Generalized Newton Algorithms for Tilt-Stable Minimizers in Nonsmooth Optimization

We propose a generalized inexact Newton method for solving the inverse eigenvalue problems, which includes the generalized Newton method as a special case. Under the nonsingularity assumption of the Jacobian matrices at the solutionc*, a convergence analysis covering both the distinct and multiple eigenvalue cases is provided and the quadratic convergence property is proved. Moreover, numerical tests are given in the last section and comparisons with the generalized Newton method are made.

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A Novel Approach for Solving Nonsmooth Optimization Problems with Application to Nonsmooth Equations

Journal of Mathematics ◽

10.1155/2013/750834 ◽

2013 ◽

Vol 2013 ◽

pp. 1-10

Author(s):

Hamid Reza Erfanian ◽

M. H. Noori Skandari ◽

A. V. Kamyad

Keyword(s):

Nonsmooth Optimization ◽

Optimization Problems ◽

Piecewise Linear ◽

Linear Optimization ◽

Optimal Solution ◽

Piecewise Linear Function ◽

Nonsmooth Equations ◽

Smooth Functions ◽

Generalized Derivative ◽

Novel Approach

We present a new approach for solving nonsmooth optimization problems and a system of nonsmooth equations which is based on generalized derivative. For this purpose, we introduce the first order of generalized Taylor expansion of nonsmooth functions and replace it with smooth functions. In other words, nonsmooth function is approximated by a piecewise linear function based on generalized derivative. In the next step, we solve smooth linear optimization problem whose optimal solution is an approximate solution of main problem. Then, we apply the results for solving system of nonsmooth equations. Finally, for efficiency of our approach some numerical examples have been presented.

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Elastohydrodynamic Squeeze Films: Effects of Viscoelasticity and Fluctuating Load

Journal of Lubrication Technology ◽

10.1115/1.3453280 ◽

1979 ◽

Vol 101 (1) ◽

pp. 74-80 ◽

Cited By ~ 12

Author(s):

S. M. Rohde ◽

D. Whicker ◽

J. F. Booker

Keyword(s):

Surface Deformation ◽

Dynamic Responses ◽

Film Rupture ◽

Fully Implicit ◽

Oscillatory Response ◽

Generalized Newton ◽

Newton Scheme ◽

Bounding Surfaces ◽

Implicit Numerical Method ◽

Practical Implications

The dynamic responses of squeeze films to fluctuating loads are compared for the cases of rigid, elastic, and viscoelastic bounding surfaces. Unexpected and interesting differences are observed. These include the oscillatory response of indenter position to a non-negative fluctuating load and the decaying load versus time response to a prescribed indenter trajectory which remains constant over a period of time. Practical implications of these responses are noted. The transient analysis procedure, while general, is applied to a simple geometrical and physical model: an isoviscous Newtonian fluid squeezed between circular, initially flat, three-element viscoelastic Winkler solids. Surface deformation rate and film rupture are incorporated. The fully implicit numerical method utilizes a generalized Newton scheme together with finite element spatial discretization.

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Generalized Newton–Raphson algorithm for high dimensional LASSO regression

Statistics and Its Interface ◽

10.4310/20-sii643 ◽

2021 ◽

Vol 14 (3) ◽

pp. 339-350

Author(s):

Yueyong Shi ◽

Jian Huang ◽

Yuling Jiao ◽

Yicheng Kang ◽

Hu Zhang

Keyword(s):

High Dimensional ◽

Lasso Regression ◽

Newton Raphson ◽

Generalized Newton ◽

Raphson Algorithm

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Nonsmooth Optimization Method for Image Restoration

Volume 3C: 15th Biennial Conference on Mechanical Vibration and Noise — Vibration Control, Analysis, and Identification ◽

10.1115/detc1995-0673 ◽

1995 ◽

Author(s):

Kazufumi Ito ◽

Karl Kunisch

Keyword(s):

Image Restoration ◽

Nonsmooth Optimization ◽

Numerical Optimization ◽

Augmented Lagrangian ◽

Energy Criterion ◽

Optimization Methods ◽

Optimization Method ◽

Augmented Lagrangian Methods ◽

Active Set ◽

First Order

Abstract In this paper we discuss applications of the numerical optimization methods for nonsmooth optimization, developed in [IK1] for the variational formulation of image restoration problems involving bounded variation type energy criterion. The Uzawa’s algorithm, first order augmented Lagrangian methods and Newton-like update using the active set strategy are described.

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