DC Algorithm Research Articles

Steering exact penalty DCA for nonsmooth DC optimisation problems with equality and inequality constraints

We propose and study a version of the DCA (Difference-of-Convex functions Algorithm) using the penalty function for solving nonsmooth DC optimisation problems with nonsmooth DC equality and inequality constraints. The method employs an adaptive penalty updating strategy to improve its performance. This strategy is based on the so-called steering exact penalty methodology and relies on solving some auxiliary convex subproblems to determine a suitable value of the penalty parameter. We present a detailed convergence analysis of the method and illustrate its practical performance by applying the method to two nonsmooth discrete optimal control problem.

Comparative study of non-convex penalties and related algorithms in compressed sensing

Non-convex penalties are widely used in sparse representation models and fields such as compressed sensing, because they can efficiently recover signals with a high sparsity level compared with the well-known ℓ1-norm. However, few classic penalties have been selected in most applications without exploring the performance of other non-convex penalties. Therefore, the objective of this study is to provide a comparative and comprehensive study on non-convex penalties. We first collected nine representative non-convex penalties and then applied two optimization frameworks: the difference of convex function algorithm (DCA) and iterative shrinkage/thresholding algorithm (ISTA) to solve the optimization problems associated with the nine non-convex penalties. The performances of the nine penalties and two algorithms were systematically compared and analyzed experimentally. We hope that these results that concern non-convex penalty selection in sparse representation models can be used as a general reference for research.

A Fault-Tolerant Bidirectional Converter for Battery Energy Storage Systems in DC Microgrids

Battery energy storage systems (BESSs) can control the power balance in DC microgrids through power injection or absorption. A BESS uses a bidirectional DC–DC converter to control the power flow to/from the grid. On the other hand, any fault occurrence in the power switches of the bidirectional converter may disturb the power balance and stability of the DC microgrid and, thus, the safe operation of the battery bank. This paper presents a fault-tolerant topology along with a fault diagnosis algorithm for a bidirectional DC–DC converter in a BESS. The proposed scheme can detect open circuit faults (OCFs) and reconfigure the topology to guarantee the safe and continuous operation of the system while it is connected to the DC microgrid. The proposed method can be extended to multi-phase structures of interleaved bidirectional DC–DC converters using only two power switches and n TRIACs to support the OCF occurrence on 2 × n switches of n legs. The proposed fault diagnosis algorithm detects OCFs only by observing the current of the inductors and does not require any sensor. Hence, the cost, weight, volume and complexity of the system is considerably reduced. Experimental results show that the reconfiguration of the converter, along with its fast fault detection, leads to fewer switches overloading and less DC voltage deviation.

A variable metric and nesterov extrapolated proximal DCA with backtracking for a composite DC program

In this paper, we consider a composite difference-of-convex (DC) program, whose objective function is the sum of a smooth convex function with Lipschitz continuous gradient, a proper closed and convex function, and a continuous concave function. This problem has many applications in machine learning and data science. The proximal DCA (pDCA), a special case of the classical difference-of-convex algorithm (DCA), as well as two Nesterov-type extrapolated DCA – ADCA (Phan et al. IJCAI:1369–1375, 2018) and pDCAe (Wen et al. Comput. Optim. Appl. 69:297–324, 2018) – can solve this problem. The algorithmic stepsizes of pDCA, pDCAe, and ADCA are fixed and determined by estimating a prior the smoothness parameter of the loss function. However, such an estimate may be hard to obtain or poor in some real-world applications. Motivated by this difficulty, we propose a variable metric and Nesterov extrapolated proximal DCA with backtracking (SPDCAe), which combines the backtracking line search procedure (not necessarily monotone) and the Nesterov's extrapolation for potential acceleration; moreover, the variable metric method is incorporated for better local approximation. Numerical simulations on sparse binary logistic regression and compressed sensing with Poisson noise demonstrate the effectiveness of our proposed method.

A new difference of anisotropic and isotropic total variation regularization method for image restoration.

Total variation (TV) regularizer has diffusely emerged in image processing. In this paper, we propose a new nonconvex total variation regularization method based on the generalized Fischer-Burmeister function for image restoration. Since our model is nonconvex and nonsmooth, the specific difference of convex algorithms (DCA) are presented, in which the subproblem can be minimized by the alternating direction method of multipliers (ADMM). The algorithms have a low computational complexity in each iteration. Experiment results including image denoising and magnetic resonance imaging demonstrate that the proposed models produce more preferable results compared with state-of-the-art methods.

SOLVING THE DAYS-OFF SCHEDULING PROBLEM USING QUADRATIC PROGRAMMING WITH CIRCULANT MATRIX

The purpose of this paper is the approach of a mathematical model dedicated to planning the consecutive days off of a company’s employees. Companies must find a flexible work schedule between employees, always considering the satisfaction of work tasks as well as guaranteeing consecutive days off. The analysis is based on solving a quadratic programming problem with binary variables. The proposed method uses the properties of the circulant symmetric matrix in the researched model, which allows the transformation of the considered problems into an equivalent separable non-convex optimization problem. A practical continuous convex relaxation approach is proposed. DC Algorithm is used to solve relaxed problems. A solved numerical example is presented.

The Boosted DC Algorithm for Linearly Constrained DC Programming

The Boosted Difference of Convex functions Algorithm (BDCA) has been recently introduced to accelerate the performance of the classical Difference of Convex functions Algorithm (DCA). This acceleration is achieved thanks to an extrapolation step from the point computed by DCA via a line search procedure. In this work, we propose an extension of BDCA that can be applied to difference of convex functions programs with linear constraints, and prove that every cluster point of the sequence generated by this algorithm is a Karush–Kuhn–Tucker point of the problem if the feasible set has a Slater point. When the objective function is quadratic, we prove that any sequence generated by the algorithm is bounded and R-linearly (geometrically) convergent. Finally, we present some numerical experiments where we compare the performance of DCA and BDCA on some challenging problems: to test the copositivity of a given matrix, to solve one-norm and infinity-norm trust-region subproblems, and to solve piecewise quadratic problems with box constraints. Our numerical results demonstrate that this new extension of BDCA outperforms DCA.

Structured Sparse Coding With the Group Log-regularizer for Key Frame Extraction

Key frame extraction based on sparse coding can reduce the redundancy of continuous frames and concisely express the entire video. However, how to develop a key frame extraction algorithm that can automatically extract a few frames with a low reconstruction error remains a challenge. In this paper, we propose a novel model of structured sparse-coding-based key frame extraction, wherein a nonconvex group log-regularizer is used with strong sparsity and a low reconstruction error. To automatically extract key frames, a decomposition scheme is designed to separate the sparse coefficient matrix by rows. The rows enforced by the nonconvex group log-regularizer become zero or nonzero, leading to the learning of the structured sparse coefficient matrix. To solve the nonconvex problems due to the log-regularizer, the difference of convex algorithm (DCA) is employed to decompose the log-regularizer into the difference of two convex functions related to the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$I_{1}$</tex> norm, which can be directly obtained through the proximal operator. Therefore, an efficient structured sparse coding algorithm with the group log-regularizer for key frame extraction is developed, which can automatically extract a few frames directly from the video to represent the entire video with a low reconstruction error. Experimental results demonstrate that the proposed algorithm can extract more accurate key frames from most SumMe videos compared to the state-of-the-art methods. Furthermore, the proposed algorithm can obtain a higher compression with a nearly 18% increase compared to sparse modeling representation selection (SMRS) and an 8% increase compared to SC-det on the VSUMM dataset.

Joint Resource Allocation for Full-Duplex Ambient Backscatter Communication: A Difference Convex Algorithm

Nowadays, Ambient Backscatter Communication (AmBC) systems have emerged as a green communication technology to enable massive self-sustainable wireless networks by leveraging Radio Frequency (RF) Energy Harvesting (EH) capability. A Full-duplex Ambient Backscatter Communication (FAmBC) network with a Full-duplex Access Point (AP), a dedicated Legacy User (LU), and several Backscatter Devices (BDs) is considered in this study. The AP with two antennas transfers downlink Orthogonal Frequency Division Multiplexing (OFDM) information and energy to the dedicated LU and several BDs, respectively, while receiving uplink backscattered information from BDs at the same time. One of the key aims in AmBC networks is to ensure fairness among BDs. To address this, we propose the Multi-objective Lexicographical Optimization Problem (MLOP), which aims to maximize the minimum BD’s throughput while enhancing overall BDs’ throughput, subject to the AP’s subcarrier power, BDs’ reflection coefficients, and backscatter time allocation. Owe to the MLOP is non-convex, we propose Difference Convex Algorithm (DCA) using Exterior Penalty Function Method (EPFM)—an inventive non-convex optimization method— to reach the optimal solution. The most critical advantage of applying this proposed approach is finding the globally optimal solution. The effectiveness of the proposed method supported by theoretical analysis confirms its superiority compared to some of the investigated suboptimal algorithms with the same computational complexity.

New Bregman proximal type algorithms for solving DC optimization problems

Difference of Convex (DC) optimization problems have objective functions that are differences between two convex functions. Representative ways of solving these problems are the proximal DC algorithms, which require that the convex part of the objective function have L-smoothness. In this article, we propose the Bregman Proximal DC Algorithm (BPDCA) for solving large-scale DC optimization problems that do not possess L-smoothness. Instead, it requires that the convex part of the objective function has the L-smooth adaptable property that is exploited in Bregman proximal gradient algorithms. In addition, we propose an accelerated version, the Bregman Proximal DC Algorithm with extrapolation (BPDCAe), with a new restart scheme. We show the global convergence of the iterates generated by BPDCA(e) to a limiting critical point under the assumption of the Kurdyka-Łojasiewicz property or subanalyticity of the objective function and other weaker conditions than those of the existing methods. We applied our algorithms to phase retrieval, which can be described both as a nonconvex optimization problem and as a DC optimization problem. Numerical experiments showed that BPDCAe outperformed existing Bregman proximal-type algorithms because the DC formulation allows for larger admissible step sizes.

Stochastic Difference-of-Convex-Functions Algorithms for Nonconvex Programming

The paper deals with stochastic difference-of-convex-functions (DC) programs, that is, optimization problems whose cost function is a sum of a lower semicontinuous DC function and the expectation of a stochastic DC function with respect to a probability distribution. This class of nonsmooth and nonconvex stochastic optimization problems plays a central role in many practical applications. Although there are many contributions in the context of convex and/or smooth stochastic optimization, algorithms dealing with nonconvex and nonsmooth programs remain rare. In deterministic optimization literature, the DC algorithm (DCA) is recognized to be one of the few algorithms able to effectively solve nonconvex and nonsmooth optimization problems. The main purpose of this paper is to present some new stochastic DCAs for solving stochastic DC programs. The convergence analysis of the proposed algorithms is carefully studied, and numerical experiments are conducted to justify the algorithms' behaviors.

RIP analysis for the weighted ℓr-ℓ1 minimization method

The weighted ℓr−ℓ1 minimization method with 0<r≤1 largely generalizes the classical ℓr minimization method and achieves very good performance in compressive sensing. However, its restricted isometry property (RIP) and high-order RIP analysis results remain unknown. In this paper, we fill in this gap by adopting newly developed analysis tools. Moreover, through a novel decomposition of the objective function into a difference of two convex functions, we propose to solve the weighted ℓr−ℓ1 minimization problem via the difference of convex functions algorithms (DCA) directly. Numerical experiments show that our DCA based weighted ℓr−ℓ1 minimization method gives satisfactory results in sparse recovery no matter whether the measurement matrix is coherent or not. For highly coherent measurements, our proposed method even outperforms the state-of-art ℓ1−ℓ2 minimization method.

Research of metering scheme key techniques for medium and low voltage DC distribution network

DC power grid has received increasing attentions recently, and with it comes the demand for highly reliable and stable metering solutions. At present, in the DC distribution projects that have been implemented, some DC metering devices have been put into use. However, from the national and industrial level, there is no clear standard for the design of the metering scheme for the medium and low-voltage DC distribution network, and there is a situation of imperfect metering device configuration. Therefore, based on the analysis of DC metering devices, algorithms and location selection, this paper takes the Hangzhou Jiangdong flexible DC distribution network as an example to analyze the error distribution caused by metering algorithm selection and device configuration. In addition, the influence of harmonics on DC measurement is analyzed to provide the suggestions for DC measurement solutions. The analysis has been verified by MATLAB/Simulink, which provides references for DC metering standards.

Blind deconvolution with non-smooth regularization via Bregman proximal DCAs

Blind deconvolution is a technique to recover an original signal without knowing a convolving filter. It is naturally formulated as a minimization of a quartic objective function under some assumption. Because its differentiable part does not have a Lipschitz continuous gradient, existing first-order methods are not theoretically supported. In this paper, we employ the Bregman-based proximal methods, whose convergence is theoretically guaranteed under the L-smooth adaptable (L-smad) property. We first reformulate the objective function as a difference of convex (DC) functions and apply the Bregman proximal DC algorithm (BPDCA). This DC decomposition satisfies the L-smad property. The method is extended to the BPDCA with extrapolation (BPDCAe) for faster convergence. When our regularizer has a sufficiently simple structure, each iteration is solved in a closed-form expression, and thus our algorithms solve large-scale problems efficiently. We also provide the stability analysis of the equilibriums and demonstrate the proposed methods through numerical experiments on image deblurring. The results show that BPDCAe successfully recovered the original image and outperformed other existing algorithms.

A parallel structured banded DC algorithm for symmetric eigenvalue problems

A parallel structured banded DC algorithm for symmetric eigenvalue problems

AI-based automatic segmentation of craniomaxillofacial anatomy from CBCT scans for automatic detection of pharyngeal airway evaluations in OSA patients

This study aims to generate and also validate an automatic detection algorithm for pharyngeal airway on CBCT data using an AI software (Diagnocat) which will procure a measurement method. The second aim is to validate the newly developed artificial intelligence system in comparison to commercially available software for 3D CBCT evaluation. A Convolutional Neural Network-based machine learning algorithm was used for the segmentation of the pharyngeal airways in OSA and non-OSA patients. Radiologists used semi-automatic software to manually determine the airway and their measurements were compared with the AI. OSA patients were classified as minimal, mild, moderate, and severe groups, and the mean airway volumes of the groups were compared. The narrowest points of the airway (mm), the field of the airway (mm2), and volume of the airway (cc) of both OSA and non-OSA patients were also compared. There was no statistically significant difference between the manual technique and Diagnocat measurements in all groups (p > 0.05). Inter-class correlation coefficients were 0.954 for manual and automatic segmentation, 0.956 for Diagnocat and automatic segmentation, 0.972 for Diagnocat and manual segmentation. Although there was no statistically significant difference in total airway volume measurements between the manual measurements, automatic measurements, and DC measurements in non-OSA and OSA patients, we evaluated the output images to understand why the mean value for the total airway was higher in DC measurement. It was seen that the DC algorithm also measures the epiglottis volume and the posterior nasal aperture volume due to the low soft-tissue contrast in CBCT images and that leads to higher values in airway volume measurement.

An Inexact Proximal DC Algorithm with Sieving Strategy for Rank Constrained Least Squares Semidefinite Programming

In this paper, the optimization problem of supervised distance preserving projection (SDPP) for data dimensionality reduction is considered, which is equivalent to a rank constrained least squares semidefinite programming (RCLSSDP). Due to the combinatorial nature of rank function, the rank constrained optimization problems are NP-hard in most cases. In order to overcome the difficulties caused by rank constraint, a difference-of-convex (DC) regularization strategy is employed, then RCLSSDP is transferred into a DC programming. For solving the corresponding DC problem, an inexact proximal DC algorithm with sieving strategy (s-iPDCA) is proposed, whose subproblems are solved by an accelerated block coordinate descent method. The global convergence of the sequence generated by s-iPDCA is proved. To illustrate the efficiency of the proposed algorithm for solving RCLSSDP, s-iPDCA is compared with classical proximal DC algorithm, proximal gradient method, proximal gradient-DC algorithm and proximal DC algorithm with extrapolation by performing dimensionality reduction experiment on COIL-20 database. From the computation time and the quality of solution, the numerical results demonstrate that s-iPDCA outperforms other methods. Moreover, dimensionality reduction experiments for face recognition on ORL and YaleB databases demonstrate that rank constrained kernel SDPP is efficient and competitive when comparing with kernel semidefinite SDPP and kernel principal component analysis in terms of recognition accuracy.

Multidimensional Information Network Big Data Mining Algorithm Relying on Finite Element Analysis.

In recent years, with the rapid development of the Internet, online social networks have been continuously integrated with traditional interpersonal networks and research on information dissemination in social networks has gradually increased. This article studies and analyzes the multidimensional information network big data mining algorithm based on the finite element analysis method. This paper firstly introduces the finite element analysis and calculation process, a finite element data mining simulation application software management system will integrate current data, calculation, and background data into one, then analyzes the data mining clustering algorithm, and conducts an experimental exploration of the influential node mining algorithm in complex networks. The experimental results show that the LIC algorithm is better than the CC algorithm, the DC algorithm, and the BC algorithm; its overall performance is improved by 30%, and the effect is better. The LIC algorithm can effectively and quickly determine the influential nodes, which is helpful for social network analysis.

A refined inertial DC algorithm for DC programming

In this paper we consider the difference-of-convex (DC) programming problems, whose objective function is the difference of two convex functions. The classical DC Algorithm (DCA) is well-known for solving this kind of problems, which generally returns a critical point. Recently, an inertial DC algorithm (InDCA) equipped with heavy-ball inertial-force procedure was proposed in de Oliveira et al. (Set-Valued and Variational Analysis 27(4):895--919, 2019), which potentially helps to improve both the convergence speed and the solution quality. Based on InDCA, we propose a refined inertial DC algorithm (RInDCA) equipped with enlarged inertial step-size compared with InDCA. Empirically, larger step-size accelerates the convergence. We demonstrate the subsequential convergence of our refined version to a critical point. In addition, by assuming the Kurdyka-{\L}ojasiewicz (KL) property of the objective function, we establish the sequential convergence of RInDCA. Numerical simulations on checking copositivity of matrices and image denoising problem show the benefit of larger step-size.

Efficient Boosted DC Algorithm for Nonconvex Image Restoration with Rician Noise

Image deblurring under Rician noise has attracted considerable attention in imaging science. Frequently appearing in medical imaging, Rician noise leads to an interesting nonconvex optimization problem, termed as the MAP-Rician model, which is based on the Maximum a Posteriori (MAP) estimation approach. As the MAP-Rician model is deeply rooted in Bayesian analysis, we want to understand its mathematical analysis carefully. Moreover, one needs to properly select a suitable algorithm for tackling this nonconvex problem to get the best performance. This paper investigates both issues. Indeed, we first present a theoretical result about the existence of a minimizer for the MAP-Rician model under mild conditions. Next, we aim to adopt an efficient boosted difference of convex functions algorithm (BDCA) to handle this challenging problem. Basically, BDCA combines the classical difference of convex functions algorithm (DCA) with a backtracking line search, which utilizes the point generated by DCA to define a search direction. In particular, we apply a smoothing scheme to handle the nonsmooth total variation (TV) regularization term in the discrete MAP-Rician model. Theoretically, using the Kurdyka--Lojasiewicz (KL) property, the convergence of the numerical algorithm can be guaranteed. We also prove that the sequence generated by the proposed algorithm converges to a stationary point with the objective function values decreasing monotonically. Numerical simulations are then reported to clearly illustrate that our BDCA approach outperforms some state-of-the-art methods for both medical and natural images in terms of image recovery capability and CPU-time cost.