DC approximation approaches for sparse optimization

•A unifying DC approximation, including all standard approximations, of the zero-norm is proposed.•The consistency between global/local minima of approximate and original problems are proved.•The equivalence between approximate and original problems are established for some approximations.•Four DCA...

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Vydáno v:	European journal of operational research Ročník 244; číslo 1; s. 26 - 46
Hlavní autoři:	Le Thi, H.A., Pham Dinh, T., Le, H.M., Vo, X.T.
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	Amsterdam Elsevier B.V 01.07.2015 Elsevier Sequoia S.A Elsevier
Témata:	Algorithms Approximation Computer Science DC approximation function DC programming and DCA Direct current Feature selection in SVM Global optimization Mathematical analysis Mathematical functions Mathematical models Mathematical problems Mathematical programming Optimization Optimization algorithms Programming Sparse optimization Studies Sparse optimization DC programming and DCA Global optimization DC approximation function Feature selection in SVM
ISSN:	0377-2217, 1872-6860
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Abstract	•A unifying DC approximation, including all standard approximations, of the zero-norm is proposed.•The consistency between global/local minima of approximate and original problems are proved.•The equivalence between approximate and original problems are established for some approximations.•Four DCA schemes are developed that cover all standard nonconvex approximation algorithms.•A careful empirical experiment for feature selection in SVM are performed. Sparse optimization refers to an optimization problem involving the zero-norm in objective or constraints. In this paper, nonconvex approximation approaches for sparse optimization have been studied with a unifying point of view in DC (Difference of Convex functions) programming framework. Considering a common DC approximation of the zero-norm including all standard sparse inducing penalty functions, we studied the consistency between global minimums (resp. local minimums) of approximate and original problems. We showed that, in several cases, some global minimizers (resp. local minimizers) of the approximate problem are also those of the original problem. Using exact penalty techniques in DC programming, we proved stronger results for some particular approximations, namely, the approximate problem, with suitable parameters, is equivalent to the original problem. The efficiency of several sparse inducing penalty functions have been fully analyzed. Four DCA (DC Algorithm) schemes were developed that cover all standard algorithms in nonconvex sparse approximation approaches as special versions. They can be viewed as, an ℓ1-perturbed algorithm/reweighted-ℓ1 algorithm / reweighted-ℓ2 algorithm. We offer a unifying nonconvex approximation approach, with solid theoretical tools as well as efficient algorithms based on DC programming and DCA, to tackle the zero-norm and sparse optimization. As an application, we implemented our methods for the feature selection in SVM (Support Vector Machine) problem and performed empirical comparative numerical experiments on the proposed algorithms with various approximation functions.
AbstractList	Sparse optimization refers to an optimization problem involving the zero-norm in objective or constraints. In this paper, nonconvex approximation approaches for sparse optimization have been studied with a unifying point of view in DC (Difference of Convex functions) programming framework. Considering a common DC approximation of the zero-norm including all standard sparse inducing penalty functions, we studied the consistency between global minimums (resp. local minimums) of approximate and original problems. We showed that, in several cases, some global minimizers (resp. local minimizers) of the approximate problem are also those of the original problem. Using exact penalty techniques in DC programming, we proved stronger results for some particular approximations, namely, the approximate problem, with suitable parameters, is equivalent to the original problem. The efficiency of several sparse inducing penalty functions have been fully analyzed. Four DCA (DC Algorithm) schemes were developed that cover all standard algorithms in nonconvex sparse approximation approaches as special versions. They can be viewed as, an ℓ1-perturbed algorithm/reweighted-ℓ1 algorithm / reweighted-ℓ2 algorithm. We offer a unifying nonconvex approximation approach, with solid theoretical tools as well as efficient algorithms based on DC programming and DCA, to tackle the zero-norm and sparse optimization. As an application, we implemented our methods for the feature selection in SVM (Support Vector Machine) problem and performed empirical comparative numerical experiments on the proposed algorithms with various approximation functions. Sparse optimization refers to an optimization problem involving the zero-norm in objective or constraints. In this paper, nonconvex approximation approaches for sparse optimization have been studied with a unifying point of view in DC (Difference of Convex functions) programming framework. Considering a common DC approximation of the zero-norm including all standard sparse inducing penalty functions, we studied the consistency between global minimums (resp. local minimums) of approximate and original problems. We showed that, in several cases, some global minimizers (resp. local minimizers) of the approximate problem are also those of the original problem. Using exact penalty techniques in DC programming, we proved stronger results for some particular approximations, namely, the approximate problem, with suitable parameters, is equivalent to the original problem. The efficiency of several sparse inducing penalty functions have been fully analyzed. Four DCA (DC Algorithm) schemes were developed that cover all standard algorithms in nonconvex sparse approximation approaches as special versions. They can be viewed as, an ...-perturbed algorithm/reweighted-... algorithm / reweighted-... algorithm. We offer a unifying nonconvex approximation approach, with solid theoretical tools as well as efficient algorithms based on DC programming and DCA, to tackle the zero-norm and sparse optimization. As an application, we implemented our methods for the feature selection in SVM (Support Vector Machine) problem and performed empirical comparative numerical experiments on the proposed algorithms with various approximation functions. (ProQuest: ... denotes formulae/symbols omitted.) Sparse optimization refers to an optimization problem involving the zero-norm in objective or constraints. In this paper, nonconvex approximation approaches for sparse optimization have been studied with a unifying point of view in DC (Difference of Convex functions) programming framework. Considering a common DC approximation of the zero-norm including all standard sparse inducing penalty functions, we studied the consistency between global minimums (resp. local minimums) of approximate and original problems. We showed that, in several cases, some global minimizers (resp. local minimizers) of the approximate problem are also those of the original problem. Using exact penalty techniques in DC programming, we proved stronger results for some particular approximations, namely, the approximate problem, with suitable parameters, is equivalent to the original problem. The efficiency of several sparse inducing penalty functions have been fully analyzed. Four DCA (DC Algorithm) schemes were developed that cover all standard algorithms in nonconvex sparse approximation approaches as special versions. They can be viewed as, an [ell] sub(1)-perturbed algorithm/reweighted-[ell] sub(1) algorithm / reweighted-[ell] sub(2) algorithm. We offer a unifying nonconvex approximation approach, with solid theoretical tools as well as efficient algorithms based on DC programming and DCA, to tackle the zero-norm and sparse optimization. As an application, we implemented our methods for the feature selection in SVM (Support Vector Machine) problem and performed empirical comparative numerical experiments on the proposed algorithms with various approximation functions. •A unifying DC approximation, including all standard approximations, of the zero-norm is proposed.•The consistency between global/local minima of approximate and original problems are proved.•The equivalence between approximate and original problems are established for some approximations.•Four DCA schemes are developed that cover all standard nonconvex approximation algorithms.•A careful empirical experiment for feature selection in SVM are performed. Sparse optimization refers to an optimization problem involving the zero-norm in objective or constraints. In this paper, nonconvex approximation approaches for sparse optimization have been studied with a unifying point of view in DC (Difference of Convex functions) programming framework. Considering a common DC approximation of the zero-norm including all standard sparse inducing penalty functions, we studied the consistency between global minimums (resp. local minimums) of approximate and original problems. We showed that, in several cases, some global minimizers (resp. local minimizers) of the approximate problem are also those of the original problem. Using exact penalty techniques in DC programming, we proved stronger results for some particular approximations, namely, the approximate problem, with suitable parameters, is equivalent to the original problem. The efficiency of several sparse inducing penalty functions have been fully analyzed. Four DCA (DC Algorithm) schemes were developed that cover all standard algorithms in nonconvex sparse approximation approaches as special versions. They can be viewed as, an ℓ1-perturbed algorithm/reweighted-ℓ1 algorithm / reweighted-ℓ2 algorithm. We offer a unifying nonconvex approximation approach, with solid theoretical tools as well as efficient algorithms based on DC programming and DCA, to tackle the zero-norm and sparse optimization. As an application, we implemented our methods for the feature selection in SVM (Support Vector Machine) problem and performed empirical comparative numerical experiments on the proposed algorithms with various approximation functions.
Author	Pham Dinh, T. Le Thi, H.A. Vo, X.T. Le, H.M.
Author_xml	– sequence: 1 givenname: H.A. surname: Le Thi fullname: Le Thi, H.A. email: hoai-an.le-thi@univ-lorraine.fr organization: Laboratory of Theoretical and Applied Computer Science EA 3097, University of Lorraine, Ile du Saulcy, Metz 57045, France – sequence: 2 givenname: T. surname: Pham Dinh fullname: Pham Dinh, T. email: pham@insa-rouen.fr organization: Laboratory of Mathematics, INSA – Rouen, University of Normandie, Saint-Etienne-du-Rouvray Cedex 76801, France – sequence: 3 givenname: H.M. surname: Le fullname: Le, H.M. email: minh.le@univ-lorraine.fr organization: Laboratory of Theoretical and Applied Computer Science EA 3097, University of Lorraine, Ile du Saulcy, Metz 57045, France – sequence: 4 givenname: X.T. surname: Vo fullname: Vo, X.T. email: xuan-thanh.vo@univ-lorraine.fr organization: Laboratory of Theoretical and Applied Computer Science EA 3097, University of Lorraine, Ile du Saulcy, Metz 57045, France
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Snippet	•A unifying DC approximation, including all standard approximations, of the zero-norm is proposed.•The consistency between global/local minima of approximate... Sparse optimization refers to an optimization problem involving the zero-norm in objective or constraints. In this paper, nonconvex approximation approaches...
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SubjectTerms	Algorithms Approximation Computer Science DC approximation function DC programming and DCA Direct current Feature selection in SVM Global optimization Mathematical analysis Mathematical functions Mathematical models Mathematical problems Mathematical programming Optimization Optimization algorithms Programming Sparse optimization Studies
Title	DC approximation approaches for sparse optimization
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