Journal ArticleDOI
Quadratic Convergence of Newton's Method for Convex Interpolation and Smoothing
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This article showed that Newton's method for convex best interpolation is locally quadratically convergent, giving an answer to a question of Irvine, Marin, and Smith [7] and strengthening a result of Andersson and Elfving [1] and our previous work.Abstract:
. In this paper, we prove that Newton's method for convex best interpolation is locally quadratically convergent, giving an answer to a question of Irvine, Marin, and Smith [7] and strengthening a result of Andersson and Elfving [1] and our previous work [5]. A damped Newton-type method is presented which has global quadratic convergence. Analogous results are obtained for the convex smoothing problem. Numerical examples are presented.read more
Citations
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Journal ArticleDOI
A Quadratically Convergent Newton Method for Computing the Nearest Correlation Matrix
Houduo Qi,Defeng Sun +1 more
TL;DR: The quadratic convergence of the proposed Newton method for the nearest correlation matrix problem is proved, which confirms the fast convergence and the high efficiency of the method.
Journal ArticleDOI
Semismoothness of solutions to generalized equations and the Moreau-Yosida regularization
TL;DR: It is proved that the semismoothness of solutions to the Moreau-Yosida regularization of a lower semicontinuous proper convex function is implied by the semistic of the metric projector over the epigraph of the convexfunction.
Journal ArticleDOI
Calibrating Least Squares Semidefinite Programming with Equality and Inequality Constraints
Yan Gao,Defeng Sun +1 more
TL;DR: This paper reformulates the least squares semidefinite programming with a large number of equality and inequality constraints as a system of semismooth equations with two level metric projection operators and designs an inexact smoothing Newton method to solve the resultingSemismooth system.
Journal ArticleDOI
Differentiability and semismoothness properties of integral functions and their applications
TL;DR: This paper study differentiability and semismoothness properties of functions defined as integrals of parameterized functions and applications of the developed theory to the problems of shape-preserving interpolation, option pricing and semi-infinite programming.
Journal ArticleDOI
On almost smooth functions and piecewise smooth functions
Liquin Qi,Paul Tseng +1 more
TL;DR: In this paper, it was shown that the B-subdifferential of an almost smooth function at a point has either one or infinitely many elements, which contrasts with piecewise smooth functions whose B-differential has only a finite number of elements.
References
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Book
A practical guide to splines
TL;DR: This book presents those parts of the theory which are especially useful in calculations and stresses the representation of splines as linear combinations of B-splines as well as specific approximation methods, interpolation, smoothing and least-squares approximation, the solution of an ordinary differential equation by collocation, curve fitting, and surface fitting.
Journal ArticleDOI
A nonsmooth version of Newton's method
TL;DR: It is shown that the gradient function of the augmented Lagrangian forC2-nonlinear programming is semismooth, and the extended Newton's method can be used in the augmentedlagrangian method for solving nonlinear programs.
Journal ArticleDOI
Solution of monotone complementarity problems with locally Lipschitzian functions
TL;DR: IfF is monotone in a neighbourhood ofx, it is proved that 0 εδθ(x) is necessary and sufficient forx to be a solution of CP(F) and the generalized Newton method is shown to be locally well defined and superlinearly convergent with the order of 1+p.
Journal ArticleDOI
Piecewise smoothness, local invertibility, and parametric analysis of normal maps
Jong-Shi Pang,Daniel Ralph +1 more
TL;DR: These properties of the Euclidean projection map are used to obtain inverse and implicit function theorems for associated normal maps, using a new characterization of invertibility of a PC1 function in terms of its B-derivative.
Journal ArticleDOI
Semismooth Karush-Kuhn-Tucker equations and convergence analysis of Newton and quasi-Newton methods for solving these equations
Liqun Qi,Houyuan Jiang +1 more
TL;DR: A mixed quasi-Newton method which converges Q-superlinearly with common symmetrical updating rules under the above conditions for the generalized Newton method is presented.