DC programming : a variant of DCA via decomposition

The multiple-input multiple-output (MIMO) system is widely used in wireless communications. For the problem of the discrete maximum-likelihood (ML) detection for the MIMO system, one can formulate it as binary quadratic programming (BQP). The general BQP problem is an NP-hard problem, which is a challenge for finding promising solutions. The variable complexity is a special considered issue. In this paper, inspired by the optimization of sparse constrains, we employ a regularization approach to deal with the binary constraints in the proposed formulation and then introduce the difference of convex functions (DC) programming to solve the formulated nonconvex cost function. A novel and robust DC algorithm is proposed. Numerical experiments show that the proposed algorithm, which is based on DC programming, can achieve accurate results with a higher convergence rate.

We consider the supervised pattern classification in the high-dimensional setting, in which the number of features is much larger than the number of observations. We present a novel approach to the sparse Fisher linear discriminant problem using the $$\ell _0$$ -norm. The resulting optimization problem is nonconvex, discontinuous and very hard to solve. We overcome the discontinuity by using appropriate approximations to the $$\ell _0$$ -norm such that the resulting problems can be formulated as difference of convex functions (DC) programs to which DC programming and DC Algorithms (DCA) are investigated. The experimental results on both simulated and real datasets demonstrate the efficiency of the proposed algorithms compared to some state-of-the-art methods.

We propose a new solution for a class of nonconvex programs involving l 0 norm. Our method is based on a reformulation of these programs as bilevel programs, in which the objective function in the first level program is a DC function, and the second level consists of finding a Karush-Kuhn-Tucker point of a linear programming problem. Exact penalty techniques are then used to reformulate the obtained programs as DC programs. The resulted problems are then handled by the local algorithm DCA in DC programming. Preliminary computational results are reported.

In the present paper, we are concerned with conditions ensuring the exact penalty for nonconvex programming. Firstly, we consider problems with concave objective and constraints. Secondly, we establish various results on error bounds for systems of DC inequalities and exact penalty, with/without error bounds, in DC programming. They permit to recast several class of difficult nonconvex programs into suitable DC programs to be tackled by the efficient DCA.

Lagrange-type duality in DC programming

DC Programming and DCA for Secrecy Rate Maximization via Null Space AF Relay Beamforming in a Wireless Network with Multiple Eavesdroppers

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.

Convex sparse regularization techniques such as l <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> -regularization enjoy robustness to random initialization and computational efficiency, but often underestimate the true solution. To mitigate the estimation bias, the generalized minimax concave (GMC) penalty (Selesnick 2017) and its linearly involved extension LiGME (Abe et al. 2020) have been proposed recently to yield improved convex sparse regularization. However, empirical results indicate that existing proximal splitting algorithms for GMC/LiGME do not necessarily achieve the convergence speed desired in some real-world applications (e.g., stream data processing).To resolve this situation, we propose a fast and well scalable algorithm (termed DC-LiGME) based on DC programming, which allows us to solve the difficult original problem via solution to a sequence of simpler subproblems. We establish convergence guarantee of the DC-LiGME algorithm and present an intuitive explanation of its fast convergence. Numerical experiments demonstrate superior convergence speed of the DC-LiGME algorithm, and suggest that mere two iterations of the algorithm suffice to stably produce satisfactory solutions (at the cost of solving three standard convex regularization models), which is a favourable feature in practice.

Optimization of Buffer Network via DC Programming

The authors formulate and implement a numerical mode-matching (NMM) method to model electrode-type resistivity tools in invaded thin beds. The authors derive the low-frequency approximation of the Maxwell's equations to obtain the partial differential equation for the potential field. The new NMM program is validated by comparing the numerical results with those obtained from other dc programs. It is found that this new program is much faster than the program using the finite-element method (FEM), and hence is useful for routine interpretation of resistivity logs and for inversion. >

In this paper, a new concave reformulation is proposed on a convex hull of some given points. Based on its properties, we attempt to solve DC Programming problems globally under uncertain environment by using Robust optimization method and CVaR method. A global optimization algorithm is developed for the Robust counterpart and CVaR model with two kinds of special convex hulls: simplex set and box set. The global solution is obtained by solving a sequence of convex relaxation programming on the original constraint sets or divided subsets with branch and bound method. Finally, numerical experiments are given for DC programs under uncertain environment with two kinds of constraints: simplex and box sets. Simulation results show the feasibility and efficiency of the proposed global optimization algorithm.

As a development of ν-support vector machine (ν-SVM), parametric-margin ν-support vector machine (Par-ν-SVM) can be useful in many cases, especially heteroscedastic noise classification problems. The present article proposes a novel and fast method to solve the primal problem of Par-ν-SVM (named as DC-Par-ν-SVM), while Par-ν-SVM maximizes the parametric-margin by solving a dual quadratic programming problem. In fact, the primal non-convex problem is converted into an unconstrained problem to express the objective function as the difference of convex functions (DC). The DC-Algorithm (DCA) based on generalized Newton’s method is proposed to solve the unconstrained problem cited. Numerical experiments performed on several artificial, real-life, UCI and NDC data sets showed the superiority of the DC-Par-ν-SVM in terms of both accuracy and learning speed.

In this paper, we consider a two-way relay network system which contains multiple cooperative relays transmitting information between two sources with attendance of an eavesdropper. A null space beamforming scheme is applied to ensure the secrecy in system. Our goal is achieving maximal secrecy sum rate with a certain level power transmit. In [1], authors propose an approach to solve Secrecy Sum Rate Maximization (SSRM) problem, whereas we use another approach based on Different of Convex Functions Algorithm (DCA).

Sparse optimization is about finding minimizers of functions characterized by a number of nonzero components as small as possible, such paradigm being of great practical relevance in Machine Learning, particularly in classification approaches based on support vector machines. By exploiting some properties of the k-norm of a vector, namely, of the sum of its k largest absolute-value components, we formulate a sparse optimization problem as a mixed-integer nonlinear program, whose continuous relaxation is equivalent to the unconstrained minimization of a difference-of-convex function. The approach is applied to Feature Selection in the support vector machine framework, and tested on a set of benchmark instances. Numerical comparisons against both the standard ell _1-based support vector machine and a simple version of the Slope method are presented, that demonstrate the effectiveness of our approach in achieving high sparsity level of the solutions without impairing test-correctness.

Discrete DC programming with convex extensible functions is studied. A natural approach for this problem is a continuous relaxation that extends the problem to a continuous domain and applies the algorithm in continuous DC programming. By employing a special form of continuous relaxation, which is named “lin-vex extension,” the optimal solution of the continuous relaxation coincides with the original discrete problem. The proposed method is demonstrated for the degree-concentrated spanning tree problem.