As far as more complex systems are being accessible for quantum chemical calculations, the reliability of the algorithms used becomes increasingly important. Trust-region strategies comprise a large family of optimization algorithms that incorporates both robustness and applicability for a great variety of problems. The objective of this work is to provide a basic algorithm and an adequate theoretical framework for the application of globally convergent trust-region methods to electronic structure calculations. Closed shell restricted Hartree-Fock calculations are addressed as finite-dimensional nonlinear programming problems with weighted orthogonality constraints. A Levenberg-Marquardt-like modification of a trust-region algorithm for constrained optimization is developed for solving this problem. It is proved that this algorithm is globally convergent. The subproblems that ensure global convergence are easy-to-compute projections and are dependent only on the structure of the constraints, thus being extendable to other problems. Numerical experiments are presented, which confirm the theoretical predictions. The structure of the algorithm is such that accelerations can be easily associated without affecting the convergence properties.

We re-analyze a Tikhonov parameter choice rule devised by Regińska (1996 SIAM J. Sci. Comput. 3 740-49) and algorithmically realized through a fast fixed-point (FP) method by Bazán (2008 Inverse Problems 24 035001). The method determines a Tikhonov parameter associated with a point near the L-corner of the maximum curvature and at which the L-curve is locally convex. In practice, it works well when the L-curve presents an L-shaped form with distinctive vertical and horizontal parts, but failures may occur when there are several local convex corners. We derive a simple and computable condition which describes the regions where the L-curve is concave/convex, while providing insight into the choice of the regularization parameter through the L-curve method or FP. Based on this, we introduce variants of the FP algorithm capable of handling the parameter choice problem even in the case where the L-curve has several local corners. The theory is illustrated both graphically and numerically, and the performance of the variants on a difficult ill-posed problem is evaluated by comparing the results with those provided by the L-curve method, generalized cross-validation and the discrepancy principle.

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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.