scholarly journals Polyhedral separation via difference of convex (DC) programming

2021 ◽  
Author(s):  
Annabella Astorino ◽  
Massimo Di Francesco ◽  
Manlio Gaudioso ◽  
Enrico Gorgone ◽  
Benedetto Manca
Keyword(s):  
Optimization Model ◽  
Supervised Classification ◽  
Dc Programming ◽  
Separation Of Sets ◽  
Difference Of Convex ◽  
Optim Theory

AbstractWe consider polyhedral separation of sets as a possible tool in supervised classification. In particular, we focus on the optimization model introduced by Astorino and Gaudioso (J Optim Theory Appl 112(2):265–293, 2002) and adopt its reformulation in difference of convex (DC) form. We tackle the problem by adapting the algorithm for DC programming known as DCA. We present the results of the implementation of DCA on a number of benchmark classification datasets.

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.

scholarly journals Analysis Sparse Representation for Nonnegative Signals Based on Determinant Measure by DC Programming

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-12 ◽  
Author(s):  
Yujie Li ◽  
Benying Tan ◽  
Atsunori Kanemura ◽  
Shuxue Ding ◽  
Wuhui Chen
Keyword(s):  
Sparse Representation ◽  
Dictionary Learning ◽  
Optimization Problem ◽  
Learning Algorithm ◽  
State Of The Art ◽  
Optimization Methods ◽  
Dc Programming ◽  
Sparse Model ◽  
Alternative Approach ◽  
Difference Of Convex

Analysis sparse representation has recently emerged as an alternative approach to the synthesis sparse model. Most existing algorithms typically employ the l0-norm, which is generally NP-hard. Other existing algorithms employ the l1-norm to relax the l0-norm, which sometimes cannot promote adequate sparsity. Most of these existing algorithms focus on general signals and are not suitable for nonnegative signals. However, many signals are necessarily nonnegative such as spectral data. In this paper, we present a novel and efficient analysis dictionary learning algorithm for nonnegative signals with the determinant-type sparsity measure which is convex and differentiable. The analysis sparse representation can be cast in three subproblems, sparse coding, dictionary update, and signal update, because the determinant-type sparsity measure would result in a complex nonconvex optimization problem, which cannot be easily solved by standard convex optimization methods. Therefore, in the proposed algorithms, we use a difference of convex (DC) programming scheme for solving the nonconvex problem. According to our theoretical analysis and simulation study, the main advantage of the proposed algorithm is its greater dictionary learning efficiency, particularly compared with state-of-the-art algorithms. In addition, our proposed algorithm performs well in image denoising.

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.

scholarly journals Cooperative Crossing Cache Placement in Cache-Enabled Device to Device-Aided Cellular Networks

2018 ◽  
Vol 8 (9) ◽  
pp. 1578 ◽  
Author(s):  
Somayeh Soleimani ◽  
Xiaofeng Tao
Keyword(s):  
Cellular Networks ◽  
Low Complexity ◽  
Network Capacity ◽  
Dc Programming ◽  
Base Stations ◽  
Device To Device ◽  
Difference Of Convex ◽  
Cache Placement ◽  
The Difference ◽  
The Impact

In cache-enabled device-to-device (D2D) -aided cellular networks, the technique of caching contents in the cooperative crossing between base stations (BSs) and devices can significantly reduce core traffic and enhance network capacity. In this paper, we propose a scheme that establishes device availability, which indicates whether a cache-enabled device can handle the transmission of the desired content within the required sending time, called the delay, while achieving optimal probabilistic caching. We also investigate the impact of transmission device availability on the effectiveness of a scenario of cooperative crossing cache placement, where content delivery traffic can be offloaded from the local cache, a D2D transmitter’s cache via a D2D link, or else directly from a BS via a cellular link, in order to maximize the offloading probability. Further, we derive the cooperation content offloading strategy while considering successful content transmission by D2D transmitters or BSs to guarantee the delay, even though reducing the delay is not the focus of this study. Finally, the proposed problem is formulated. Owing to the non-convexity of the optimization problem, it can be rewritten as a minimization of the difference between the convex functions; thus, it can be solved by difference of convex (DC) programming using a low-complexity algorithm. Simulation results show that the proposed cache placement scheme improves the offloading probability by 13.5% and 23% compared to Most Popular Content (MPC) scheme, in which both BSs and devices cache the most popular content and Coop. BS/D2D caching scheme, in which each BS tier and user tier applies cooperative content caching separately.

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.

Improved set-membership guaranteed state estimation for uncertain non-linear systems

2017 ◽  
Vol 40 (5) ◽  
pp. 1471-1480 ◽  
Author(s):  
Fuqiang You ◽  
Hualu Zhang ◽  
Fuli Wang
Keyword(s):  
State Estimation ◽  
Dc Programming ◽  
Discrete Time Systems ◽  
Set Membership ◽  
Non Linear ◽  
Difference Of Convex ◽  
The Difference ◽  
Guaranteed State Estimation ◽  
Time Systems ◽  
Outer Bound

This paper presents a new improved set-membership algorithm for guaranteed state estimation for non-linear discrete-time systems with an unknown but bounded description of disturbance and noise. In this paper, the non-linear model is linearized first and the difference of convex (DC) programming method is used to outer-bound the linearization error, which is regarded as a disturbance of the system, and the improved set-membership algorithm guaranteed state estimation for the linearization system. An example is given to illustrate the proposed 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.

scholarly journals Infrared Small Target Detection via Non-Convex Rank Approximation Minimization Joint l2,1 Norm

2018 ◽  
Vol 10 (11) ◽  
pp. 1821 ◽  
Author(s):  
Landan Zhang ◽  
Lingbing Peng ◽  
Tianfang Zhang ◽  
Siying Cao ◽  
Zhenming Peng
Keyword(s):  
State Of The Art ◽  
Infrared Image ◽  
Dc Programming ◽  
Background Suppression ◽  
Convex Model ◽  
L1 Norm ◽  
Alternating Direction ◽  
Difference Of Convex ◽  
Rank Approximation ◽  
Small Targets

To improve the detection ability of infrared small targets in complex backgrounds, a novel method based on non-convex rank approximation minimization joint l2,1 norm (NRAM) was proposed. Due to the defects of the nuclear norm and l1 norm, the state-of-the-art infrared image-patch (IPI) model usually leaves background residuals in the target image. To fix this problem, a non-convex, tighter rank surrogate and weighted l1 norm are instead utilized, which can suppress the background better while preserving the target efficiently. Considering that many state-of-the-art methods are still unable to fully suppress sparse strong edges, the structured l2,1 norm was introduced to wipe out the strong residuals. Furthermore, with the help of exploiting the structured norm and tighter rank surrogate, the proposed model was more robust when facing various complex or blurry scenes. To solve this non-convex model, an efficient optimization algorithm based on alternating direction method of multipliers (ADMM) plus difference of convex (DC) programming was designed. Extensive experimental results illustrate that the proposed method not only shows superiority in background suppression and target enhancement, but also reduces the computational complexity compared with other baselines.

An optimization model of self-paced tracking

1984 ◽  
Author(s):  
M. A. Montazer ◽  
Colin G. Drury
Keyword(s):  
Optimization Model
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