This paper proposes a new robust truncated L-2-norm twin support vector machine ((TSVM)-S-2), where the truncated L-2-norm is used to measure the empirical risk to make the classifiers more robust when encountering lo...
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This paper proposes a new robust truncated L-2-norm twin support vector machine ((TSVM)-S-2), where the truncated L-2-norm is used to measure the empirical risk to make the classifiers more robust when encountering lots of outliers. Meanwhile, chance constraints are also employed to specify false positive and false negative error rates. (TSVM)-S-2 considers a pair of chance constrained nonconvex nonsmooth problems. To solve these difficult problems, we propose an efficient iterative method for (TSVM)-S-2 based on difference of convex functions (dc) programs and dc algorithms (dcA). Experiments on benchmark data sets and artificial data sets demonstrate the significant virtues of (TSVM)-S-2 in terms of robustness and generalization performance.
In this paper, we study characterizations of differentiability for real-valued functions based on generalized differentiation. These characterizations provide the mathematical foundation for Nesterov's smoothing t...
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In this paper, we study characterizations of differentiability for real-valued functions based on generalized differentiation. These characterizations provide the mathematical foundation for Nesterov's smoothing techniques in infinite dimensions. As an application, we provide a simple approach to image reconstructions based on Nesterov's smoothing and algorithms for minimizing differences of convex (dc) functions that involve the regularization.
In this article, we study a merit function based on sub-additive functions for solving the non-linear complementarity problem (NCP). This leads to consider an optimization problem that is equivalent to the NCP. In the...
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In this article, we study a merit function based on sub-additive functions for solving the non-linear complementarity problem (NCP). This leads to consider an optimization problem that is equivalent to the NCP. In the case of a concave NCP this optimization problem is a Difference of Convex (dc) program and we can therefore use dc algorithm to locally solve it. We prove that in the case of a concave monotone NCP, it is sufficient to compute a stationary point of the optimization problem to obtain a solution of the complementarity problem. In the case of a general NCP, assuming that a dc decomposition of the complementarity problem is known, we propose a penalization technique to reformulate the optimization problem as a dc program and prove that local minima of this penalized problem are also local minima of the merit problem. Numerical results on linear complementarity problems, absolute value equations and non-linear complementarity problems show that our method is promising.
The paper deals with a transportation network protection problem. The aim is to limit losses due to disasters by choosing an optimal retrofiting plan. The mathematical model given by Lu, Gupte, Huang [11] is a mixed i...
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ISBN:
(纸本)9783030383640;9783030383633
The paper deals with a transportation network protection problem. The aim is to limit losses due to disasters by choosing an optimal retrofiting plan. The mathematical model given by Lu, Gupte, Huang [11] is a mixed integer non linear optimization problem. Existing solution methods are complicated and their computing time is long. Hence, it is necessary to develop efficient solution methods for the considered model. Our approach is based on dc (difference of two convex functions) programming and dc algorithm (dcA). The original model is first reformulated as a dc program by using exact penalty techniques. We then apply dcA to solve the resulting problem. Numerical results on a small network are reported to see the behavior of dcA. It shows that dcA is fast and the proposed approach is promissing.
Train scheduling plays an important role in the operation of railways systems. This work focuses on a model of scheduling in which one minimizes the total travel time of trains in a single track railways network. The ...
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ISBN:
(纸本)9783030383640;9783030383633
Train scheduling plays an important role in the operation of railways systems. This work focuses on a model of scheduling in which one minimizes the total travel time of trains in a single track railways network. The model can be written in the form of a mixed 0-1 linear program which has the worst case exponential complexity to calculate the optimal solution. In this paper, we propose a computationally efficient approach to solve the train scheduling problem. Our approach is based on a so-called Difference of Convex functions algorithm (dcA) to provide good feasible solutions with finite convergence. The algorithm is tested on three different railway network topologies including one topology introduced in [18] and two practical topologies in Northern Vietnam. The numerical results are encouraging and demonstrate the efficiency of the approach.
In this work, two novel formulations for embedded feature selection are presented. A second-order cone programming approach for Support Vector Machines is extended by adding a second regularizer to encourage feature e...
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In this work, two novel formulations for embedded feature selection are presented. A second-order cone programming approach for Support Vector Machines is extended by adding a second regularizer to encourage feature elimination. The one- and the zero norm penalties are used in combination with the Tikhonov regularization under a robust setting designed to correctly classify instances, up to a predefined error rate, even for the worst data distribution. The use of the zero norm leads to a nonconvex formulation, which is solved by using Difference of Convex (dc) functions, extending dc programming to second-order cones. Experiments on high-dimensional microarray datasets were performed, and the best performance was obtained with our approaches compared with well-known feature selection methods for Support Vector Machines. (C) 2017 Elsevier Inc. All rights reserved.
In this paper we address the problem of visualizing in a bounded region a set of individuals, which has attached a dissimilarity measure and a statistical value, as convex objects. This problem, which extends the stan...
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In this paper we address the problem of visualizing in a bounded region a set of individuals, which has attached a dissimilarity measure and a statistical value, as convex objects. This problem, which extends the standard Multidimensional Scaling Analysis, is written as a global optimization problem whose objective is the difference of two convex functions (dc). Suitable dc decompositions allow us to use the Difference of Convex algorithm (dcA) in a very efficient way. Our algorithmic approach is used to visualize two real-world datasets.
The goal of affine matrix rank minimization problem is to reconstruct a low-rank or approximately low-rank matrix under linear constraints. In general, this problem is combinatorial and NP-hard. In this paper, a nonco...
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The goal of affine matrix rank minimization problem is to reconstruct a low-rank or approximately low-rank matrix under linear constraints. In general, this problem is combinatorial and NP-hard. In this paper, a nonconvex fraction function is studied to approximate the rank of a matrix and translate this NP-hard problem into a transformed affine matrix rank minimization problem. The equivalence between these two problems is established, and we proved that the uniqueness of the global minimizer of transformed affine matrix rank minimization problem also solves affine matrix rank minimization problem if some conditions are satisfied. Moreover, we also proved that the optimal solution to the transformed affine matrix rank minimization problem can be approximately obtained by solving its regularization problem for some proper smaller lambda > 0. Lastly, the dc algorithm is utilized to solve the regularization transformed affine matrix rank minimization problem and the numerical experiments on image inpainting problems show that our method performs effectively in recovering low-rank images compared with some state-of-art algorithms. (c) 2018 Elsevier B.V. All rights reserved.
In this paper, we study nearest prototype classifiers, which classify data instances into the classes to which their nearest prototypes belong. We propose a maximum-margin model for nearest prototype classifiers. To p...
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In this paper, we study nearest prototype classifiers, which classify data instances into the classes to which their nearest prototypes belong. We propose a maximum-margin model for nearest prototype classifiers. To provide the margin, we define a class-wise discriminant function for instances by the negatives of distances of their nearest prototypes of the class. Then, we define the margin by the minimum of differences between the discriminant function values of instances with respect to the classes they belong to and the values of the other classes. The optimization problem corresponding to the maximum-margin model is a difference of convex functions (dc) program. It is solved using a dc algorithm, which is a k-means-like algorithm, i.e., the members and positions of prototypes are alternately optimized. Through a numerical study, we analyze the effects of hyperparameters of the maximum-margin model, especially considering the classification performance.
Motivated by a class of applied problems arising from physical layer based security in a digital communication system, in particular, by a secrecy sum-rate maximization problem, this paper studies a nonsmooth, differe...
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Motivated by a class of applied problems arising from physical layer based security in a digital communication system, in particular, by a secrecy sum-rate maximization problem, this paper studies a nonsmooth, difference-of-convex (dc) minimization problem. The contributions of this paper are (i) clarify several kinds of stationary solutions and their relations;(ii) develop and establish the convergence of a novel algorithm for computing a d-stationary solution of a problem with a convex feasible set that is arguably the sharpest kind among the various stationary solutions;(iii) extend the algorithm in several directions including a randomized choice of the subproblems that could help the practical convergence of the algorithm, a distributed penalty approach for problems whose objective functions are sums of dc functions, and problems with a specially structured (nonconvex) dc constraint. For the latter class of problems, a pointwise Slater constraint qualification is introduced that facilitates the verification and computation of a B(ouligand)-stationary point.
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