Newton's Method for Solving Inclusions Using Set-Valued Approximations

International audienceResults on stability of both local and global metric regularity under set-valued perturbations are presented. As an application, we study (super)linear convergence of a Newton- type iterative process for solving generalized equations. We investigate several iterative schemes such as the inexact Newton’s method, the nonsmooth Newton’s method for semismooth functions, the inexact proximal point algorithm, etc. Moreover, we also cover a forward-backward splitting algorithm for finding a zero of the sum of two multivalued (not necessarily monotone) operators. Finally, a globalization of the Newton’s method is discussed

On a semismooth* Newton method for solving generalized equations

In the paper, a Newton-type method for the solution of generalized equations (GEs) is derived, where the linearization concerns both the single-valued and the multivalued part of the considered GE. The method is based on the new notion of semismoothness\ast, which, together with a suitable regularity condition, ensures the local superlinear convergence. An implementable version of the new method is derived for a class of GEs, frequently arising in optimization and equilibrium models. © 2021 Society for Industrial and Applied Mathematic

On Matrix Nearness Problems: Distance to Delocalization

This paper introduces two new matrix nearness problems that are intended to generalize the distance to instability and the distance to stability. They are named the distance to delocalization and the distance to localization due to their applicability in analyzing the robustness of eigenvalues with respect to arbitrary localization sets (domains) in the complex plane. For the open left-half plane or the unit circle, the distance to the nearest unstable/stable matrix is obtained as a special case. Then, following the theoretical framework of Hermitian functions and the Lyapunov-type localization approach, we present a new Newton-type algorithm for the distance to delocalization (D2D) and study its implementations using both an explicit and an implicit computation of the desired singular values. Since our investigations are motivated by several practical applications, we will illustrate our approach on some of them. Furthermore, in the special case when the distance to delocalization becomes the distance to instability, we will validate our algorithms against the state of the art computational method

The theory and some applications of Pták's method of non-discrete mathematical induction

Bibliography: p. 79-80.The aim of this thesis is three-fold: (1) to develop the theory of small functions; (2) to synthesize Pták's work presented in his papers [10], [11], ..., [16] into a coherent body of knowledge; (3) to elaborate on Pták's work (i) by providing small function generalizations of Banach's Fixed Point Theorem and Edelstein's Extended Contraction Principle; (ii) by connecting the Induction Theorem to Baire's Category Theorem and Cantor's Intersection Theorem. Throughout the exposition the editorial "we" is to be understood in the sense of Halmos [ 18]; "we" means "the author and the reader"

The construction of Chebyshev approximations in the complex plane

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Model parameter estimation of atherosclerotic plaque mechanical properties : calculus-based and heuristic algorithms

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Mechanical Engineering, 2004.Includes bibliographical references (p. 123-133).A sufficient understanding of the pathology that leads to cardiovascular disease is currently deficient. Atherosclerosis is a complex disease that is believed to be initiated and promoted by linked biochemical and biomechanical pathways. This thesis focuses on studying plaque biomechanics because (i) there is a dearth of data on the mechanical behavior of soft arterial tissue yet (ii) it is the biomechanics that is able to provide invaluable insight into patient-specific disease evolution and plaque vulnerability. Arterial elasticity reconstruction is a venture that combines imaging, elastography, and computational modeling in an effort to build maps of an artery's material properties, ultimately to identify plaques exhibiting stress concentrations and to pinpoint rupture-prone locales. The inverse elasticity problem was explored extensively and two solution methods are demonstrated. The first is a version of the traditional linear perturbation Gauss-Newton method, which contingent on an appropriate regularization scheme, was able to reconstruct both homogeneous and inhomogeneous distributions including hard and spatially continuous inclusions. The second was an attempt to tackle the inherent and problem-specific limitations associated with such gradient-based searches. With a model reduction of the discrete elasticity parameters into lumped values, such as the plaque components, more robust and adaptive strategies become feasible. A novel combined finite element modeling-genetic algorithm system was implemented that is easily implemented, manages multiple regions of far-reaching modulus, is globally convergent, shows immunity to ill-conditioning, and is expandable to more complex material models(cont.) and geometries. The implementation of both provides flexibility in the endeavor of arterial elasticity reconstruction as well as potential complementary and joint efforts.by Ahmad S. Khalil.S.M

Nondifferentiable Optimization: Motivations and Applications

IIASA has been involved in research on nondifferentiable optimization since 1976. The Institute's research in this field has been very productive, leading to many important theoretical, algorithmic and applied results. Nondifferentiable optimization has now become a recognized and rapidly developing branch of mathematical programming. To continue this tradition and to review developments in this field IIASA held this Workshop in Sopron (Hungary) in September 1984. This volume contains selected papers presented at the Workshop. It is divided into four sections dealing with the following topics: (I) Concepts in Nonsmooth Analysis; (II) Multicriteria Optimization and Control Theory; (III) Algorithms and Optimization Methods; (IV) Stochastic Programming and Applications