scholarly journals An Augmented Lagrangian Algorithm for Solving Semiinfinite Programming

2012 ◽  
Vol 2012 ◽  
pp. 1-11
Author(s):  
Qian Liu ◽  
Changyu Wang
Keyword(s):  
Global Convergence ◽  
Augmented Lagrangian ◽  
Zero Point ◽  
Accumulation Point ◽  
Convergence Result ◽  
Boundedness Condition ◽  
Perturbation Theorem ◽  
Lower Semi ◽  
Optimal Value ◽  
Coercive Condition

We present a smooth augmented Lagrangian algorithm for semiinfinite programming (SIP). For this algorithm, we establish a perturbation theorem under mild conditions. As a corollary of the perturbation theorem, we obtain the global convergence result, that is, any accumulation point of the sequence generated by the algorithm is the solution of SIP. We get this global convergence result without any boundedness condition or coercive condition. Another corollary of the perturbation theorem shows that the perturbation function at zero point is lower semi-continuous if and only if the algorithm forces the sequence of objective function convergence to the optimal value of SIP. Finally, numerical results are given.

The stochastic equation Yt+1 = AtYt + Bt with non-stationary coefficients

2001 ◽  
Vol 38 (1) ◽  
pp. 80-94 ◽  
Author(s):  
Ulrich Horst
Keyword(s):  
Global Convergence ◽  
Random Environment ◽  
Linear Relation ◽  
Stochastic Equation ◽  
Stationary Process ◽  
Sufficient Conditions ◽  
Convergence Result ◽  
Stationary Case ◽  
Finite Dimensional ◽  
Long Run

In this paper, we consider the stochastic sequence {Yt}t∊ℕ defined recursively by the linear relation Yt+1 = AtYt + Bt in a random environment which is described by the non-stationary process {(At, Bt)}t∊ℕ. We formulate sufficient conditions on the environment which ensure that the finite-dimensional distributions of {Yt}t∊ℕ converge weakly to the finite-dimensional distributions of a unique stationary process. If the driving sequence {(At, Bt)}t∊ℕ becomes stationary in the long run, then we can establish a global convergence result. This extends results of Brandt (1986) and Borovkov (1998) from the stationary to the non-stationary case.

Block-preconditioners for the incompressible Navier–Stokes equations discretized by a finite volume method

2017 ◽  
Vol 25 (2) ◽  
Author(s):  
Xin He ◽  
Cornelis Vuik ◽  
Christiaan Klaij
Keyword(s):  
Reynolds Number ◽  
Finite Volume Method ◽  
Finite Volume ◽  
Augmented Lagrangian ◽  
Dimensionless Parameter ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Optimal Value ◽  
Volume Method

Abstract The modified augmented Lagrangian preconditioner has attracted much attention in solving nondimensional Navier–Stokes equations discretized by the finite element method. In industrial applications the governing equations are often in dimensional form and discretized using the finite volume method. This paper assesses the capability of this preconditioner for dimensional Navier–Stokes equations in the context of the finite volume method. Two main contributions are made. First, this paper introduces a new dimensionless parameter that is involved in the modified augmented Lagrangian preconditioner. Second, with a number of academic test problems this paper reveals that the convergence of both nonlinear and linear iterations depend on this dimensionless parameter. A way to choose the optimal value of the dimensionless parameter is suggested and it is found that the optimal value is dependent of the Reynolds number, instead of the fluid’s properties, e.g., density and dynamic viscosity. The outcomes of this paper yield a potential rule to choose the optimal dimensionless parameter in practice, namely, correspondingly increasing with enlarging the Reynolds number.

scholarly journals New Convergence Properties of the Primal Augmented Lagrangian Method

2011 ◽  
Vol 2011 ◽  
pp. 1-14 ◽  
Author(s):  
Jinchuan Zhou ◽  
Xunzhi Zhu ◽  
Lili Pan ◽  
Wenling Zhao
Keyword(s):  
Optimization Problem ◽  
Augmented Lagrangian ◽  
Augmented Lagrangian Method ◽  
Accumulation Point ◽  
Lagrangian Method ◽  
Iterative Sequence ◽  
Convergence Properties ◽  
Primal Problem ◽  
Kkt Point ◽  
Convergence Results

New convergence properties of the proximal augmented Lagrangian method is established for continuous nonconvex optimization problem with both equality and inequality constrains. In particular, the multiplier sequences are not required to be bounded. Different convergence results are discussed dependent on whether the iterative sequence{xk}generated by algorithm is convergent or divergent. Furthermore, under certain convexity assumption, we show that every accumulation point of{xk}is either a degenerate point or a KKT point of the primal problem. Numerical experiments are presented finally.

scholarly journals A Fast Image Restoration Algorithm Based on a Fixed Point and Optimization Method

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 378 ◽  
Author(s):  
Adisak Hanjing ◽  
Suthep Suantai
Keyword(s):  
Fixed Point ◽  
Weak Convergence ◽  
Image Restoration ◽  
Optimization Method ◽  
Convergence Result ◽  
Fixed Point Algorithm ◽  
Convergence Behavior ◽  
Lower Semi ◽  
Restoration Algorithm ◽  
Nonexpansive Operators

In this paper, a new accelerated fixed point algorithm for solving a common fixed point of a family of nonexpansive operators is introduced and studied, and then a weak convergence result and the convergence behavior of the proposed method is proven and discussed. Using our main result, we obtain a new accelerated image restoration algorithm, called the forward-backward modified W-algorithm (FBMWA), for solving a minimization problem in the form of the sum of two proper lower semi-continuous and convex functions. As applications, we apply the FBMWA algorithm to solving image restoration problems. We analyze and compare convergence behavior of our method with the others for deblurring the image. We found that our algorithm has a higher efficiency than the others in the literature.

scholarly journals An inexact Levenberg-Marquardt method for large sparse nonlinear least squres

1985 ◽  
Vol 26 (4) ◽  
pp. 387-403 ◽  
Author(s):  
S. J. Wright ◽  
J. N. Holt
Keyword(s):  
Global Convergence ◽  
Least Squares ◽  
Jacobian Matrix ◽  
Numerical Test ◽  
Convergence Result ◽  
Test Results ◽  
Linear Least Squares ◽  
Optimal Point ◽  
Marquardt Method ◽  
Levenberg Marquardt

AbstractA method for solving problems of the form is presented. The approach of Levenberg and Marquardt is used, except that the linear least squares subproblem arising at each iteration is not solved exactly, but only to within a certain tolerance. The method is most suited to problems in which the Jacobian matrix is sparse. Use is made of the iterative algorithm LSQR of Paige and Saunders for sparse linear least squares.A global convergence result can be proven, and under certain conditions it can be shown that the method converges quadratically when the sum of squares at the optimal point is zero.Numerical test results for problems of varying residual size are given.

scholarly journals A Global Convergence Result for a Higher Order Difference Equation

2007 ◽  
Vol 2007 ◽  
pp. 1-7 ◽  
Author(s):  
Bratislav D. Iricanin
Keyword(s):  
Global Convergence ◽  
Difference Equation ◽  
Bounded Solution ◽  
Convergence Result ◽  
Higher Order ◽  
Unbounded Solution ◽  
Order Difference ◽  
Initial Values ◽  
Order Difference Equation

Letf(z1,…,zk)∈C(Ik,I)be a given function, whereIis (bounded or unbounded) subinterval ofℝ, andk∈ℕ. Assume thatf(y1,y2,…,yk)≥f(y2,…,yk,y1)ify1≥max{y2,…,yk},f(y1,y2,…,yk)≤f(y2,…,yk,y1)ify1≤min{y2,…,yk}, andfis non- decreasing in the last variablezk. We then prove that every bounded solution of an autonomous difference equation of orderk, namely,xn=f(xn−1,…,xn−k),n=0,1,2,…,with initial valuesx−k,…,x−1∈I, is convergent, and every unbounded solution tends either to+∞or to−∞.

scholarly journals On the Convergence of Bounded Solutions of Non Homogeneous Gradient-like Systems

2017 ◽  
Vol 1 (1) ◽  
pp. 61
Author(s):  
Phuong Minh Tran ◽  
Nhan Thanh Nguyen
Keyword(s):  
Lyapunov Function ◽  
Accumulation Point ◽  
Convergence Result ◽  
Time Behavior ◽  
Bounded Solutions ◽  
Creative Commons ◽  
Angle Condition ◽  
Łojasiewicz Inequality ◽  
Long Time ◽  
Lojasiewicz Inequality

We study the long time behavior of the bounded solutions of non homogeneous gradient-like system which admits a strict Lyapunov function. More precisely, we show that any bounded solution of the gradient-like system converges to an accumulation point as time goes to infinity under some mild hypotheses. As in homogeneous case, the key assumptions for this system are also the angle condition and the Kurdyka-Lojasiewicz inequality. The convergence result will be proved under a L1 -condition of the perturbation term. Moreover, if the Lyapunov function satisfies a Lojasiewicz inequality then the rate of convergence will be even obtained.This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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