{'fraction_non_alphanumeric': 0.027210884353741496, 'fraction_numerical': 0.01020408163265306, 'mean_word_length': 5.035897435897436, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Maintaining factorized KKT systems subject to rank-one updates of Hessians and Jacobians

For use in a total quasi-Newton NLP code [Griewank, A. and Walther, A., 2002, On constrained optimization by adjoint based quasi-Newton methods. Optimization Methods and Software, 17, 869–889.], we describe a QR-based nullspace factorization of KKT matrices. We illustrate the linear algebra in detail and present a theory for maintaining factorized matrices after low-rank updates. Each update of the whole system is incorporated with a computational effort that grows only quadratically with respect to the number of variables and active constraints. Furthermore, our method is special in making use of quasi-Newton updates for the constraint Jacobian approximation, instead of the usual way of using the exact derivative or divided differences. To avoid singularity or blow-up of the KKT matrix, we limit the variations of its determinant to a certain factor and dampen or augment the updates if necessary. We maintain the reduced Hessian positive definite, so that the resulting quasi-Newton steps in the primal and dual variables are downhill for suitably weighted merit functions.