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

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.

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

Neural Computation ◽

10.1162/neco_a_00490 ◽

2013 ◽

Vol 25 (10) ◽

pp. 2776-2807 ◽

Cited By ~ 8

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.

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

Mathematical Problems in Engineering ◽

10.1155/2016/7473041 ◽

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.

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