Quadratic minimum

When r < 0, there is a quadratic minimum at the origin. At the bifurcation value r = 0, the minimum becomes a much flatter quartic. For r > 0, a local maximum appears at the origin, and a symmetric pair of minima occur to either side of it. ... 

Most gradient optimization methods rely on a quadratic model of the potential surface. The minimum condition for the... 

Figure B3.5.1. Contour line representation of a quadratic surface and part of a steepest descent path zigzagging toward the minimum.

Compared with the Morse potential, Hooke s law performs reasonably well in the equilibrium area near If, where the shape of the Morse function is more or less quadratic (see Figure 7-9 in the minimum-energy region). To improve the performance of the harmonic potential for non-equilibrium bond lengths also, higher-order terms can be added to the potential according to Eq. (21). 

HyperChein has two synch ron ons transit meth ods im piemen ted. The linear synchronous transit method (LST) searches for a maximum along a linear path between reactants and products. It may happen that this method will end up with a structure having two or more negative eigenvalues. The quadratic synchronous transit method (QSTlisan improvement of LST approach and searches for a maximum along a parabola connecting reactants and products, and for a minimum in all directions perpendicular to the parabola. 

Fig. 5.8 The minimum in a line search may be found more effectively by fitting an analytical function such as a quadratic to the initial set of three points (1, 2 and 3). A better estimate of the minimum can then be found by fitting a new function to the points 1, 2 and 4 and finding its minimum. (Figure adapted from Press W H, B P Flannery,...

This formula is exact for a quadratic function, but for real problems a line search may be desirable. This line search is performed along the vector — x. . It may not be necessary to locate the minimum in the direction of the line search very accurately, at the expense of a few more steps of the quasi-Newton algorithm. For quantum mechanics calculations the additional energy evaluations required by the line search may prove more expensive than using the more approximate approach. An effective compromise is to fit a function to the energy and gradient at the current point x/t and at the point X/ +i and determine the minimum in the fitted function. 

Quantitative controllable variables are ftequentiy related to the response (or performance) variable by some assumed statistical relationship or model. The minimum number of conditions or levels per variable is determined by the form of the assumed model. For example, if a straight-line relationship can be assumed, two levels (or conditions) may be sufficient for a quadratic relationship a minimum of three levels is required. However, it is often desirable to include some added points, above the minimum needed, so as to allow assessment of the adequacy of the assumed model. 

There are several reasons that Newton-Raphson minimization is rarely used in mac-romolecular studies. First, the highly nonquadratic macromolecular energy surface, which is characterized by a multitude of local minima, is unsuitable for the Newton-Raphson method. In such cases it is inefficient, at times even pathological, in behavior. It is, however, sometimes used to complete the minimization of a structure that was already minimized by another method. In such cases it is assumed that the starting point is close enough to the real minimum to justify the quadratic approximation. Second, the need to recalculate the Hessian matrix at every iteration makes this algorithm computationally expensive. Third, it is necessary to invert the second derivative matrix at every step, a difficult task for large systems. 

Figure 10. Divisions of the Em map for the quadratic Hamiltonian in Eq. (39). Va, Vb, and Vx are the energies of the two isomers and that of the saddle point between them. The solid lines are relative equilibria, and the cusp points, a , indicate points at which the secondary minimum and the saddle point in Fig. 9 merge at a point of inflection.

Figure 3 gives two examples of L and L closeness of two functions. The L closeness leaves open the possibility that in a small region of the input space (with, therefore, small contribution to the overall error) the two functions can be considerably different. This is not the case for L closeness, which guarantees some minimal proximity of the two functions. Such a proximity is important when, as in this case, one of the functions is used to predict the behavior of the other, and the accuracy of the prediction has to be established on a pointwise basis. In these cases, the L error criterion (4) and its equivalent [Eq. (6)] are superior. In fact, L closeness is a much stricter requirement than L closeness. It should be noted that whereas the minimization of Eq. (3) is a quadratic problem and is guaranteed to have a unique solution, by minimizing the IT expected risk [Eq. (4)], one may yield many solutions with the same minimum error. With respect to their predictive accuracy, however, all these solutions are equivalent and, in addition, we have already retreated from the requirement to find the one and only real function. Therefore, the multiplicity of the best solutions is not a problem. 

In order to solve Eq. (34), we use the method of characteristics and consider a family of classical trajectories on the inverted potential q(p, x), p(P, x), where P is an (A — 1)-dimensional parameter to characterize the trajectory and x is the time running for the infinite interval along the trajectory, where x = — oo corresponds to the minimum of the potential q(p, —oo) = q ,p(p, —oo) = 0. The solution we want is the trajectory that connects the two potential minima and along which the action becomes minimum. This is called the instanton trajectory and belongs to the above mentioned family qo(x) = q(Po At q close to the potential minimum q , the momentum p(q) is linear with respect to the deviation (q — q ) and Wo(q) is quadratic. 

---