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 On the Convergence of a Smooth Penalty Algorithm without Computing Global Solutions

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Bingzhuang Liu ◽  
Changyu Wang ◽  
Wenling Zhao
Keyword(s):  
Penalty Function ◽  
Optimization Problem ◽  
Constraint Qualification ◽  
Global Solutions ◽  
Accumulation Point ◽  
Smooth Functions ◽  
Convex Optimization Problem ◽  
Primal Problem ◽  
Kkt Point ◽  
Penalty Algorithm

We consider a smooth penalty algorithm to solve nonconvex optimization problem based on a family of smooth functions that approximate the usual exact penalty function. At each iteration in the algorithm we only need to find a stationary point of the smooth penalty function, so the difficulty of computing the global solution can be avoided. Under a generalized Mangasarian-Fromovitz constraint qualification condition (GMFCQ) that is weaker and more comprehensive than the traditional MFCQ, we prove that the sequence generated by this algorithm will enter the feasible solution set of the primal problem after finite times of iteration, and if the sequence of iteration points has an accumulation point, then it must be a Karush-Kuhn-Tucker (KKT) point. Furthermore, we obtain better convergence for convex optimization problem.

scholarly journals An augmented Lagrangian method for solving total variation (TV)-based image registration model

2020 ◽  
Vol 14 ◽  
pp. 174830262097353
Author(s):  
Noppadol Chumchob ◽  
Ke Chen
Keyword(s):  
Image Registration ◽  
Augmented Lagrangian ◽  
Numerical Solutions ◽  
Augmented Lagrangian Method ◽  
Closed Form Solution ◽  
Numerical Algorithms ◽  
Lagrangian Method ◽  
Form Solution ◽  
The Other ◽  
Other Hand

Variational methods for image registration basically involve a regularizer to ensure that the resulting well-posed problem admits a solution. Different choices of regularizers lead to different deformations. On one hand, the conventional regularizers, such as the elastic, diffusion and curvature regularizers, are able to generate globally smooth deformations and generally useful for many applications. On the other hand, these regularizers become poor in some applications where discontinuities or steep gradients in the deformations are required. As is well-known, the total (TV) variation regularizer is more appropriate to preserve discontinuities of the deformations. However, it is difficult in developing an efficient numerical method to ensure that numerical solutions satisfy this requirement because of the non-differentiability and non-linearity of the TV regularizer. In this work we focus on computational challenges arising in approximately solving TV-based image registration model. Motivated by many efficient numerical algorithms in image restoration, we propose to use augmented Lagrangian method (ALM). At each iteration, the computation of our ALM requires to solve two subproblems. On one hand for the first subproblem, it is impossible to obtain exact solution. On the other hand for the second subproblem, it has a closed-form solution. To this end, we propose an efficient nonlinear multigrid (NMG) method to obtain an approximate solution to the first subproblem. Numerical results on real medical images not only confirm that our proposed ALM is more computationally efficient than some existing methods, but also that the proposed ALM delivers the accurate registration results with the desired property of the constructed deformations in a reasonable number of iterations.

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