In this paper, we import interval method to the iteration for computing Moore-Penrose inverse of the full row (or column) rank matrix. Through modifying the classical Newton iteration by interval method, we can get better numerical results. The convergence of the interval iteration is proven. We also give some numerical examples to compare interval iteration with classical Newton iteration.

We analyze controllability properties of the inverse iteration and the QR-algorithm equipped with a shifting parameter as a control input. In the case of the inverse iteration with real shifts the theory of universally regular controls may be used to obtain necessary and suucient conditions for complete controllability in terms of the solvability of a matrix equation. Partial results on conditi...

The present paper applies the recently proposed Augmented Lagrangian Alternating Direction Inexact Newton (ALADIN) method to solve non-convex AC Optimal Power Flow Problems (OPF) in a distributed fashion. In contrast to the often used Alternaring Direction of Multipliers Method (ADMM), ALADIN guarantees locally quadratic convergence for AC-OPF. Numerical results for IEEE 5–300 bus test cases in...

Convex quadratic semidefinite programming (QSDP) has been widely applied in solving engineering and scientific problems such as nearest correlation problems and nearest Euclidean distance matrix problems. In this paper, we study an inexact primal-dual infeasible path-following algorithm for QSDP problems of the form: minX{12X • Q(X) + C •X : A(X) = b, X 0}, where Q is a self-adjoint positive se...

In this work, we propose an inexact projected gradient-like method for solving smooth constrained vector optimization problems. In the unconstrained case, we retrieve the steepest descent method introduced by Graña Drummond and Svaiter. In the constrained setting, the method we present extends the exact one proposed by Graña Drummond and Iusem, since it admits relative errors on the search dire...

We consider primal-dual IP methods where the linear system arising at each iteration is formulated in the reduced (augmented) form and solved approximately. Focusing on the iterates close to a solution, we analyze the accuracy of the so-called inexact step, i.e., the step that solves the unreduced system, when combining the effects of both different levels of accuracy in the inexact computation...

We propose a class of regularized Hermitian and skew-Hermitian splitting methods for the solution of large, sparse linear systems in saddle-point form. These methods can be used as stationary iterative solvers or as preconditioners for Krylov subspace methods. We establish unconditional convergence of the stationary iterations and we examine the spectral properties of the corresponding precondi...

In this paper, we analyze the iteration-complexity of Generalized Forward–Backward (GFB) splitting algorithm, as proposed in [2], for minimizing a large class of composite objectives f` řn i“1 hi on a Hilbert space, where f has a Lipschitzcontinuous gradient and the hi’s are simple (i.e. their proximity operators are easy to compute). We derive iterationcomplexity bounds (pointwise and ergodic)...

We consider distributed convex optimization problems originated from sample average approximation of stochastic optimization, or empirical risk minimization in machine learning. We assume that each machine in the distributed computing system has access to a local empirical loss function, constructed with i.i.d. data sampled from a common distribution. We propose a communication-efficient distri...

We adapt the inverse iteration method for symmetric matrices to some nonlinear PDE eigenvalue problems. In particular, for p ∈ (1,∞) and a given domain Ω ⊂ Rn, we analyze a scheme that allows us to approximate the smallest value the ratio ∫ Ω |Dψ| pdx/ ∫ Ω |ψ| pdx can assume for functions ψ that vanish on ∂Ω. The scheme in question also provides a natural way to approximate minimizing ψ. Our an...