Global Convergence of Damped Newton's Method for Nonsmooth Equations, via the Path Search

A natural damping of Newton's method for nonsmooth equations is presented. This damping, via the path search instead of the traditional line search, enlarges the domain of convergence of Newton's method and therefore is said to be globally convergent. Convergence behavior is like that of line search damped Newton's method for smooth equations, including Q-quadratic convergence rates under appropriate conditions. Applications of the path search include damping Robinson-Newton's method for nonsmooth normal equations corresponding to nonlinear complementarity problems and variational inequalities, hence damping both Wilson's method (sequential quadratic programming) for nonlinear programming and Josephy-Newton's method for generalized equations. Computational examples from nonlinear programming are given

Deflation for semismooth equations

Variational inequalities can in general support distinct solutions. In this paper we study an algorithm for computing distinct solutions of a variational inequality, without varying the initial guess supplied to the solver. The central idea is the combination of a semismooth Newton method with a deflation operator that eliminates known solutions from consideration. Given one root of a semismooth residual, deflation constructs a new problem for which a semismooth Newton method will not converge to the known root, even from the same initial guess. This enables the discovery of other roots. We prove the effectiveness of the deflation technique under the same assumptions that guarantee locally superlinear convergence of a semismooth Newton method. We demonstrate its utility on various finite- and infinite-dimensional examples drawn from constrained optimization, game theory, economics and solid mechanics.Comment: 24 pages, 3 figure

Sensor network deployment as least squares problems.

Xu, Yang.Thesis (M.Phil.)--Chinese University of Hong Kong, 2011.Includes bibliographical references (leaves 99-104).Abstracts in English and Chinese.Chapter 1 --- Introduction --- p.1Chapter 1.1 --- Background of Sensors and Sensor Networks --- p.2Chapter 1.2 --- Introduction to Coverage Problems --- p.6Chapter 1.3 --- Literature Review --- p.8Chapter 1.3.1 --- Deterministic Deployment Methods --- p.9Chapter 1.3.2 --- Dynamic Deployment Methods --- p.10Chapter 1.4 --- A Brief Introduction to Least Squares Analysis --- p.13Chapter 1.5 --- Thesis Outline --- p.15Chapter 2 --- Mobile Sensor Network Deployment Problem --- p.18Chapter 2.1 --- Sensor Coverage Models --- p.18Chapter 2.1.1 --- Binary Sensor Models --- p.19Chapter 2.1.2 --- Attenuated and Truncated Attenuated Disk Models --- p.20Chapter 2.2 --- Problem Statement --- p.23Chapter 3 --- Coverage Optimization as Nonlinear Least Squares Problems --- p.26Chapter 3.1 --- Introduction --- p.26Chapter 3.2 --- Network Deployment as Least Squares Problems --- p.28Chapter 3.2.1 --- Assignment of Sample Points --- p.28Chapter 3.2.2 --- Least Squares Function --- p.30Chapter 3.2.3 --- Gauss-Newton Method --- p.33Chapter 3.2.4 --- Solutions --- p.36Chapter 3.3 --- Extension to Binary Sensor Models --- p.39Chapter 3.3.1 --- Restrictions of Subgradient Methods --- p.40Chapter 3.3.2 --- Sigmoid Functions --- p.42Chapter 3.4 --- Convergence and Multiple Minima Issues --- p.44Chapter 3.4.1 --- Convergence --- p.44Chapter 3.4.2 --- Multiple Minima --- p.48Chapter 3.5 --- Stopping Criteria --- p.52Chapter 3.6 --- Summary --- p.53Chapter 4 --- Experimental Results --- p.55Chapter 4.1 --- Introduction --- p.55Chapter 4.2 --- Numerical Examples --- p.56Chapter 4.2.1 --- Examples of Attenuated Disk Models --- p.57Chapter 4.2.2 --- Examples of Binary Sensor Models --- p.63Chapter 4.3 --- Performance Metrics of Mobile Sensor Deployment Schemes --- p.68Chapter 4.4 --- Comparison to Existing Methods --- p.74Chapter 4.5 --- Summary --- p.81Chapter 5 --- Conclusions --- p.83Chapter 5.1 --- Conclusions --- p.83Chapter 5.2 --- Future Research Directions --- p.85Appendices --- p.87Chapter A --- An Overview of Existing Deployment Methods --- p.88Chapter A.1 --- Potential Fields and Virtual Forces --- p.88Chapter A.2 --- Distributed Self-Spreading Algorithm --- p.92Chapter A.3 --- VD-Based Deployment Algorithm --- p.96Bibliography --- p.9

Implementation of a continuation method for nonlinear complementarity problems via normal maps

Ankara : Department of Industrial Engineering and Institute of Engineering and Sciences, Bilkent Univ., 1997.Thesis (Master's) -- Bilkent University, 1997.Includes bibliographical references.In this thesis, a continuation method for nonlinear complementarity problems via normal maps that is developed by Chen, Harker and Pinar [8] is implemented. This continuation method uses the smooth function to approximate the normal map reformulation of nonlinear complementarity problems. The algorithm is implemented and tested with two different plussmoothing functions namely interior point plus-smooth function and piecewise quadratic plus-smoothing function. These two functions are compared. Testing of the algorithm is made with several known problems.Erkan, AliM.S