scholarly journals New Nonsmooth Equations-Based Algorithms for -Norm Minimization and Applications

2012 ◽  
Vol 2012 ◽  
pp. 1-14
Author(s):  
Lei Wu ◽  
Zhe Sun
Keyword(s):  
Minimization Problem ◽  
Numerical Results ◽  
Nonsmooth Equations ◽  
Mild Conditions ◽  
Minimization Problems ◽  
Norm Minimization ◽  
Globally Convergent

Recently, Xiao et al. proposed a nonsmooth equations-based method to solve the -norm minimization problem (2011). The advantage of this method is its simplicity and lower storage. In this paper, based on new nonsmooth equations reformulation, we investigate new nonsmooth equations-based algorithms for solving -norm minimization problems. Under mild conditions, we show that the proposed algorithms are globally convergent. The preliminary numerical results demonstrate the effectiveness of the proposed algorithms.

scholarly journals Minimization problems for certain structured matrices

2015 ◽  
Vol 30 ◽  
pp. 613-631 ◽  
Author(s):  
Zhongyun Liu ◽  
Rui Ralha ◽  
Yulin Zhang ◽  
Carla Ferreira
Keyword(s):  
Minimization Problem ◽  
Numerical Results ◽  
Frobenius Norm ◽  
Structured Matrices ◽  
Minimization Problems ◽  
Symplectic Matrices ◽  
Unitary Conjugate

For given Z, B ∈ C^{n\times k}, the problem of finding A ∈ C^{n\times n}, in some prescribed class W, that minimizes ||AZ − B|| (Frobenius norm) has been considered by different authors for distinct classes W. Here, this minimization problem is studied for two other classes, which include the symmetric Hamiltonian, symmetric skew-Hamiltonian, real orthogonal symplectic and unitary conjugate symplectic matrices. The problem of minimizing ||A − A˜||, where A˜ is given and A is a solution of the previous problem, is also considered (as has been done by others, for different classes W). The key idea of this contribution is the reduction of each one of the above minimization problems to two independent subproblems in orthogonal subspaces of C^{n\times n}. This is possible due to the special structures under consideration. Matlab codes are developed, and numerical results of some tests are presented.

scholarly journals Medical Image Denoising Via Matrix Norm Minimization Problems

2021 ◽  
Vol 24 (2) ◽  
pp. 72-77
Author(s):  
Zainab Abd-Alzahra ◽  
◽  
Basad Al-Sarray ◽  
Keyword(s):  
Minimization Problem ◽  
Matrix Completion ◽  
Nuclear Norm ◽  
Nuclear Norm Minimization ◽  
Minimization Problems ◽  
Completion Problem ◽  
Norm Minimization ◽  
Matrix Completion Problem ◽  
The Matrix ◽  
The Given

This paper presents the matrix completion problem for image denoising. Three problems based on matrix norm are performing: Spectral norm minimization problem (SNP), Nuclear norm minimization problem (NNP), and Weighted nuclear norm minimization problem (WNNP). In general, images representing by a matrix this matrix contains the information of the image, some information is irrelevant or unfavorable, so to overcome this unwanted information in the image matrix, information completion is used to comperes the matrix and remove this unwanted information. The unwanted information is handled by defining {0,1}-operator under some threshold. Applying this operator on a given matrix keeps the important information in the image and removing the unwanted information by solving the matrix completion problem that is defined by P. The quadratic programming use to solve the given three norm-based minimization problems. To improve the optimal solution a weighted exponential is used to compute the weighted vector of spectral that use to improve the threshold of optimal low rank that getting from solving the nuclear norm and spectral norm problems. The result of applying the proposed method on different types of images is given by adopting some metrics. The results showed the ability of the given methods.

scholarly journals A Projected Forward-Backward Algorithm for Constrained Minimization with Applications to Image Inpainting

Mathematics ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 890
Author(s):  
Suthep Suantai ◽  
Kunrada Kankam ◽  
Prasit Cholamjiak
Keyword(s):  
Image Processing ◽  
Minimization Problem ◽  
Convex Functions ◽  
Lower Semicontinuous ◽  
Image Inpainting ◽  
Constrained Minimization ◽  
Convex Minimization Problem ◽  
Mild Conditions ◽  
Constrained Minimization Problem ◽  
Weak Convergence Theorem

In this research, we study the convex minimization problem in the form of the sum of two proper, lower-semicontinuous, and convex functions. We introduce a new projected forward-backward algorithm using linesearch and inertial techniques. We then establish a weak convergence theorem under mild conditions. It is known that image processing such as inpainting problems can be modeled as the constrained minimization problem of the sum of convex functions. In this connection, we aim to apply the suggested method for solving image inpainting. We also give some comparisons to other methods in the literature. It is shown that the proposed algorithm outperforms others in terms of iterations. Finally, we give an analysis on parameters that are assumed in our hypothesis.

scholarly journals An Efficient Method for Convex Constrained Rank Minimization Problems Based on DC Programming

2016 ◽  
Vol 2016 ◽  
pp. 1-13
Author(s):  
Wanping Yang ◽  
Jinkai Zhao ◽  
Fengmin Xu
Keyword(s):  
Objective Function ◽  
Minimization Problem ◽  
Dc Programming ◽  
Rank Minimization ◽  
Closed Form Solutions ◽  
Minimization Problems ◽  
Difference Of Convex Functions ◽  
Convex Objective Function ◽  
Difference Of Convex ◽  
Cut Problems

The constrained rank minimization problem has various applications in many fields including machine learning, control, and signal processing. In this paper, we consider the convex constrained rank minimization problem. By introducing a new variable and penalizing an equality constraint to objective function, we reformulate the convex objective function with a rank constraint as a difference of convex functions based on the closed-form solutions, which can be reformulated as DC programming. A stepwise linear approximative algorithm is provided for solving the reformulated model. The performance of our method is tested by applying it to affine rank minimization problems and max-cut problems. Numerical results demonstrate that the method is effective and of high recoverability and results on max-cut show that the method is feasible, which provides better lower bounds and lower rank solutions compared with improved approximation algorithm using semidefinite programming, and they are close to the results of the latest researches.

A Penalty-Free Method with Trust Region for Nonlinear Semidefinite Programming

2015 ◽  
Vol 32 (01) ◽  
pp. 1540006 ◽  
Author(s):  
Zhongwen Chen ◽  
Shicai Miao
Keyword(s):  
Semidefinite Programming ◽  
Objective Function ◽  
Penalty Function ◽  
Trust Region ◽  
Numerical Results ◽  
New Method ◽  
Penalty Parameter ◽  
Nonlinear Semidefinite Programming ◽  
Globally Convergent ◽  
Nonlinear Objective Function

In this paper, we propose a class of new penalty-free method, which does not use any penalty function or a filter, to solve nonlinear semidefinite programming (NSDP). So the choice of the penalty parameter and the storage of filter set are avoided. The new method adopts trust region framework to compute a trial step. The trial step is then either accepted or rejected based on the some acceptable criteria which depends on reductions attained in the nonlinear objective function and in the measure of constraint infeasibility. Under the suitable assumptions, we prove that the algorithm is well defined and globally convergent. Finally, the preliminary numerical results are reported.

A new piecewise quadratic approximation approach for L0 norm minimization problem

2018 ◽  
Vol 62 (1) ◽  
pp. 185-204 ◽  
Author(s):  
Qian Li ◽  
Yanqin Bai ◽  
Changjun Yu ◽  
Ya-xiang Yuan
Keyword(s):  
Minimization Problem ◽  
Quadratic Approximation ◽  
Approximation Approach ◽  
Norm Minimization

An algorithm for the total, or partial, factorization of a polynomial

1977 ◽  
Vol 82 (3) ◽  
pp. 427-437 ◽  
Author(s):  
M. R. Farmer ◽  
G. Loizou
Keyword(s):  
Numerical Results ◽  
Search Procedure ◽  
Multiple Zeros ◽  
Globally Convergent ◽  
Simultaneous Iteration ◽  
Accelerate Convergence ◽  
Iteration Function ◽  
Convergent Algorithm

AbstractA globally convergent algorithm is presented for the total, or partial, factorization of a polynomial. Firstly, a circle is found containing all the zeros. Secondly, a search procedure locates smaller circles, each containing a zero, and the multiplicities are then calculated. Thirdly, a simultaneous Iteration Function is used to accelerate convergence. The Iteration Function is chosen from a class of such functions derived herein to deal with the general case of multiple zeros; various properties of these functions are also discussed. Finally, sample numerical results are given which demon-strate the effectiveness of the algorithm.

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