scholarly journals A Decomposition Algorithm for Convex Nondifferentiable Minimization with Errors

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Yuan Lu ◽  
Li-Ping Pang ◽  
Jie Shen ◽  
Xi-Jun Liang
Keyword(s):  
Exact Solution ◽  
Unconstrained Minimization ◽  
Decomposition Algorithm ◽  
Approximate Evaluation ◽  
Type Method ◽  
Numerical Tests ◽  
Inexact Data ◽  
Unconstrained Minimization Problem ◽  
Nondifferentiable Minimization ◽  
Approximate Subgradients

A decomposition algorithm based on proximal bundle-type method with inexact data is presented for minimizing an unconstrained nonsmooth convex functionf. At each iteration, only the approximate evaluation offand its approximate subgradients are required which make the algorithm easier to implement. It is shown that every cluster of the sequence of iterates generated by the proposed algorithm is an exact solution of the unconstrained minimization problem. Numerical tests emphasize the theoretical findings.

A penalty method for nonlinear programming

2019 ◽  
Vol 53 (1) ◽  
pp. 29-38
Author(s):  
Larbi Bachir Cherif ◽  
Bachir Merikhi
Keyword(s):  
Nonlinear Programming ◽  
Convex Programming ◽  
Penalty Method ◽  
Line Search ◽  
Penalty Methods ◽  
Descent Direction ◽  
Type Method ◽  
Numerical Tests ◽  
Line Searches ◽  
Majorant Function

This paper presents a variant of logarithmic penalty methods for nonlinear convex programming. If the descent direction is obtained through a classical Newton-type method, the line search is done on a majorant function. Numerical tests show the efficiency of this approach versus classical line searches.

scholarly journals A Wavelet Method for the Cauchy Problem for the Helmholtz Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-18
Author(s):  
Fang-Fang Dou ◽  
Chu-Li Fu
Keyword(s):  
Cauchy Problem ◽  
Exact Solution ◽  
Helmholtz Equation ◽  
Error Estimates ◽  
Fixed Frequency ◽  
Wavelet Method ◽  
Numerical Tests ◽  
The Cauchy Problem ◽  
Ill Posed

We consider a Cauchy problem for the Helmholtz equation at a fixed frequency. The problem is severely ill posed in the sense that the solution (if it exists) does not depend continuously on the data. We present a wavelet method to stabilize the problem. Some error estimates between the exact solution and its approximation are given, and numerical tests verify the efficiency and accuracy of the proposed method.

scholarly journals Multipoint Fractional Iterative Methods with (2α + 1)th-Order of Convergence for Solving Nonlinear Problems

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 452
Author(s):  
Giro Candelario ◽  
Alicia Cordero ◽  
Juan R. Torregrosa
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Newton Method ◽  
Fractional Derivatives ◽  
Order Of Convergence ◽  
Recent Literature ◽  
Nonlinear Problems ◽  
Type Method ◽  
Numerical Tests ◽  
Multipoint Method

In the recent literature, some fractional one-point Newton-type methods have been proposed in order to find roots of nonlinear equations using fractional derivatives. In this paper, we introduce a new fractional Newton-type method with order of convergence α + 1 and compare it with the existing fractional Newton method with order 2 α . Moreover, we also introduce a multipoint fractional Traub-type method with order 2 α + 1 and compare its performance with that of its first step. Some numerical tests and analysis of the dependence on the initial estimations are made for each case, including a comparison with classical Newton ( α = 1 of the first step of the class) and classical Traub’s scheme ( α = 1 of fractional proposed multipoint method). In this comparison, some cases are found where classical Newton and Traub’s methods do not converge and the proposed methods do, among other advantages.

A NONLINEAR BACKWARD PARABOLIC PROBLEM: REGULARIZATION BY QUASI-REVERSIBILITY AND ERROR ESTIMATES

2011 ◽  
Vol 04 (01) ◽  
pp. 145-161 ◽  
Author(s):  
Nguyen Huy Tuan ◽  
Pham Hoang Quan ◽  
Dang Duc Trong ◽  
Nguyen Do Minh Nhat
Keyword(s):  
Exact Solution ◽  
Parabolic Equation ◽  
Small Parameter ◽  
Parabolic Problem ◽  
Unbounded Operator ◽  
Numerical Tests ◽  
Nonlinear Parabolic ◽  
Time Problem ◽  
Ill Posed ◽  
Well Posed

In this paper, we consider an inverse time problem for a nonlinear parabolic equation in the form ut + Au(t) = f(t, u(t)), u(T) = φ, where A is a positive self-adjoint unbounded operator and f is a Lipschitz function. As known, it is ill-posed. Using a quasi-reversibility method, we shall construct regularization solutions depended on a small parameter ϵ. We show that the regularized problem is well-posed and that their solution uϵ(t) converges on [0, T] to the exact solution u(t). This paper extends the work by Dinh Nho Hao et al. [8] to nonlinear ill-posed problems. Some numerical tests illustrate that the proposed method is feasible and effective.

NUMEROV-TYPE METHODS FOR OSCILLATORY LINEAR INITIAL VALUE PROBLEMS

2009 ◽  
Vol 20 (03) ◽  
pp. 383-398 ◽  
Author(s):  
I. TH. FAMELIS
Keyword(s):  
Initial Value Problems ◽  
New Method ◽  
Phase Lag ◽  
Initial Value ◽  
Oscillatory Solutions ◽  
Type Method ◽  
Numerical Tests ◽  
Oscillating Solutions ◽  
Algebraic Order ◽  
High Phase

We present a new explicit Numerov-type method for the solution of second-order linear initial value problems with oscillating solutions. The new method attains algebraic order seven at a cost of six function evaluations per step. The method has the characteristic of zero dissipation and high phase-lag order making it suitable for the solution of problems with oscillatory solutions. The numerical tests in a variety of problems justify our effort.

scholarly journals Global Convergence of a New Nonmonotone Filter Method for Equality Constrained Optimization

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Ke Su ◽  
Wei Liu ◽  
Xiaoli Lu
Keyword(s):  
Optimization Problems ◽  
Trust Region Method ◽  
Trust Region ◽  
Filter Method ◽  
Type Method ◽  
Numerical Tests ◽  
Starting Point ◽  
Filter Type ◽  
Acceptability Criterion ◽  
The Given

A new nonmonotone filter trust region method is introduced for solving optimization problems with equality constraints. This method directly uses the dominated area of the filter as an acceptability criterion for trial points and allows the dominated area decreasing nonmonotonically. Compared with the filter-type method, our method has more flexible criteria and can avoid Maratos effect in a certain degree. Under reasonable assumptions, we prove that the given algorithm is globally convergent to a first order stationary point for all possible choices of the starting point. Numerical tests are presented to show the effectiveness of the proposed algorithm.

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