Inexact restoration method for minimization problems arising in electronic structure calculations

Abstract

An inexact restoration (IR) approach is presented to solve a matricial optimization problem arising in electronic structure calculations. The solution of the problem is the closed-shell density matrix and the constraints are represented by a Grassmann manifold. One of the mathematical and computational challenges in this area is to develop methods for solving the problem not using eigenvalue calculations and having the possibility of preserving sparsity of iterates and gradients. The inexact restoration approach enjoys local quadratic convergence and global convergence to stationary points and does not use spectral matrix decompositions, so that, in principle, large-scale implementations may preserve sparsity. Numerical experiments show that IR algorithms are competitive with current algorithms for solving closed-shell Hartree-Fock equations and similar mathematical problems, thus being a promising alternative for problems where eigenvalue calculations are a limiting factor.

Sequential equality-constrained optimization for nonlinear programming

“…In Algorithm 3.1 (Step 4.4) the acceptance of the trial step (see (17) and (18)) is conditioned to the decrease of two different merit functions: the Lagrangian L(·, λ k ) and the Augmented Lagrangian L se…”

“…Conditions (17) and (18) in Algorithm 3.1 are the heuristic "practical" versions of (21) and (22), respectively. Algorithm 3.1 may be considered as an heuristic Newton-based procedure for solving the equality-constrained minimization problem (8).…”

“…Each iteration of the FIR algorithm, as other Inexact-Restoration methods, has two phases: Feasibility and Optimality. In the Feasibility Phase the algorithm aims to improve feasibility and in the Optimality Phase the algorithm minimizes approximately a Lagrangian approximation subject to the linearization of the constraints [7,15,17,19,21,22,23,24,26,27]. As a consequence, a trial point is obtained which is accepted as new iterate or not according to the value of a Sharp Lagrangian merit function [15,26].…”

Assessing the reliability of general-purpose Inexact Restoration methods

“…See [3,7,8,14,15,13,20,21,22,23,28,29]. The idea of IR methods is that, at each iteration, feasibility and optimality are addressed in different phases.…”

“…In the Restoration Phase the algorithms aim to improve feasibility and in the Optimization Phase they aim to improve optimality, preserving a linear approximation of feasibility. These algorithms have been successfully used in applications in which there exist a natural way to improve (or even obtain) feasibility (see [3,13,22,23] among others).…”