A globalization strategy for multigrid schemes solving optimal control problems is presented. This approach searches for possible negative eigenvalues of the reduced Hessian considered at the coarsest grid of the multigrid process. If negative eigenvalues are detected, a globalization step in the direction of negative curvature is performed to escape undesired maxima or saddle points. It is sho...

In this paper we present a Newton method to jointly approximately diagonalize a set of positive definite Hermitian matrices. To this end, we derive the local gradient and Hessian of the underlying cost function in closed form. The algorithm is derived for the complex case and can also update a non-square diagonalization matrix. We analyze the cost function at the critical points and show its re...

It has been known since the early 1970s that the Hessian matrices in quasi– Newton methods can be updated by variational means, in several different ways. The usual formulation of these variational problems uses a coordinate system, and the symmetry of the Hessian matrices are enforced as explicit constraints. As a result, the variational problems seem complicated. In this paper, we give a very...

We consider a combined IPM-SQP method to solve smooth nonlinear optimization problems, which may possess a large number of variables and a sparse Jacobian matrix of the constraints. Basically, the algorithm is a sequential quadratic programming (SQP) method, where the quadratic programming subproblem is solved by a primal-dual interior point method (IPM). A special feature of the algorithm is t...

The reaction path is a key concept in the theoretical description of a chemical reaction. The intrinsic reaction coordinate is defined as the steepest descent path in mass-weighted Cartesian coordinates that connects the transition state to reactants and products on the potential energy surface. Recently, a new Hessian based predictor-corrector reaction path following algorithm was presented th...

Quasi-Newton algorithms for unconstrained nonlinear minimization generate a sequence of matrices that can be considered as approximations of the objective function second derivatives. This paper gives conditions under which these approximations can be proved to converge globally to the true Hessian matrix, in the case where the Symmetric Rank One update formula is used. The rate of convergence ...

In tackling the problem of minimizing the deformation of a loaded structure, by varying the shape of the original structure, past second order optimization efforts have focused on general Newton techniques like the Davidon-Fletcher-Powell (DFP) update formula and the BroydenFletcher-Goldfarb-Shanno (BFGS) formula which iteratively build estimates of the structure's Hessian. This thesis bypasses...

The BFGS and DFP updates are perhaps the most successful Hessian and inverse Hessian approximations respectively for unconstrained minimization problems. This paper describes these methods in terms of two successive steps: rank reduction and rank restoration. From rank subtractivity and a powerful spectral result, the first step must necessarily result in a positive semidefinite matrix; and the...

We study the spectra of the molecular orbital Hessian (stability matrix) and random-phase approximation (RPA) Hamiltonian of broken-symmetry Hartree-Fock solutions, focusing on zero eigenvalue modes. After all negative eigenvalues are removed from the Hessian by following their eigenvectors downhill, one is left with only positive and zero eigenvalues. Zero modes correspond to orbital rotations...

The paper resolves the problem concerning the rate of convergence of the working set based MPRGP (modified proportioning with reduced gradient projection) algorithm with a long steplength of the reduced projected gradient step. The main results of this paper are the formula for the R-linear rate of convergence of MPRGP in terms of the spectral condition number of the Hessian matrix and the proo...