Method minimization algorithm

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... 

Order 2 minimization algorithms, which use the second derivative (curvamre) as well as the first derivative (slope) of the potential function, exhibit in many cases improved rate of convergence. For a molecule of N atoms these methods require calculating the 3N X 3N Hessian matrix of second derivatives (for the coordinate set at step k)... 

In non-metric MDS the analysis takes into account the measurement level of the raw data (nominal, ordinal, interval or ratio scale see Section 2.1.2). This is most relevant for sensory testing where often the scale of scores is not well-defined and the differences derived may not represent Euclidean distances. For this reason one may rank-order the distances and analyze the rank numbers with, for example, the popular method and algorithm for non-metric MDS that is due to Kruskal [7]. Here one defines a non-linear loss function, called STRESS, which is to be minimized ... 

Nelder and Mead (1965) described a more efficient (but more complex) version of the simplex method that permitted the geometric figures to expand and contract continuously during the search. Their method minimized a function of n variables using (n + 1) vertices of a flexible polyhedron. Details of the method together with a computer code to execute the algorithm can be found in Avriel (1976). 

In practice, initial guesses of the fitting parameters (e.g. pre-exponential factors and decay times in the case of a multi-exponential decay) are used to calculate the decay curve the latter is reconvoluted with the instrument response for comparison with the experimental curve. Then, a minimization algorithm (e.g. Marquardt method) is employed to search the parameters giving the best fit. At each step of the iteration procedure, the calculated decay is reconvoluted with the instrument response. Several softwares are commercially available. 

The best-fitting set of parameters can be found by minimization of the objective function (Section 13.2.8.2). This can be performed only by iterative procedures. For this purpose several minimization algorithms can be applied, for example, Simplex, Gauss-Newton, and the Marquardt methods. It is not the aim of this chapter to deal with non-linear curve-fitting extensively. For further reference, excellent papers and books are available [18]. 

This projected velocity Verlet algoiithm has been found to be an efficient and simple minimization algorithm for many of the methods discussed here. 

There Eire other Hessian updates but for minimizations the BFGS update is the most successful. Hessism update techniques are usually combined with line search vide infra) and the resulting minimization algorithms are called quasi-Newton methods. In saddle point optimizations we must allow the approximate Hessian to become indefinite and the PSB update is therefore more appropriate. 

Much of this progress has been centered around the discovery that energy derivatives can often be explicitly evaluated [3-7], freeing potential energy searches from inefficient minimization algorithms such as uniaxial methods, simplex methods and even numerical derivative methods [8]. 

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... 

It is beyond the scope of this short review to list every available molecular mechanics program. Only a selected few programs are mentioned here, without descriptive details of the potential functions, minimization algorithms, or comparative evaluations. Both the CHARMM and AMBER force fields use harmonic potential functions to calculate protein structures. They were developed in the laboratories of Karplus and Kollman, respectively, and work remarkably well. The CFF and force fields use more complex potential functions. Both force fields were developed in commercial settings and based extensively or exclusively on results obtained from quantum mechanics. Unlike the other molecular mechanics methods, the OPLS force field was parameterized by Jorgensen to simulate solution phase phenomena. 

---