Some computational convergent iterative algorithms to solve nonlinear problems

2021 ◽  
Author(s):  
Mohsen Rabbani ◽  
Ji Huan He ◽  
Murat Düz
Keyword(s):  
Iterative Algorithms ◽  
Nonlinear Problems

scholarly journals Solving the Variational Inequality Problem Defined on Intersection of Finite Level Sets

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Songnian He ◽  
Caiping Yang
Keyword(s):  
Hilbert Space ◽  
Variational Inequality ◽  
Strong Convergence ◽  
Convex Functions ◽  
Level Sets ◽  
Variational Inequality Problem ◽  
Real Hilbert Space ◽  
Iterative Algorithms ◽  
Nonlinear Problems ◽  
Original Domain

Consider the variational inequalityVI(C,F)of finding a pointx*∈Csatisfying the property〈Fx*,x-x*〉≥0, for allx∈C, whereCis the intersection of finite level sets of convex functions defined on a real Hilbert spaceHandF:H→His anL-Lipschitzian andη-strongly monotone operator. Relaxed and self-adaptive iterative algorithms are devised for computing the unique solution ofVI(C,F). Since our algorithm avoids calculating the projectionPC(calculatingPCby computing several sequences of projections onto half-spaces containing the original domainC) directly and has no need to know any information of the constantsLandη, the implementation of our algorithm is very easy. To prove strong convergence of our algorithms, a new lemma is established, which can be used as a fundamental tool for solving some nonlinear problems.

Efficiency evaluation of structural nonlinear analysis method based on the Woodbury formula

2019 ◽  
Vol 36 (4) ◽  
pp. 1082-1100
Author(s):  
Gang Li ◽  
Shuo Jia ◽  
Hong-Nan Li
Keyword(s):  
Nonlinear Analysis ◽  
Time Complexity ◽  
Linear Equations ◽  
Iterative Algorithms ◽  
Nonlinear Problems ◽  
Factorization Method ◽  
Point Method ◽  
Efficiency Evaluation ◽  
Analysis Method ◽  
Content Type

Purpose The purpose of this paper is to make a theoretical comprehensive efficiency evaluation of a nonlinear analysis method based on the Woodbury formula from the efficiency of the solution of linear equations in each incremental step and the selected iterative algorithms. Design/methodology/approach First, this study employs the time complexity theory to quantitatively compare the efficiency of the Woodbury formula and the LDLT factorization method which is a commonly used method to solve linear equations. Moreover, the performance of iterative algorithms also significantly effects the efficiency of the analysis. Thus, the three-point method with a convergence order of eight is employed to solve the equilibrium equations of the nonlinear analysis method based on the Woodbury formula, aiming to improve the iterative performance of the Newton–Raphson (N–R) method. Findings First, the result shows that the asymptotic time complexity of the Woodbury formula is much lower than that of the LDLT factorization method when the number of inelastic degrees of freedom (IDOFs) is much less than that of DOFs, indicating that the Woodbury formula is more efficient for local nonlinear problems. Moreover, the time complexity comparison of the N–R method and the three-point method indicates that the three-point method is more efficient than the N–R method for local nonlinear problems with large-scale structures or a larger ratio of IDOFs number to the DOFs number. Originality/value This study theoretically evaluates the efficiency of nonlinear analysis method based on the Woodbury formula, and quantitatively shows the application condition of the comparative methods. The comparison result provides a theoretical basis for the selection of algorithms for different nonlinear problems.

scholarly journals Internal Perturbation Projection Algorithm for the Extended Split Equality Problem and the Extended Split Equality Fixed Point Problem

2020 ◽  
Vol 2020 ◽  
pp. 1-15
Author(s):  
Meixia Li ◽  
Xueling Zhou ◽  
Wenchao Wang
Keyword(s):  
Fixed Point ◽  
Iterative Algorithms ◽  
Nonlinear Problems ◽  
Fixed Point Problem ◽  
Projection Algorithm ◽  
Feasibility Problem ◽  
Point Problem ◽  
Split Equality Problem ◽  
Equality Problem ◽  
Internal Perturbation

In this article, we study the extended split equality problem and extended split equality fixed point problem, which are extensions of the convex feasibility problem. For solving the extended split equality problem, we present two self-adaptive stepsize algorithms with internal perturbation projection and obtain the weak and the strong convergence of the algorithms, respectively. Furthermore, based on the operators being quasinonexpansive, we offer an iterative algorithm to solve the extended split equality fixed point problem. We introduce a way of selecting the stepsize which does not need any prior information about operator norms in the three algorithms. We apply our iterative algorithms to some convex and nonlinear problems. Finally, several numerical results are shown to confirm the feasibility and efficiency of the proposed algorithms.

scholarly journals New algorithms for discrete-time optimal control problems

1984 ◽  
Vol 26 (2) ◽  
pp. 129-145
Author(s):  
Nikola B. Nedeljković
Keyword(s):  
Optimal Control ◽  
Discrete Time ◽  
Optimal Control Problems ◽  
Iterative Algorithms ◽  
Nonlinear Problems ◽  
Linear Constraints ◽  
Time Optimal Control ◽  
Control Problems ◽  
Linear Quadratic ◽  
Time Optimal

AbstractThe paper presents new demonstrably convergent first-order iterative algorithms for unconstrained discrete-time optimal control problems. The algorithms, which solve the linear-quadratic problem in one iterative step, are particularly suited for solving nonlinear problems with linear constraints via penalty function methods. The proofs of the reduction of cost at each iteration and convergence of the algorithms are provided.

New Iterative Algorithms for Solving the Mixed Antenna Synthesis Problem

2002 ◽  
Vol 58 (9-10) ◽  
pp. 9
Author(s):  
Efim Grigor'evich Zelkin ◽  
Victor Filippovich Kravchenko ◽  
Miklhail Alekseevich Basarab
Keyword(s):  
Iterative Algorithms ◽  
Synthesis Problem ◽  
Antenna Synthesis

A comparative study of iterative algorithms for the Euler equations of gasdynamics

1989 ◽  
Author(s):  
DANIEL DORNEY ◽  
GEORGE DULIKRAVICH ◽  
KI LEE
Keyword(s):  
Comparative Study ◽  
Euler Equations ◽  
Iterative Algorithms
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