We introduce a partial proximal point algorithm for solving nuclear norm regularized and semidefinite matrix least squares problems with linear equality constraints. For the inner subproblems, we show that the positive definiteness of the generalized Hessian of the objective function for the inner subproblems is equivalent to the constraint nondegeneracy of the corresponding primal problem, whi...

We introduce a novel method to compute a rank m approximation of the inverse of the Hessian matrix in the distributed regime. By leveraging the differences in gradients and parameters of multiple Workers, we are able to efficiently implement a distributed approximation of the Newton-Raphson method. We also present preliminary results which underline advantages and challenges of secondorder meth...

We investigate two simple sufficient criteria for positive invariance of sets in the domain of n-dimensional nonlinear autonomous discrete time systems. These criteria are derived from the exact Taylor expansion with linear and quadratic remainder terms. By a simple example we demonstrate that systems exist for which positive invariance can be established with the second order criterion but not...

We introduce a novel method to compute a rank m approximation of the inverse of the Hessian matrix in the distributed regime. By leveraging the differences in gradients and parameters of multiple Workers, we are able to efficiently implement a distributed approximation of the Newton-Raphson method. We also present preliminary results which underline advantages and challenges of secondorder meth...

Large scale optimization problems are ubiquitous in machine learning and data analysis and there is a plethora of algorithms for solving such problems. Many of these algorithms employ sub-sampling, as a way to either speed up the computations and/or to implicitly implement a form of statistical regularization. In this paper, we consider second-order iterative optimization algorithms, i.e., thos...

This article proposes a new, more efficient method to compute the minus two log likelihood, its gradient, and the Hessian for structural equation models (SEMs) in reticular action model (RAM) notation. The method exploits the beneficial aspect of RAM notation that the matrix derivatives used in RAM are sparse. For an SEM with K variables, P parameters, and P' entries in the symmetrical or asymm...

This paper presents a fast and powerful method for the computation of eigenvalue bounds for Hessian matrices ∇2φ(x) of nonlinear wice continuously differentiable functions φ : U ⊆ R → R on hyperrectangles B ⊂ U . The method is based on a recently proposed procedure [9] for an efficient computation of spectral bounds using extended codelists. Both that approach and the one presented here substan...