Hessian evaluation

When a large optimization problem is involved, it is essential to exploit the sparsity of the Hessian to reduce both memory allocation and computational time. The following alternatives can be used  

This does not explicitly require the Hessian, just the gradient evaluation at a single point along the axis p. 

When an algorithm requires the evaluation ofthe Hessian, preserving Hessian fectori-zation for certain iterations, especially in the final steps of the search, may be advisable. 

In the BzzMath library, the BzzMatrixSparseSymmetricLocked class allows the gradient, the elements of the Hessian diagonal and the nonzero elements on the left-hand side Hessian matrix to be evaluated numerically by the functions 

The function BuildGradientAndPositiveHessian ensures that the Hessian is positive definite by adequately increasing diagonal elements similarly to Gill-Murray s method (see Section 3.6.1). 

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To discuss the form and cost of analytic gradient and Hessian evaluations, we consider the simple case of Hartree-Fock (HF) calculations. In nearly all chemical applications of HF theory, the molecular orbitals (MOs) are represented by a linear combination of atomic orbitals (LCAO). In the context of most electronic structure methods, the LCAO approximation employs a more convenient set of basis functions such as contracted Gaussians, rather than using actual atomic orbitals. Taken together, the collection of basis functions used to represent the atomic orbitals comprises a basis set. 

This family of methods does not solve problems relating to Hessian evaluation and linear system solution (problems no. 9 and 10 of Section 3.5). 

Compare the Hessian evaluate in X to the Hessian updated using the relation (13.18). 

Chapter 4 has been devoted to large-scale unconstrained optimization problems, where problems related to the management of matrix sparsity and the ordering of rows and columns are broached. Hessian evaluation, Newton and inexact Newton methods are discussed. 

This equation is similar to Eq. (44), although the additive term/(7 ) has been omitted because it is a function of temperature only. t/Min represents the local minimum energy of or L, and Det(//Mi ) is the determinant of the Hessian evaluated at this local minimum. The specification of a thermodynamic temperature (p = l/fc T) is required as an additional input parameter. 

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