Hessian matrix definition

It can be shown from a Taylor series expansion that if/(x) has continuous second partial derivatives, /(x) is concave if and only if its Hessian matrix is negative-semidefinite. For/(x) to be strictly concave, H must be negative-definite. For /(x) to be convex H(x) must be positive-semidefinite and for/(x) to be strictly convex, H(x) must be positive-definite. 

As indicated in Table 4.2, the eigenvalues of the Hessian matrix of fix) indicate the shape of a function. For a positive-definite symmetric matrix, the eigenvectors (refer to Appendix A) form an orthonormal set. For example, in two dimensions, if the eigenvectors are Vj and v2, v[v2 =0 (the eigenvectors are perpendicular to each other). The eigenvectors also correspond to the directions of the principal axes of the contours of fix). 

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... 

Marquardt (1963), Levenberg (1944), and others have suggested that the Hessian matrix of fix) be modified on each stage of the search as needed to ensure that the modified H(x),H(x), is positive-definite and well conditioned. The procedure adds elements to the diagonal elements of H(x)... 

Is it necessary that the Hessian matrix of the objective function always be positive-definite in an unconstrained minimization problem ... 

Show how to make the Hessian matrix of the following objective function positive-definite at x = [1 l]r by using Marquardt s method ... 

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 condition number of the Hessian matrix of the objective function is an important measure of difficulty in unconstrained optimization. By definition, the smallest a condition number can be is 1.0. A condition number of 105 is moderately large, 109 is large, and 1014 is extremely large. Recall that, if Newton s method is used to minimize a function/, the Newton search direction s is found by solving the linear equations... 

To check the sufficiency conditions, we examine the Hessian matrix of W (after substituting p and / ) to see if it is positive-definite. 

This form is convenient in that the active inequality constraints can now be replaced in the QP by all of the inequalities, with the result that Sa is determined directly from the QP solution. Finally, since second derivatives may often be hard to calculate and a unique solution is desired for the QP problem, the Hessian matrix, is approximated by a positive definite matrix, B, which is constructed by a quasi-Newton formula and requires only first-derivative information. Thus, the Newton-type derivation for (2) leads to a nonlinear programming algorithm based on the successive solution of the following QP subproblem ... 

SHELXL (Sheldrick and Schneider, 1997) is often viewed as a refinement program for high-resolution data only. Although it undoubtedly offers features needed for that resolution regime (optimization of anisotropic temperature factors, occupancy refinement, full matrix least squares to obtain standard deviations from the inverse Hessian matrix, flexible definitions for NCS, easiness to describe partially... 

Now, since the Hessian is the second derivative matrix, it is real and symmetric, and therefore hermitian. Thus, all its eigenvalues are real, and it is positive definite if all its eigenvalues are positive. We find that minimization amounts to finding a solution to g(x)=0 in a region where the Hessian is positive definite. Convergence properties of iterative methods to solve this equation have earlier been studied in terms of the Jacobian. We now find that for this type of problems the Jacobian is in fact a Hessian matrix. 

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. 

Newton s method and quasi-Newton techniques make use of second-order derivative information. Newton s method is computationally expensive because it requires analytical first-and second-order derivative information, as well as matrix inversion. Quasi-Newton methods rely on approximate second-order derivative information (Hessian) or an approximate Hessian inverse. There are a number of variants of these techniques from various researchers most quasi-Newton techniques attempt to find a Hessian matrix that is positive definite and well-conditioned at each iteration. Quasi-Newton methods are recognized as the most powerful unconstrained optimization methods currently available. 

The algorithm also ensures that a regular quadratic programme with a positive definite Hessian matrix is obtained at each step, provided that this is so for the initial point. Thus, although a "first-phase" procedure may sometimes be required to locate such a point, no rescue procedure is needed subsequently. 

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. 

The Hessian matrix is a generalization in R of the concept of curvature of a function. The positive-definiteness of the Hessian is a generalized notion of positive curvature. Thus, the properties of H are very important in formulating minimum-seeking algorithms. 

Condition (2c) requires the Hessian matrix to be positive definite that is, the eigenvalues of G are all greater than... 

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. 

The minimum on such a quadratics surface can readily be identified, it is either a minimum within the region, with the gradient vector g null and the Hessian matrix G positive definite or the minimum is on the boundary of the region. 

Standard representation of the TS in organic chemistry textbooks is the point of maximum energy on the reaction coordinate. More precise is the definition provided in Section 1.6 the TS is the col, a point where aU the gradients vanish, and all of the eigenvalues of the Hessian matrix are positive except one, which corresponds to the reaction coordinate. In statistical kinetic theories, a slightly different definition of the TS is required. 

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