Recovering a large matrix from limited measurements is a challenging task arising in many real applications, such as image inpainting, compressive sensing, and medical imaging, and these kinds of problems are mostly f...
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Recovering a large matrix from limited measurements is a challenging task arising in many real applications, such as image inpainting, compressive sensing, and medical imaging, and these kinds of problems are mostly formulated as low-rank matrix approximation problems. Due to the rank operator being nonconvex and discontinuous, most of the recent theoretical studies use the nuclear norm as a convex relaxation and the low-rank matrix recovery problem is solved through minimization of the nuclear norm regularized problem. However, a major limitation of nuclear norm minimization is that all the singular values are simultaneously minimized and the rank may not be well approximated (Hu et al., 2013). Correspondingly, in this paper, we propose a new multistage algorithm, which makes use of the concept of Truncated Nuclear Norm Regularization (TNNR) proposed by Hu et al., 2013, and iterative support detection (ISD) proposed by Wang and Yin, 2010, to overcome the above limitation. Besides matrix completion problems considered by Hu et al., 2013, the proposed method can be also extended to the general low-rank matrix recovery problems. Extensive experiments well validate the superiority of our new algorithms over other state-of-the-art methods.
In this paper, we propose a structured trust-region algorithm combining with filter technique to minimize the sum of two general functions with general constraints. Specifically, the new iterates are generated in the ...
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In this paper, we propose a structured trust-region algorithm combining with filter technique to minimize the sum of two general functions with general constraints. Specifically, the new iterates are generated in the Gauss-Seidel type iterative procedure, whose sizes are controlled by a trust-region type parameter. The entries in the filter are a pair: one resulting from feasibility;the other resulting from optimality. The global convergence of the proposed algorithm is proved under some suitable assumptions. Some preliminary numerical results show that our algorithm is potentially efficient for solving general nonconvex optimization problems with separable structure.
We present an extension of a previously proposed approach based on the method of moments for solving the optimal control problem for a switching system considering now a continuous external input. This method is based...
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We present an extension of a previously proposed approach based on the method of moments for solving the optimal control problem for a switching system considering now a continuous external input. This method is based on the transformation of a nonlinear, nonconvex optimal control problem, into an equivalent optimal control problem with linear and convex structure, which allows us to obtain an equivalent convex formulation more appropriate to be solved by high-performance numerical computing. Finally, the design of optimal logic-based controllers for networked systems with a dynamic topology is presented as an application of this work.
In order to restore the high quality image, we propose a compound regularization method which combines a new higher-order extension of total variation (TV+TV2) and a nonconvex sparseness-inducing penalty. Considering ...
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In order to restore the high quality image, we propose a compound regularization method which combines a new higher-order extension of total variation (TV+TV2) and a nonconvex sparseness-inducing penalty. Considering the presence of varying directional features in images, we employ the shearlet transform to preserve the abundant geometrical information of the image. The nonconvex sparseness-inducing penalty approach increases robustness to noise and image nonsparsity. In what follows, we present the numerical solution of the proposed model by employing the split Bregman iteration and a novel p-shrinkage operator. And finally, we perform numerical experiments for image denoising, image deblurring, and image reconstructing from incomplete spectral samples. The experimental results demonstrate the efficiency of the proposed restoration method for preserving the structure details and the sharp edges of image.
During the last 40 years, simplicial partitioning has been shown to be highly useful, including in the field of nonlinear optimization, specifically global optimization. In this article, we consider results on the exh...
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During the last 40 years, simplicial partitioning has been shown to be highly useful, including in the field of nonlinear optimization, specifically global optimization. In this article, we consider results on the exhaustivity of simplicial partitioning schemes. We consider conjectures on this exhaustivity which seem at first glance to be true (two of which have been stated as true in published articles). However, we will provide counter-examples to these conjectures. We also provide a new simplicial partitioning scheme, which provides a lot of freedom, whilst guaranteeing exhaustivity.
Low-rank representation (LRR) intends to find the representation with lowest rank of a given data set, which can be formulated as a rank-minimisation problem. Since the rank operator is non-convex and discontinuous, m...
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Low-rank representation (LRR) intends to find the representation with lowest rank of a given data set, which can be formulated as a rank-minimisation problem. Since the rank operator is non-convex and discontinuous, most of the recent works use the nuclear norm as a convex relaxation. It is theoretically shown that, under some conditions, the Frobenius-norm-based optimisation problem has a unique solution that is also a solution of the original LRR optimisation problem. In other words, it is feasible to apply the Frobenius norm as a surrogate of the non-convex matrix rank function. This replacement will largely reduce the time costs for obtaining the lowest-rank solution. Experimental results show that the method (i.e. fast LRR (fLRR)) performs well in terms of accuracy and computation speed in image clustering and motion segmentation compared with nuclear-norm-based LRR algorithm.
The aim of this paper is to propose a solution method for the minimization of a class of generalized linear functions on a flow polytope. The problems will be solved by means of a network algorithm, based on graph ope...
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The aim of this paper is to propose a solution method for the minimization of a class of generalized linear functions on a flow polytope. The problems will be solved by means of a network algorithm, based on graph operations, which lies within the class of the so-called 'optimal level solutions' parametric methods. The use of the network structure of flow polytopes, allows to obtain good algorithm performances and small numerical errors. Results of a computational test are also provided.
Automatic discovery of community structures in complex networks is a fundamental task in many disciplines, including physics, biology, and the social sciences. The most used criterion for characterizing the existence ...
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Automatic discovery of community structures in complex networks is a fundamental task in many disciplines, including physics, biology, and the social sciences. The most used criterion for characterizing the existence of a community structure in a network is modularity, a quantitative measure proposed by Newman and Girvan (2004). The discovery community can be formulated as the so-called modularity maximization problem that consists of finding a partition of nodes of a network with the highest modularity. In this letter, we propose a fast and scalable algorithm called DCAM, based on DC (difference of convex function) programming and DCA (DC algorithms), an innovative approach in nonconvex programming framework for solving the modularity maximization problem. The special structure of the problem considered here has been well exploited to get an inexpensive DCA scheme that requires only a matrix-vector product at each iteration. Starting with a very large number of communities, DCAM furnishes, as output results, an optimal partition together with the optimal number of communities c*, that is, the number of communities is discovered automatically during DCAM's iterations. Numerical experiments are performed on a variety of real-world network data sets with up to 4,194,304 nodes and 30,359,198 edges. The comparative results with height reference algorithms show that the proposed approach outperforms them not only on quality and rapidity but also on scalability. Moreover, it realizes a very good trade-off between the quality of solutions and the run time.
Schedule optimization is crucial to reduce energy consumption of flexible manufacturing systems (FMSs) with shared resources and route flexibility. Based on the weighted p-timed Petri Net (WTPN) models of FMS, this pa...
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Schedule optimization is crucial to reduce energy consumption of flexible manufacturing systems (FMSs) with shared resources and route flexibility. Based on the weighted p-timed Petri Net (WTPN) models of FMS, this paper considers a scheduling problem which minimizes both productive and idle energy consumption subjected to general production constraints. The considered problem is proven to be a nonconvex mixed integer nonlinear program (MINLP). A new reachability graph (RG)-based discrete dynamic programming (DP) approach is proposed for generating near energy-optimal schedules within adequate computational time. The nonconvex MINLP is sampled, and the reduced RG is constructed such that only reachable paths are retained for computation of the energy-optimal path. Each scheduling subproblem is linearized, and each optimal substructure is computed to store in a routing table. It is proven that the sampling-induced error is bounded, and this upper bound can be reduced by increasing the sampling frequency. Experiment results on an industrial stamping system show the effectiveness of our proposed scheduling method in terms of computational complexity and deviation from optimality.
In this paper, we combine the new global optimization method proposed by Jiao [H. Jiao, A branch and bound algorithm for globally solving a class of nonconvex programming problems, Nonlinear Anal. 70 (2) (2008) 1113-1...
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In this paper, we combine the new global optimization method proposed by Jiao [H. Jiao, A branch and bound algorithm for globally solving a class of nonconvex programming problems, Nonlinear Anal. 70 (2) (2008) 1113-1123] with a suitable deleting technique to propose a new accelerating global optimization algorithm for solving a class of nonconvex programming problems (NP). This technique offers a possibility to cut away a large part of the currently investigated region in which the global optimal solution of NP does not exist, and can be seen as an accelerating device for the global optimization algorithm of the nonconvex programming problems. Compared with the method in the above cited reference, numerical results show that the computational efficiency is obviously improved by using this new technique in the number of iterations, the required list length and the overall execution time of the algorithm. (C) 2009 Elsevier Ltd. All rights reserved.
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