Pub Date : 1900-01-01DOI: 10.23952/jano.4.2022.1.03
{"title":"Vectorization in nonconvex set optimization","authors":"","doi":"10.23952/jano.4.2022.1.03","DOIUrl":"https://doi.org/10.23952/jano.4.2022.1.03","url":null,"abstract":"","PeriodicalId":205734,"journal":{"name":"Journal of Applied and Numerical Optimization","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114027916","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1900-01-01DOI: 10.23952/jano.3.2021.1.03
{"title":"Data-compatibility of algorithms for constrained convex optimization","authors":"","doi":"10.23952/jano.3.2021.1.03","DOIUrl":"https://doi.org/10.23952/jano.3.2021.1.03","url":null,"abstract":"","PeriodicalId":205734,"journal":{"name":"Journal of Applied and Numerical Optimization","volume":"49 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124084913","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1900-01-01DOI: 10.23952/jano.2.2020.3.05
Bing Tan, S. Cho
. In this paper, we investigate an inertial Mann-like algorithm for fixed points of nonexpansive mappings in Hilbert spaces and obtain strong convergence results under some mild assumptions. Based on this, we derive a forward-backward algorithm involving Tikhonov regularization terms, which converges strongly to the solution of the monotone inclusion problem. We demonstrate the advantages of our algorithms comparing with some existing ones in the literature via split feasibility problem, variational inequality problem and signal recovery problem.
{"title":"An inertial Mann-like algorithm for fixed points of nonexpansive mappings in Hilbert spaces","authors":"Bing Tan, S. Cho","doi":"10.23952/jano.2.2020.3.05","DOIUrl":"https://doi.org/10.23952/jano.2.2020.3.05","url":null,"abstract":". In this paper, we investigate an inertial Mann-like algorithm for fixed points of nonexpansive mappings in Hilbert spaces and obtain strong convergence results under some mild assumptions. Based on this, we derive a forward-backward algorithm involving Tikhonov regularization terms, which converges strongly to the solution of the monotone inclusion problem. We demonstrate the advantages of our algorithms comparing with some existing ones in the literature via split feasibility problem, variational inequality problem and signal recovery problem.","PeriodicalId":205734,"journal":{"name":"Journal of Applied and Numerical Optimization","volume":"136 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115568493","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1900-01-01DOI: 10.23952/jano.1.2019.2.04
E. Köbis, N. Popovici
We propose new algorithms for computing all minimal elements of a nonempty finite family of sets in a real linear space, with respect to a preorder relation defined on the power set of that space. These algorithms are based on a set-valued counterpart of the well-known Graef-Younes reduction procedure, originally conceived for vector optimization. One of our algorithms consists of two subsequent (forward-backward) reduction procedures, similarly to the classical Jahn-Graef-Younes method. Another algorithm involves a pre-sorting procedure with respect to a strongly increasing real-valued function, followed by a single (forward) reduction procedure. Numerical experiments in MATLAB allow us to compare our algorithms for special test families of line segments with respect to `-type, u-type and s-type preorder relations, currently used in set optimization.
{"title":"Computing minimal elements of finite families of sets w.r.t. preorder relations in set optimization","authors":"E. Köbis, N. Popovici","doi":"10.23952/jano.1.2019.2.04","DOIUrl":"https://doi.org/10.23952/jano.1.2019.2.04","url":null,"abstract":"We propose new algorithms for computing all minimal elements of a nonempty finite family of sets in a real linear space, with respect to a preorder relation defined on the power set of that space. These algorithms are based on a set-valued counterpart of the well-known Graef-Younes reduction procedure, originally conceived for vector optimization. One of our algorithms consists of two subsequent (forward-backward) reduction procedures, similarly to the classical Jahn-Graef-Younes method. Another algorithm involves a pre-sorting procedure with respect to a strongly increasing real-valued function, followed by a single (forward) reduction procedure. Numerical experiments in MATLAB allow us to compare our algorithms for special test families of line segments with respect to `-type, u-type and s-type preorder relations, currently used in set optimization.","PeriodicalId":205734,"journal":{"name":"Journal of Applied and Numerical Optimization","volume":"27 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114339972","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1900-01-01DOI: 10.23952/jano.2.2020.2.05
Dingfang Hou, Jing Zhao, Xing-Jun Wang
In this paper, we use the dual variable to propose a new iterative algorithm for solving the split common fixed-point problem of quasi-nonexpansive mappings in real Hilbert spaces. Under suitable conditions, we establish a weak convergence theorem of the proposed algorithm and obtain a related result for the split common fixed-point problem of firmly quasi-nonexpansive mappings. Some numerical experiments are given to illustrate the efficiency of the proposed iterative algorithm.
{"title":"Weak convergence of a primal-dual algorithm for split common fixed-point problems in Hilbert spaces","authors":"Dingfang Hou, Jing Zhao, Xing-Jun Wang","doi":"10.23952/jano.2.2020.2.05","DOIUrl":"https://doi.org/10.23952/jano.2.2020.2.05","url":null,"abstract":"In this paper, we use the dual variable to propose a new iterative algorithm for solving the split common fixed-point problem of quasi-nonexpansive mappings in real Hilbert spaces. Under suitable conditions, we establish a weak convergence theorem of the proposed algorithm and obtain a related result for the split common fixed-point problem of firmly quasi-nonexpansive mappings. Some numerical experiments are given to illustrate the efficiency of the proposed iterative algorithm.","PeriodicalId":205734,"journal":{"name":"Journal of Applied and Numerical Optimization","volume":"2 2","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120844289","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1900-01-01DOI: 10.23952/jano.3.2021.1.09
L. Xiaotian, Linju Cai, LI Jingchao, YU Carisakwokwai, H. Yaohua
. Clustering is one of fundamental tasks in unsupervised learning and plays a very important role in various application areas. This paper aims to present a survey of five types of clustering methods in the perspective of optimization methodology, including center-based methods, convex clustering, spectral clustering, subspace clustering, and optimal transport based clustering. The connection between optimization methodology and clustering algorithms is not only helpful to advance the understanding of the principle and theory of existing clustering algorithms, but also useful to inspire new ideas of efficient clustering algorithms. Preliminary numerical experiments of various clustering algorithms for datasets of various shapes are provided to show the preference and specificity of each algorithm.
{"title":"A survey of clustering methods via optimization methodology","authors":"L. Xiaotian, Linju Cai, LI Jingchao, YU Carisakwokwai, H. Yaohua","doi":"10.23952/jano.3.2021.1.09","DOIUrl":"https://doi.org/10.23952/jano.3.2021.1.09","url":null,"abstract":". Clustering is one of fundamental tasks in unsupervised learning and plays a very important role in various application areas. This paper aims to present a survey of five types of clustering methods in the perspective of optimization methodology, including center-based methods, convex clustering, spectral clustering, subspace clustering, and optimal transport based clustering. The connection between optimization methodology and clustering algorithms is not only helpful to advance the understanding of the principle and theory of existing clustering algorithms, but also useful to inspire new ideas of efficient clustering algorithms. Preliminary numerical experiments of various clustering algorithms for datasets of various shapes are provided to show the preference and specificity of each algorithm.","PeriodicalId":205734,"journal":{"name":"Journal of Applied and Numerical Optimization","volume":"55 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124125916","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1900-01-01DOI: 10.23952/jano.3.2021.3.07
{"title":"A note on minimax optimization problems with an infinite number of constraints","authors":"","doi":"10.23952/jano.3.2021.3.07","DOIUrl":"https://doi.org/10.23952/jano.3.2021.3.07","url":null,"abstract":"","PeriodicalId":205734,"journal":{"name":"Journal of Applied and Numerical Optimization","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124478230","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1900-01-01DOI: 10.23952/jano.4.2022.1.09
{"title":"Solvability of convex optimization problems on a *-continuous closed convex set","authors":"","doi":"10.23952/jano.4.2022.1.09","DOIUrl":"https://doi.org/10.23952/jano.4.2022.1.09","url":null,"abstract":"","PeriodicalId":205734,"journal":{"name":"Journal of Applied and Numerical Optimization","volume":"97 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124677054","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1900-01-01DOI: 10.23952/jano.1.2019.2.07
V. Babenko, V. Babenko
We consider the problem of approximation of unbounded positively homogeneous operators in L-spaces using Lipschitz operators. We study its connection to the problem of computing modulus of continuity of the unbounded operator on the class of elements, as well as, to the problem of optimal recovery of an unbounded operator by a Lipschitz one on the class of elements given with an error. Moreover, in L-spaces and for positively homogeneous operators, the connection of the above-mentioned problems with inequalities of Landau Kolmogorov type is studied. As applications, we consider the problem of approximation of unbounded operator, that for functions with values in some L-space puts in a correspondence Hukuhara-type derivatives, by Lipschitz operators. In addition, we compute the modulus of continuity of this operator and obtain exact Landau-Kolmogorov type inequalities. Further, we solve the problem of the optimal recovery of this operator on the class of functions that have Hukuhara-type derivative with the given majorant of the modulus of continuity (in the case of optimal recovery, elements of this class are given with an error).
{"title":"Best approximation, optimal recovery, and Landau inequalities for derivatives of Hukuhara-type in function L-spaces","authors":"V. Babenko, V. Babenko","doi":"10.23952/jano.1.2019.2.07","DOIUrl":"https://doi.org/10.23952/jano.1.2019.2.07","url":null,"abstract":"We consider the problem of approximation of unbounded positively homogeneous operators in L-spaces using Lipschitz operators. We study its connection to the problem of computing modulus of continuity of the unbounded operator on the class of elements, as well as, to the problem of optimal recovery of an unbounded operator by a Lipschitz one on the class of elements given with an error. Moreover, in L-spaces and for positively homogeneous operators, the connection of the above-mentioned problems with inequalities of Landau Kolmogorov type is studied. As applications, we consider the problem of approximation of unbounded operator, that for functions with values in some L-space puts in a correspondence Hukuhara-type derivatives, by Lipschitz operators. In addition, we compute the modulus of continuity of this operator and obtain exact Landau-Kolmogorov type inequalities. Further, we solve the problem of the optimal recovery of this operator on the class of functions that have Hukuhara-type derivative with the given majorant of the modulus of continuity (in the case of optimal recovery, elements of this class are given with an error).","PeriodicalId":205734,"journal":{"name":"Journal of Applied and Numerical Optimization","volume":"56 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124747319","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1900-01-01DOI: 10.23952/jano.1.2019.3.03
T. Q. Bao, A. Soubeyran, T. Q. Bao, A. Soubeyran
This paper is devoted to new versions of Ekeland’s variational principle in set optimization with domination structure, where set optimization is an extension of vector optimization from vector-valued functions to set-valued maps using Kuroiwa’s set-less relations to compare one entire image set with another whole image set, and where domination structure is an extension of ordering cone in vector optimization; it assigns each element of the image space to its own domination set. We use Gerstewitz’s nonlinear scalarization function to convert a set-valued map into an extended real-valued function and the idea of the proof of Dancs-Hegedüs-Medvegyev’s fixed-point theorem. Our setting is applicable to dynamic processes of changing jobs in which the cost function does not satisfy the symmetry axiom of metrics and the class of set-valued maps acting from a quasimetric space into a real linear space. The obtained result is new even in simpler settings.
{"title":"Variational principles in set optimization with domination structures and application to changing jobs","authors":"T. Q. Bao, A. Soubeyran, T. Q. Bao, A. Soubeyran","doi":"10.23952/jano.1.2019.3.03","DOIUrl":"https://doi.org/10.23952/jano.1.2019.3.03","url":null,"abstract":"This paper is devoted to new versions of Ekeland’s variational principle in set optimization with domination structure, where set optimization is an extension of vector optimization from vector-valued functions to set-valued maps using Kuroiwa’s set-less relations to compare one entire image set with another whole image set, and where domination structure is an extension of ordering cone in vector optimization; it assigns each element of the image space to its own domination set. We use Gerstewitz’s nonlinear scalarization function to convert a set-valued map into an extended real-valued function and the idea of the proof of Dancs-Hegedüs-Medvegyev’s fixed-point theorem. Our setting is applicable to dynamic processes of changing jobs in which the cost function does not satisfy the symmetry axiom of metrics and the class of set-valued maps acting from a quasimetric space into a real linear space. The obtained result is new even in simpler settings.","PeriodicalId":205734,"journal":{"name":"Journal of Applied and Numerical Optimization","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129778664","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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