SGA plus local minimization, and real-valued vs. binary valued chromosomes. He also examined the use of incest prevention methods, which allow only dissimilar parents to mate and that forbid duplicate children to be formed. His conclusions for this application are (1) SGA does not work well, (2) adding incest prevention makes the algorithm much more robust, and (3) adding local gradient minimization is very beneficial and not too expensive in this case. 

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