Steepest descent minimization

On the Energy minimization preferences box, specify the number of steps/ cycles and method of minimization (Steepest descent or Conjugate gradients), angles/bonds to be minimized, and conditions for terminating the minimization. Be sure to check the radio button for Lock nonselected residues and the box for Show energy report. 

The advan tage ol a conjugate gradien t m iniim/er is that it uses th e minim i/ation history to calculate the search direction, and converges t asLer Lhan the steepest descent technique. It also contains a scaling factor, b, for determining step si/e. This makes the step si/es optimal when compared to the steepest descent lechniciue. 

Another difference from steepest descent is that a one-diinen-sional minimization is performed in each search direction. Aline mmimi/ation is made along a direction h until a minlmnni energy is found at anew point i-i-l then the search direction is updated and a search down the new direction h ] is made. This... 

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. 

A conjugate gradient method differs from the steepest descent technique by using both the current gradient and the previous search direction to drive the minimization. A conjugate gradient method is a first order minimizer. 

Example Compare the steps of a conjugate gradient minimization with the steepest descent method. Amolecular system can reach a potential minimum after the second step if the first step proceeds from A to B. If the first step is too large, placing the system at D, the second step still places the system near the minimum(E) because the optimizer remembers the penultimate step. 

Another difference from steepest descent is that a one-dimensional minimization is performed in each search direction. Aline minimization is made along a direction hj until a minimum... 

Figure 5 A comparison of steepest descent (SD) minimization and conjugated gradients (CG) minimization of the same protein.

In the light of the path-integral representation, the density matrix p Q-,Q-,p) may be semi-classically represented as oc exp[ —Si(Q )], where Si(Q ) is the Eucledian action on the -periodic trajectory that starts and ends at the point Q and visits the potential minimum Q = 0 for r = 0. The one-dimensional tunneling rate, in turn, is proportional to exp[ —S2(Q-)], where S2 is the action in the barrier for the closed straight trajectory which goes along the line with constant Q. The integral in (4.32) may be evaluated by the method of steepest descents, which leads to an optimum value of Q- = Q. This amounts to minimization of the total action Si -i- S2 over the positions of the bend point Q. ... 

This corresponds to a steepest descent minimization with a fixed step As. As discussed in Section 14.1, such an approach tends to oscillate around the true path, and consequently requires a small step size for following the IRC accurately. 

SOME RELEVANT COMPUTER PROGRAMS 4.A. Steepest Descent Minimization Program... 

We confine ourselves here to the minimal residual method and the method of steepest descent relating to two-layer schemes. As usual, the explicit scheme is considered first ... 

If the matrix A is positive definite, i.e. it is symmetric and has positive eigenvalues, the solution of the linear equation system is equivalent to the minimization of the bilinear form given in Eq. (64). One of the best established methods for the solution of minimization problems is the method of steepest descent. The term steepest descent alludes to a picture where the cost function F is visualized as a land-... 

The rate of convergence of the Steepest Descent method is first order. The basic difficulty with steepest descent is that the method is too sensitive to the scaling of S(k), so that convergence is very slow and oscillations in the k-space can easily occur. In general a well scaled problem is one in which similar changes in the variables lead to similar changes in the objective function (Kowalik and Osborne, 1968). For these reasons, steepest descent/ascent is not a viable method for the general purpose minimization of nonlinear functions, ft is of interest only for historical and theoretical reasons. 

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