Coordinate descent methods

Minimization methods that incorporate only function values generally involve some systematic method to search the conformational space. In coordinate descent methods, the search directions are the standard basis vectors. A sweep through these n search vectors produces a sequential modification of one function variable at a time. Through repeated sweeping of the n-dimensional space, a local minimum might ultimately be found. Unfortunately, this strategy is inefficient and not reliable.3 4... 

DUPLEX Brian E. Hingerty hingertybe oml.gov Powell s coordinate descent method (no derivatives)... 

The steepest descent method is a first order minimizer. It uses the first derivative of the potential energy with respect to the Cartesian coordinates. The method moves down the steepest slope of the interatomic forces on the potential energy surface. The descent is accomplished by adding an increment to the coordinates in the direction of the negative gradient of the potential energy, or the force. 

In terms of the refinement illustrated here, the function Y is the function Fix], The first point, Xj, is represented by all variable bond lengths, bond angles, conformation angles, chain position parameters, coordinates of the solvent of crystallization, etc., of the initial model. All other points X2,. .., Xj, represent trial values for the same n variables within the desired interval limits and subject to any other constraints, such as coupling of variables or hydrogen bond formation. Clearly, the number and type of variables, and their limits and constraints are easily changed in this procedure, as is the form of the function. The search procedure is also relatively rapid and does not suffer from a slowdown in the vicinity of the minimum, as may occur in steep-est-descent methods. 

Pattern Search. This method differs from the steepest descent technique in that a coordinate is left at its more favourable value when the next one is changed. If the initial stepsize is too large to find a lower point on the energy surface (a pattern ), the stepsize is decreased. This procedure is reported to give lower energies than the steepest descent method and should thus give a better approach of the true minimum. 

In the steepest-descent method, a displacement opposite to the potential energy gradient (i.e., in the direction of the force) is added to the coordinates at each step. This can be written... 

Then a steepest-descent method can be used in locating that (local or global) minimum that is closest to the starting geometry. This amounts to changing all nuclear coordinates simultaneously according to... 

The steepest descents method moves in the direction parallel to the net force, which in our geographical analogy corresponds to walking straight downhill. For 3N Cartesian coordinates this direction is most conveniently represented by a 3N-dimensional unit vector, sj. Thus ... 

Fig. 6. Failure of coordinate driving method due to a long valley not leading to the transition structure. The dotted curve is the steepest-descent path from the saddle point.

Fig. 5.29 Method for correcting the path followed by a steepest descents algorithm to generate the intrinsic reaction coordinate. The solid line shows the real path and the dotted line shows the algorithmic approximation to it. (Figure redrawn from Gonzalez C and H B Schlegel 1988. An Improved Algorithm for Reaction Path Following. Journal of Chemical Physics 90 2154-2161.)...

As explained above, the QM/MM-FE method requires the calculation of the MEP. The MEP for a potential energy surface is the steepest descent path that connects a first order saddle point (transition state) with two minima (reactant and product). Several methods have been recently adapted by our lab to calculate MEPs in enzymes. These methods include coordinate driving (CD) [13,19], nudged elastic band (NEB) [20-25], a second order parallel path optimizer method [25, 26], a procedure that combines these last two methods in order to improve computational efficiency [27],... 

The search for the optimum usually starts from the coordinates in the plane of the first two eigenvectors. However, to avoid the iteration (usually done with the method of steepest descent) stopping at a relative minimum, it is advisable to repeat the search from a different starting position, such as that given by the coordinates of two original variables. 

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