A quasi-Newton proximal splitting method

A new result in convex analysis on the calculation of proximity operators in certain scaled norms is derived. We describe efficient implementations of the proximity calculation for a useful class of functions; the implementations exploit the piece-wise linear nature of the dual problem. The second part of the paper applies the previous result to acceleration of convex minimization problems, and leads to an elegant quasi-Newton method. The optimization method compares favorably against state-of-the-art alternatives. The algorithm has extensive applications including signal processing, sparse recovery and machine learning and classification

A Unified Convergence Analysis for Some Two-Point Type Methods for Nonsmooth Operators

The aim of this paper is the approximation of nonlinear equations using iterative methods. We present a unified convergence analysis for some two-point type methods. This way we compare specializations of our method using not necessarily the same convergence criteria. We consider both semilocal and local analysis. In the first one, the hypotheses are imposed on the initial guess and in the second on the solution. The results can be applied for smooth and nonsmooth operators.Research of the first and third authors supported in part by Programa de Apoyo a la investigación de la fundación Séneca-Agencia de Ciencia y Tecnología de la Región de Murcia 20928/PI/18 and by MTM2015-64382-P. Research of the fourth and fifth authors supported by Ministerio de Economía y Competitividad under grant MTM2014-52016-C2-1P. This research received no external funding

On Quasi-Newton Forward--Backward Splitting: Proximal Calculus and Convergence

We introduce a framework for quasi-Newton forward--backward splitting algorithms (proximal quasi-Newton methods) with a metric induced by diagonal $\pm$ rank-$r$ symmetric positive definite matrices. This special type of metric allows for a highly efficient evaluation of the proximal mapping. The key to this efficiency is a general proximal calculus in the new metric. By using duality, formulas are derived that relate the proximal mapping in a rank-$r$ modified metric to the original metric. We also describe efficient implementations of the proximity calculation for a large class of functions; the implementations exploit the piece-wise linear nature of the dual problem. Then, we apply these results to acceleration of composite convex minimization problems, which leads to elegant quasi-Newton methods for which we prove convergence. The algorithm is tested on several numerical examples and compared to a comprehensive list of alternatives in the literature. Our quasi-Newton splitting algorithm with the prescribed metric compares favorably against state-of-the-art. The algorithm has extensive applications including signal processing, sparse recovery, machine learning and classification to name a few.Comment: arXiv admin note: text overlap with arXiv:1206.115

Stabilization of Unstable Procedures: The Recursive Projection Method

Fixed-point iterative procedures for solving nonlinear parameter dependent problems can converge for some interval of parameter values and diverge as the parameter changes. The Recursive Projection Method (RPM), which stabilizes such procedures by computing a projection onto the unstable subspace is presented. On this subspace a Newton or special Newton iteration is performed, and the fixed-point iteration is used on the complement. As continuation in the parameter proceeds, the projection is efficiently updated, possibly increasing or decreasing the dimension of the unstable subspace. The method is extremely effective when the dimension of the unstable subspace is small compared to the dimension of the system. Convergence proofs are given and pseudo-arclength continuation on the unstable subspace is introduced to allow continuation past folds. Examples are presented for an important application of the RPM in which a “black-box” time integration scheme is stabilized, enabling it to compute unstable steady states. The RPM can also be used to accelerate iterative procedures when slow convergence is due to a few slowly decaying modes

Simulation of Piecewise Smooth Differential Algebraic Equations with Application to Gas Networks

Zuweilen wird gefördertes Erdgas als eine Brückentechnologie noch eine Weile erhalten bleiben, aber unsere Gasnetzinfrastruktur hat auch in einer Ära post-fossiler Brennstoffe eine Zukunft, um Klima-neutral erzeugtes Methan, Ammoniak oder Wasserstoff zu transportieren. Damit die Dispatcher der Zukunft, in einer sich fortwährend dynamisierenden Marktsituation, mit sich beständig wechselnden Kleinstanbietern, auch weiterhin einen sicheren Gasnetzbetrieb ermöglichen und garantieren können, werden sie auf moderne, schnelle Simulations- sowie performante Optimierungstechnologie angewiesen sein. Der Schlüssel dazu liegt in einem besseren Verständnis zur numerischen Behandlung nicht differenzierbarer Funktionen und diese Arbeit möchte einen Beitrag hierzu leisten. Wir werden stückweise differenzierbare Funktionen in sog. Abs-Normalen Form betrachten. Durch einen Prozess, der Abs-Linearisierung genannt wird, können wir stückweise lineare Approximationsmodelle erster Ordnung, mittels Techniken der algorithmischen Differentiation erzeugen. Jene Modelle können über Matrizen und Vektoren mittels gängiger Software-Bibliotheken der numerischen linearen Algebra auf Computersystemen ausgedrückt, gespeichert und behandelt werden. Über die Generalisierung der Formel von Faà di Bruno können auch Splinefunktionen höherer Ordnung generiert werden, was wiederum zu Annäherungsmodellen mit besserer Güte führt. Darauf aufbauend lassen sich gemischte Taylor-Kollokationsmethoden, darunter die mit Ordnung zwei konvergente generalisierte Trapezmethode, zur Integration von Gasnetzen, in Form von nicht glatten Algebro-Differentialgleichungssystemen, definieren. Numerische Experimente demonstrieren das Potential. Da solche implizite Integratoren auch nicht lineare und in unserem Falle zugleich auch stückweise differenzierbare Gleichungssysteme erzeugen, die es als Unterproblem zu lösen gilt, werden wir uns auch die stückweise differenzierbare, sowie die stückweise lineare Newtonmethode betrachten.As of yet natural gas will remain as a bridging technology, but our gas grid infrastructure does have a future in a post-fossil fuel era for the transportation of carbon-free produced methane, ammonia or hydrogen. In order for future dispatchers to continue to enable and guarantee safe gas network operations in a continuously changing market situation with constantly switching micro-suppliers, they will be dependent on modern, fast simulation as well as high-performant optimization technology. The key to such a technology resides in a better understanding of the numerical treatment of non-differentiable functions and this work aims to contribute here. We will consider piecewise differentiable functions in so-called abs-normal form. Through a process called abs-linearization, we can generate piecewise linear approximation models of order one, using techniques of algorithmic differentiation. Those models can be expressed, stored and treated numerically as matrices and vectors via common software libraries of numerical linear algebra. Generalizing the Faà di Bruno's formula yields higher order spline functions, which in turn leads to even higher order approximation models. Based on this, mixed Taylor-Collocation methods, including the generalized trapezoidal method converging with an order of two, can be defined for the integration of gas networks represented in terms of non-smooth system of differential algebraic equations. Numerical experiments will demonstrate the potential. Since those implicit integrators do generate non-linear and, in our case, piecewise differentiable systems of equations as sub-problems, it will be necessary to consider the piecewise differentiable, as well as the piecewise linear Newton method in advance

New sufficient convergence conditions for the secant method

summary:We provide new sufficient conditions for the convergence of the secant method to a locally unique solution of a nonlinear equation in a Banach space. Our new idea uses “Lipschitz-type” and center-“Lipschitz-type” instead of just “Lipschitz-type” conditions on the divided difference of the operator involved. It turns out that this way our error bounds are more precise than the earlier ones and under our convergence hypotheses we can cover cases where the earlier conditions are violated

Semilocal Convergence of the Extension of Chun's Method

[EN] In this work, we use the technique of recurrence relations to prove the semilocal convergence in Banach spaces of the multidimensional extension of Chun's iterative method. This is an iterative method of fourth order, that can be transferred to the multivariable case by using the divided difference operator. We obtain the domain of existence and uniqueness by taking a suitable starting point and imposing a Lipschitz condition to the first Frechet derivative in the whole domain. Moreover, we apply the theoretical results obtained to a nonlinear integral equation of Hammerstein type, showing the applicability of our results.This research was supported by PGC2018-095896-B-C22 (MCIU/AEI/FEDER, UE) and FONDOCYT 027-2018 Republica Dominicana.Cordero Barbero, A.; Maimó, JG.; Martínez Molada, E.; Torregrosa Sánchez, JR.; Vassileva, MP. (2021). Semilocal Convergence of the Extension of Chun's Method. Axioms. 10(3):1-11. https://doi.org/10.3390/axioms10030161S11110