DC programming and DCA for enhancing physical layer security via cooperative jamming

The explosive development of computational tools these days is threatening security of cryptographic algorithms, which are regarded as primary traditional methods for ensuring information security. The physical layer security approach is introduced as a method for both improving confidentiality of the secret key distribution in cryptography and enabling the data transmission without relaying on higher-layer encryption. In this paper, the cooperative jamming paradigm - one of the techniques used in the physical layer is studied and the resulting power allocation problem with the aim of maximizing the sum of secrecy rates subject to power constraints is formulated as a nonconvex optimization problem. The objective function is a so-called DC (Difference of Convex functions) function, and some constraints are coupling. We propose a new DC formulation and develop an efficient DCA (DC Algorithm) to deal with this nonconvex program. The DCA introduces the elegant concept of approximating the original nonconvex program by a sequence of convex ones: at each iteration of DCA requires solution of a convex subproblem. The main advantage of the proposed approach is that it leads to strongly convex quadratic subproblems with separate variables in the objective function, which can be tackled by both distributed and centralized methods. One of the major contributions of the paper is to develop a highly efficient distributed algorithm to solve the convex subproblem. We adopt the dual decomposition method that results in computing iteratively the projection of points onto a very simple structural set which can be determined by an inexpensive procedure. The numerical results show the efficiency and the superiority of the new DCA based algorithm compared with existing approaches.     

A continuous DC programming approach for resource allocation in OFDMA/TDD wireless networks

The next generation broadband wireless networks deploys OFDM/OFDMA as the enabling technologies for broadband data transmission with QoS capabilities. Many optimization problems have arisen in the conception of such a network. This article studies an optimization problem in resource allocation. By using mathematical modeling technique we formulate the considered problem as a pure integer linear program. This problem is reformulated as a DC (Difference of Convex functions) program via an exact penalty technique. We then propose a continuous approach for its resolution. Our approach is based on DC programming and DCA (DC Algorithm). It works in a continuous domain, but provides integer solutions. To check globality of computed solutions, a global method combining DCA with a well adapted Branch-and-Bound (B&B) algorithm is investigated. Preliminary numerical results are reported to show the efficiency of the proposed method with respect to the standard Branch-and-Bound algorithm.     

DC programming and DCA for solving Brugnano–Casulli piecewise linear systems

Piecewise linear optimization is one of the most frequently used optimization models in practice, such as transportation, finance and supply-chain management. In this paper, we investigate a particular piecewise linear optimization that is optimizing the norm of piecewise linear functions (NPLF). Specifically, we are interested in solving a class of Brugnano–Casulli piecewise linear systems (PLS), which can be reformulated as an NPLF problem. Speaking generally, the NPLF is considered as an optimization problem with a nonsmooth, nonconvex objective function. A new and efficient optimization approach based on DC (Difference of Convex functions) programming and DCA (DC Algorithms) is developed. With a suitable DC formulation, we design a DCA scheme, named ℓ₁-DCA, for the problem of optimizing the ℓ₁-norm of NPLF. Thanks to particular properties of the problem, we prove that under some conditions, our proposed algorithm converges to an exact solution after a finite number of iterations. In addition, when a nonglobal solution is found, a numerical procedure is introduced to find a feasible point having a smaller objective value and to restart ℓ₁-DCA at this point. Several numerical experiments illustrate these interesting convergence properties. Moreover, we also present an application to the free-surface hydrodynamic problem, where the correct numerical modeling often requires to have the solution of special PLS, with the aim of showing the efficiency of the proposed method.     

基于DC规划的鲁棒模糊核聚类算法

对以径向基核函数和欧拉核函数为代表的鲁棒模糊核聚类算法进行非凸优化,以改善聚类算法目标函数非凸导致的局部解问题.采用凸差规划(DCP)将目标函数转化为2个凸函数之差的形式,减缓局部解的不良性,提高聚类性能.采用凸差算法(DCA)优化求解DCP问题,能快速搜索到相对更优的解,并保持聚类的鲁棒性.在UCI数据集上的实验验证基于DCP的鲁棒模糊核聚类算法对大规模数据集表现出相对更优的聚类性能.     

A sparse logistic regression framework by difference of convex functions programming

Feature selection for logistic regression (LR) is still a challenging subject. In this paper, we present a new feature selection method for logistic regression based on a combination of the zero-norm and l₂-norm regularization. However, discontinuity of the zero-norm makes it difficult to find the optimal solution. We apply a proper nonconvex approximation of the zero-norm to derive a robust difference of convex functions (DC) program. Moreover, DC optimization algorithm (DCA) is used to solve the problem effectively and the corresponding DCA converges linearly. Compared with traditional methods, numerical experiments on benchmark datasets show that the proposed method reduces the number of input features while maintaining accuracy. Furthermore, as a practical application, the proposed method is used to directly classify licorice seeds using near-infrared spectroscopy data. The simulation results in different spectral regions illustrates that the proposed method achieves equivalent classification performance to traditional logistic regressions yet suppresses more features. These results show the feasibility and effectiveness of the proposed method.     

应用突函数差异算法进行癌症分类的基因选择(英文)

研究了有关癌症分类的基因选择问题。开发了集成的基于平滑剪切绝对偏差罚分的SVM—特征选择方法,直接最小化分类器的性能。为解决优化问题,应用了突函数差异算法(difference of convex functionsal-gorithms,DCA)这一进行非突连续优化的通用框架,致使连续线性规划算法有限收敛。真实数据集上的先验实验表明算法达到了预想目标:在压缩大量属性的同时,保持了较小分类差错。     

A variant of the SOM algorithm and its interpretation in the viewpoint of social influence and learning

The conventional self-organizing feature map (SOM) algorithm is usually interpreted as a computational model, which can capture main features of computational maps in the brain. In this paper, we present a variant of the SOM algorithm called the SOM-based optimization (SOMO) algorithm. The development of the SOMO algorithm was motivated by exploring the possibility of applying the SOM algorithm in continuous optimization problems. Through the self-organizing process, good solutions to an optimization problem can be simultaneously explored and exploited by the SOMO algorithm. In our opinion, the SOMO algorithm not only can be regarded as a biologically inspired computational model but also may be regarded as a new approach to a model of social influence and social learning. Several simulations are used to illustrate the effectiveness of the proposed optimization algorithm.     

Nonconvex dynamic spectrum allocation for cognitive radio networks via particle swarm optimization and simulated annealing

Dynamic spectrum access is a promising technique designed to meet the challenge of rapidly growing demands for broadband access in cognitive radio networks. By utilizing the allocated spectrum, cognitive radio devices can provide high throughput and low latency communications. This paper introduces an efficient dynamic spectrum allocation algorithm in cognitive radio networks based on the network utility maximization framework. The objective function in this optimization problem is always nonconvex, which makes the problem difficult to solve. Prior works on network resource optimization always transformed the nonconvex optimization problem into a convex one under some strict assumptions, which do not meet the actual networks. We solve the nonconvex optimization problem directly using an improved particle swarm optimization (PSO) method. Simulated annealing (SA), combined with PSO to form the PSOSA algorithm, overcomes the inherent defects and disadvantages of these two individual components. Simulations show that the proposed solution achieves significant throughput compared with existing approaches, and it is efficient in solving the nonconvex optimization problem.     

A DC programming approach for planning a multisensor multizone search for a target

In this paper, we consider a well-known problem in the general area of search theory: planning a multisensor in multizone search so as to maximize the probability of detection of a target under a given resource effort to be shared. We propose a new optimization model that is a nonlinear mixed 0–1 programming problem. This problem is then reformulated as a DC (Difference of Convex) functions program via an exact penalty technique. DC programming and DCA (DC algorithm) have been investigated for solving the resulting DC program. Numerical experiments demonstrate the efficiency and the superiority of the proposed algorithm in comparison with the existing method.     

Heuristic Kalman Algorithm for Solving Optimization Problems

The main objective of this paper is to present a new optimization approach, which we call heuristic Kalman algorithm (HKA). We propose it as a viable approach for solving continuous nonconvex optimization problems. The principle of the proposed approach is to consider explicitly the optimization problem as a measurement process designed to produce an estimate of the optimum. A specific procedure, based on the Kalman method, was developed to improve the quality of the estimate obtained through the measurement process. The efficiency of HKA is evaluated in detail through several nonconvex test problems, both in the unconstrained and constrained cases. The results are then compared to those obtained via other metaheuristics. These various numerical experiments show that the HKA has very interesting potentialities for solving nonconvex optimization problems, notably concerning the computation time and the success ratio.      

蜂窝网络下基于max-min公平性的D2D功率分配

针对多个终端直通通信（D2D）用户共享多个蜂窝用户资源的公平性问题,在保证蜂窝用户速率的前提下,提出了基于最大最小公平性（max-min fairness）的功率分配算法。该算法首先将非凸优化问题转化为含凸函数的差（DC）规划问题,然后采用凸近似的全局优化算法和对分算法对D2D实现功率优化。仿真结果表明,与只采用凸近似的全局优化算法相比,所提算法收敛性更优,同时最大化了瓶颈用户的速率。     

Nonsmooth DC programming approach to clusterwise linear regression: optimality conditions and algorithms

The clusterwise linear regression problem is formulated as a nonsmooth nonconvex optimization problem using the squared regression error function. The objective function in this problem is represented as a difference of convex functions. Optimality conditions are derived, and an algorithm is designed based on such a representation. An incremental approach is proposed to generate starting solutions. The algorithm is tested on small to large data sets.     

New and efficient DCA based algorithms for minimum sum-of-squares clustering

The purpose of this paper is to develop new efficient approaches based on DC (Difference of Convex functions) programming and DCA (DC Algorithm) to perform clustering via minimum sum-of-squares Euclidean distance. We consider the two most widely used models for the so-called Minimum Sum-of-Squares Clustering (MSSC in short) that are a bilevel programming problem and a mixed integer program. Firstly, the mixed integer formulation of MSSC is carefully studied and is reformulated as a continuous optimization problem via a new result on exact penalty technique in DC programming. DCA is then investigated to the resulting problem. Secondly, we introduce a Gaussian kernel version of the bilevel programming formulation of MSSC, named GKMSSC. The GKMSSC problem is formulated as a DC program for which a simple and efficient DCA scheme is developed. A regularization technique is investigated for exploiting the nice effect of DC decomposition and a simple procedure for finding good starting points of DCA is developed. The proposed DCA schemes are original and very inexpensive because they amount to computing, at each iteration, the projection of points onto a simplex and/or onto a ball, and/or onto a box, which are all determined in the explicit form. Numerical results on real word datasets show the efficiency, the scalability of DCA and its great superiority with respect to k-means and kernel k-means, standard methods for clustering.     

一种基于混沌粒子群优化的OFDM系统资源分配算法

针对无线多用户正交频分复用（OFDM）系统中功率分配问题,提出一种基于效用函数最大化框架的资源分配算法.在实际网络环境中,此类最优化算法为非凸的,利用经典最优化方法很难解决.为此,将智能优化中的粒子群方法应用到非凸优化算法设计中,并针对粒子群优化容易陷入局部极值点的问题,将Logistic混沌搜索嵌入PSO算法中,提出混沌粒子群算法.与同类算法相比,所提出算法不仅有效解决了非凸性问题,而且可以使系统具有更好的性能.     

基于粒子群的CDMA功率和速率联合控制算法

提出了基于效用函数的CDMA网络下行链路的功率和速率联合控制最优化算法.在这类算法中,效用函数为非凸函数,经典的最优化理论很难解决这类问题.将粒子群优化方法应用于算法的非凸性设计,并通过仿真算例证明了该算法能有效解决非凸优化问题,且可保证系统的公平性.     

Visiting convex regions in a polygonal map

This paper is concerned with a variant of the multi-goal path planning in which goals are represented as convex polygons. The problem is to find a closed shortest path in a polygonal map such that all goals are visited. The proposed solution is based on a self-organizing map (SOM) algorithm for the traveling salesman problem. Neurons’ weights are considered as nodes inside the polygonal domain and connected nodes represent a path that evolves according to the proposed adaptation rules. In addition, a reference algorithm based on the solution of the traveling salesman problem and the consecutive touring polygons problem is provided to find high quality solutions of the created set of problems. The problems are designed to represent various inspection and patrolling tasks and can form a kind of benchmark set for multi-goal path planning algorithms. The performance of the algorithms is examined in this problem set, which includes an instance of the watchman route problem with restricted visibility range. The proposed SOM based algorithms provide a unified approach to solve various visibility based routing problems in polygonal maps while they provide a competitive quality of solutions to the reference algorithm with significantly lower computational requirements.     

A new nonconvex approach for image restoration with Gamma noise

In this paper, a new total generalized variational(TGV) model for restoring images with multiplicative noise is proposed, which contains a nonconvex fidelity term and a TGV term. We use a difference of convex functions algorithm (DCA) to deal with the proposed model. For multiplicative noise removal, there exist many models and algorithms, most of which focus on convex approximation so that numerical algorithms with guaranteed convergence can be designed. Unlike these algorithms, we use the DCA algorithm to remove multiplicative noise. By numerical experiments, it is shown that the proposed approach leads to a better solution compared with the gradient projection algorithm for solving the classic multiplicative noise removal models. We prove that the sequence generated by the DCA algorithm converges to a stationary point, which satisfies the first order optimality condition. Finally, we demonstrate the performance of our whole scheme by numerical examples. A comparison with other methods is provided as well. Numerical results demonstrate that the proposed algorithm significantly outperforms some previous methods for multiplicative Gamma noise removal.     

Visualizing market segmentation using self-organizing maps and Fuzzy Delphi method – ADSL market of a telecommunication company

A common problem for marketing strategists is how to appropriately segment the market and select segment-specific marketing strategies. This paper presents a novel approach which integrates Fuzzy Delphi method, self-organizing maps (SOM) and a visualization technique to cluster customers according to their various characteristic variables and visualize segments by producing colorful market maps. These market maps not only help the managers to see fully visualized clusters of market but also reveal mutual non-linear correlations between different customers’ characteristic variables. In this research we studied ADSL service market of an Iranian Telecommunication Company. SOM algorithm and visualizing technique were implemented in MATLAB environment to produce market maps of data set.     

Large-scale convex optimal control problems: time decomposition,incentive coordination, and parallel algorithm

A parallel algorithm based on time decomposition and incentive coordination is developed for long-horizon optimal control problems. This is done by first decomposing the original problem into subproblems with shorter time horizon, and then using the incentive coordination scheme to coordinate the interaction of subproblems. For strictly convex problems it is proved that the decomposed problem with linear incentive coordination is equivalent to the original problem, in the sense that each optimal solution of the decomposed problem produces one global optimal solution of the original problem and vice versa. In other words, linear incentive terms are sufficient in this case and impose no additional computation burden on the subproblems. The high-level parameter optimization problem is shown to be nonconvex, despite the uniqueness of the optimal solution and the convexity of the original problem. Nevertheless, the high-level problem has no local minimum, even though it is nonconvex. A parallel algorithm based on a prediction method is developed, and a numerical example is used to demonstrate the feasibility of the approach