Hessian matrix local

The second energy derivatives with respect to the x, y, and z directions of centers a and b (for example, the x, y component for centers a and b is Hax,by = (3 E/dxa3yb)o) form the Hessian matrix H. The elements of H give the local curvatures of the energy surface along the 3N cartesian directions. 

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. 

Instead of a formal development of conditions that define a local optimum, we present a more intuitive kinematic illustration. Consider the contour plot of the objective function fix), given in Fig. 3-54, as a smooth valley in space of the variables X and x2. For the contour plot of this unconstrained problem Min/(x), consider a ball rolling in this valley to the lowest point offix), denoted by x. This point is at least a local minimum and is defined by a point with a zero gradient and at least nonnegative curvature in all (nonzero) directions p. We use the first-derivative (gradient) vector Vf(x) and second-derivative (Hessian) matrix V /(x) to state the necessary first- and second-order conditions for unconstrained optimality ... 

Steepest descent can terminate at any type of stationary point, that is, at any point where the elements of the gradient of /(x) are zero. Thus you must ascertain if the presumed minimum is indeed a local minimum (i.e., a solution) or a saddle point. If it is a saddle point, it is necessary to employ a nongradient method to move away from the point, after which the minimization may continue as before. The stationary point may be tested by examining the Hessian matrix of the objective function as described in Chapter 4. If the Hessian matrix is not positive-definite, the stationary point is a saddle point. Perturbation from the stationary point followed by optimization should lead to a local minimum x. ... 

Difficulty 3 can be ameliorated by using (properly) finite difference approximation as substitutes for derivatives. To overcome difficulty 4, two classes of methods exist to modify the pure Newton s method so that it is guaranteed to converge to a local minimum from an arbitrary starting point. The first of these, called trust region methods, minimize the quadratic approximation, Equation (6.10), within an elliptical region, whose size is adjusted so that the objective improves at each iteration see Section 6.3.2. The second class, line search methods, modifies the pure Newton s method in two ways (1) instead of taking a step size of one, a line search is used and (2) if the Hessian matrix H(x ) is not positive-definite, it is replaced by a positive-definite matrix that is close to H(x ). This is motivated by the easily verified fact that, if H(x ) is positive-definite, the Newton direction... 

The Kuhn-Tucker necessary conditions are satisfied at any local minimum or maximum and at saddle points. If (x, A, u ) is a Kuhn-Tucker point for the problem (8.25)-(8.26), and the second-order sufficiency conditions are satisfied at that point, optimality is guaranteed. The second order optimality conditions involve the matrix of second partial derivatives with respect to x (the Hessian matrix of the... 

If no active constraints occur (so x is an unconstrained stationary point), then (8.32a) must hold for all vectors y, and the multipliers A and u are zero, so V L = V /. Hence (8.32a) and (8.32b) reduce to the condition discussed in Section 4.5 that if the Hessian matrix of the objective function, evaluated at x, is positive-definite and x is a stationary point, then x is a local unconstrained minimum of/. 

The second order sufficiency conditions show that the first two of these three Kuhn-Tucker points are local minima, and the third is not. The Hessian matrix of the Lagrangian function is... 

Using DFT calculations to predict a phonon density of states is conceptually similar to the process of finding localized normal modes. In these calculations, small displacements of atoms around their equilibrium positions are used to define finite-difference approximations to the Hessian matrix for the system of interest, just as in Eq. (5.3). The mathematics involved in transforming this information into the phonon density of states is well defined, but somewhat more complicated than the results we presented in Section 5.2. Unfortunately, this process is not yet available as a routine option in the most widely available DFT packages (although these calculations are widely... 

The direction given by —H(0s) lVU 0s) is a descent direction only when the Hessian matrix is positive definite. For this reason, the Newton-Raphson algorithm is less robust than the steepest descent method hence, it does not guarantee the convergence toward a local minimum. On the other hand, when the Hessian matrix is positive definite, and in particular in a neighborhood of the minimum, the algorithm converges much faster than the first-order methods. 

Most modern minimization methods are designed to find local minima in the function by search techniques characteristically they assume very little knowledge of the detailed analytic properties of the function to be minimized, other than the fact that a minimum exists and therefore that, close enough to the minimum, the matrix of the second derivatives of the function with respect to the minimizing variables (the hessian matrix) is positive definite. 

In molecular quantum mechanics, the analytical calculation of G is very time consuming. Furthermore, as discussed later, the Hessian should be positive definite to ensure a step in the direction of the local minimum. One solution to this later problem is to precondition the Hessian matrix and this is discussed for the restricted step methods. The Quasi-Newton methods, presented next, provides alternative solution to both of these problems. 

If p denotes the number of negative eigenvalues of the local Hessian matrix H(r), then point r is said to belong to a domain Djj of the contour surface G(a). A local curvature analysis along the surface generates a subdivision into various curvature domains. For the three possible p values of 0, I, and 2, one obtains... 

The local canonical curvatures can be compared to a reference curvature parameter b [156,199]. For each point r of the molecular surface G(a) a number X = x(r,b) is defined as the number of local canonical curvatures [the number of eigenvalues of the local Hessian matrix H(r) that are less than this reference value b. The special case of b=0 allows one to relate this cla.ssification of points to the concept of ordinary convexity. If b=0, then p is the number of negative eigenvalues, also called the index of critical point r. As mentioned previously, in this special case the values 0, 1, or 2 for p(r,0) indicate that at the point r the molecular surface G(a) is locally concave, saddle-type, or convex, respectively [199]. 

The local curvature properties of the surface G(m) in each point r of the surface are given by the eigenv ues of the local Hessian matrix. Moreover, for a defined reference curvature b, the number p,(r, b) is defined as the number of local canonical curvatures (Hessian matrix eigenvalues) that are less than b. Usually b is chosen equal to zero and therefore the number p(r, 0) can take values 0,1, or 2 indicating that at the point r the molecular surface is locally concave, saddle-type, or convex, respectively. The three disjoint subsets Ao, Ai, and A2 are the collections of the surface points at which the molecular surface is locally concave, saddle-type, or convex, respectively the maximum connected components of these subsets Ao, Aj, and A2 are the surface domains denoted by Do,, Diand D2, where the index k refers to an ordering of these domains, usually according to decreasing surface size. 

---