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.

Solving nonmonotone affine variational inequalities problem by DC programming and DCA

2018 ◽  
Vol 13 (03) ◽  
pp. 2050067
Author(s):  
Zahira Kebaili ◽  
Mohamed Achache
Keyword(s):  
Variational Inequalities ◽  
Numerical Experiments ◽  
Dc Programming ◽  
Quadratic Program ◽  
Test Problems ◽  
Speed Of Convergence ◽  
Difference Of Convex Functions ◽  
Convex Quadratic Program ◽  
Difference Of Convex

In this paper, we consider an optimization model for solving the nonmonotone affine variational inequalities problem (AVI). It is formulated as a DC (Difference of Convex functions) program for which DCA (DC Algorithms) are applied. The resulting DCA are simple: it consists of solving successive convex quadratic program. Numerical experiments on several test problems illustrate the efficiency of the proposed approach in terms of the quality of the obtained solutions and the speed of convergence.

scholarly journals Optimal correction of infeasible equations system as Ax + B|x|= b using ℓ p-norm regularization

2022 ◽  
Vol 40 ◽  
pp. 1-16
Author(s):  
Fakhrodin Hashemi ◽  
Saeed Ketabchi
Keyword(s):  
Convex Functions ◽  
Regularization Method ◽  
High Performance ◽  
Optimization Problem ◽  
Dc Programming ◽  
Smoothing Technique ◽  
Optimal Correction ◽  
Difference Of Convex Functions ◽  
Difference Of Convex ◽  
Nonsmooth Optimization Problem

Optimal correction of an infeasible equations system as Ax + B|x|= b leads into a non-convex fractional problem. In this paper, a regularization method(ℓp-norm, 0 < p < 1), is presented to solve mentioned fractional problem. In this method, the obtained problem can be formulated as a non-convex and nonsmooth optimization problem which is not Lipschitz. The objective function of this problem can be decomposed as a difference of convex functions (DC). For this reason, we use a special smoothing technique based on DC programming. The numerical results obtained for generated problem show high performance and the effectiveness of the proposed method.

A Redistributed Bundle Algorithm Based on Local Convexification Models for Nonlinear Nonsmooth DC Programming

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Jie Shen ◽  
Na Xu ◽  
Fang-Fang Guo ◽  
Han-Yang Li ◽  
Pan Hu
Keyword(s):  
Cutting Plane ◽  
Dc Programming ◽  
Bundle Method ◽  
Difference Of Convex Functions ◽  
Bundle Algorithm ◽  
Difference Of Convex ◽  
Proximal Bundle Method ◽  
Local Convexification ◽  
Original Objective ◽  
Linearization Errors

Abstract For nonlinear nonsmooth DC programming (difference of convex functions), we introduce a new redistributed proximal bundle method. The subgradient information of both the DC components is gathered from some neighbourhood of the current stability center and it is used to build separately an approximation for each component in the DC representation. Especially we employ the nonlinear redistributed technique to model the second component of DC function by constructing a local convexification cutting plane. The corresponding convexification parameter is adjusted dynamically and is taken sufficiently large to make the ”augmented” linearization errors nonnegative. Based on above techniques we obtain a new convex cutting plane model of the original objective function. Based on this new approximation the redistributed proximal bundle method is designed and the convergence of the proposed algorithm to a Clarke stationary point is proved. A simple numerical experiment is given to show the validity of the presented algorithm.

Block Clustering Based on Difference of Convex Functions (DC) Programming and DC Algorithms

2013 ◽  
Vol 25 (10) ◽  
pp. 2776-2807 ◽  
Author(s):  
Hoai Minh Le ◽  
Hoai An Le Thi ◽  
Tao Pham Dinh ◽  
Van Ngai Huynh
Keyword(s):  
Convex Functions ◽  
Optimization Problem ◽  
Dc Programming ◽  
Clustering Problem ◽  
Block Classification ◽  
Dc Algorithm ◽  
Difference Of Convex Functions ◽  
Dc Program ◽  
Difference Of Convex ◽  
Block Clustering

We investigate difference of convex functions (DC) programming and the DC algorithm (DCA) to solve the block clustering problem in the continuous framework, which traditionally requires solving a hard combinatorial optimization problem. DC reformulation techniques and exact penalty in DC programming are developed to build an appropriate equivalent DC program of the block clustering problem. They lead to an elegant and explicit DCA scheme for the resulting DC program. Computational experiments show the robustness and efficiency of the proposed algorithm and its superiority over standard algorithms such as two-mode K-means, two-mode fuzzy clustering, and block classification EM.

Uniqueness Guarantee of Solutions of Tensor Tubal-Rank Minimization Problem

2020 ◽  
Vol 27 ◽  
pp. 540-544
Author(s):  
Feng Zhang ◽  
Jingyao Hou ◽  
Jianjun Wang ◽  
Wendong Wang
Keyword(s):  
Minimization Problem ◽  
Rank Minimization

Online Learning Based on Online DCA and Application to Online Classification

2020 ◽  
Vol 32 (4) ◽  
pp. 759-793 ◽  
Author(s):  
Hoai An Le Thi ◽  
Vinh Thanh Ho
Keyword(s):  
Online Learning ◽  
State Of The Art ◽  
Classification Algorithms ◽  
Data Sets ◽  
Prediction Problem ◽  
Dc Algorithm ◽  
Difference Of Convex Functions ◽  
Online Classification ◽  
Learning Techniques ◽  
Difference Of Convex

We investigate an approach based on DC (Difference of Convex functions) programming and DCA (DC Algorithm) for online learning techniques. The prediction problem of an online learner can be formulated as a DC program for which online DCA is applied. We propose the two so-called complete/approximate versions of online DCA scheme and prove their logarithmic/sublinear regrets. Six online DCA-based algorithms are developed for online binary linear classification. Numerical experiments on a variety of benchmark classification data sets show the efficiency of our proposed algorithms in comparison with the state-of-the-art online classification algorithms.

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