Local minimization

The forces in a protein molecule are modeled by the gradient of the potential energy V(s, x) in dependence on a vector s encoding the amino acid sequence of the molecule and a vector x containing the Cartesian coordinates of all essential atoms of a molecule. In an equilibrium state x, the forces (s, x) vanish, so x is stationary and for stability reasons we must have a local minimizer. The most stable equilibrium state of a molecule is usually the... 

ICM [64] pseudo-Brownian sampling and local minimization mbced force field and empirical score... 

Generating Inherent Structures of Liquids Comparison of Local Minimization Algorithms. 

One can show (Ericksen, 1975) that the homogenous state w(x) = Wi corresponding to a deeper minimum of g(w Oo) is the global minimizer of the functional (2.1) another homogenous configuration w(x) = Wo (the metastable state), is only a weak local minimizer... 

Ball, J.M., and James, R.D., 1996, Local minimizers and phase transformations, Preprint. 

A common difficulty accompanying the representation of the conformational space by sampling is the redundancy among conformers. Usually many hundreds or thousands of conformers are generated and then reprocessed to pick out families representative of the whole space. After a local minimization, many of these conformers may fall into the same conformation, reproduced several times. Therefore, Catalyst focuses on the coverage of all possible bioactive conformations of a compound compared with methods that represent conformational space as a collection (clusters) of local minima. 

Figure 8 Descent structure of local minimization algorithms.

Figure 26 Stereo diagrams illustrating (top) a compact conformation of N-acetyl-Ala19-NHCH3 after local minimization of the complete energy function to reach this particular local minimum, and (bottom) the global-minimum (right-handed a helix) structure attained by first optimizing the local electrostatic interactions and then carrying out a complete energy minimization.188...

In view of the advantages just described, the Lamarckian GA method incorporating local minimization of R is now our standard approach for tackling structure solution from powder diffraction data. 

The first real-life application of Eq. (7.1-7) was made by Mezaki and Butt (1968) to a four-component chemical reaction system. They estimated the six kinetic parameters at each temperature by minimizing n(0). and also by minimizing the total sum of squares, tr [n(6)]. The parameter estimates by the two methods differed significantly, and the minimum-]n estimates were more consistent with the literature. The greater steepness of the n(6) function gave faster convergence toward 6. Computation of a grid of values around the endpoint confirmed the local minimization of ln(e). ... 

ICM MC minimization, Pseudo-Brownian sampling and local minimization [83]... 

Exploiting the heuristic that many local minimizers of constraints lie at vertices of the parameter space through vertex searches and local searches from promising vertices. 

Among the various MC procedures designed to solve the phase problem from powder data, we quote the pioneering work of Harris et al., the paper by Andreev et al.," where the mathematical procedures for generating flexible molecular structures has been clearly described, and the contribution by Tremayne et al., implemented in the computer program OCTOPUS,where a local minimization at each MC move is introduced to increase the efficiency... 

It is important to realize that throughout each local minimization cycle the densities at nearest-neighbor sites of site i represented by the set remain fixed at the initial values assigned to them at the beginning of the local cycle. The iterative solution of Eq. (D.14) is halted if [see Eq. (D.7)]... 

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