Pub Date : 2023-05-12DOI: 10.1080/01630563.2023.2208867
Yingrang Xu, S. Li
Abstract In this article, optimality conditions on (weak) efficient solutions in multiobjective optimization problems are investigated by using the image space analysis. A class of strong separation functions is constructed by oriented distance functions. Simultaneously, a generalized Lagrange function is introduced by the class of strong separation functions. Then, generalized Karush-Kuhn-Tucker (KKT for short) necessary optimality conditions are established without constraint qualifications or regularity conditions. Under the suitable assumptions, Lagrangian-type sufficient optimality conditions are also characterized. Moreover, the difference between strong separation and weak separation methods is explained.
{"title":"Optimality Conditions for Multiobjective Optimization Problems via Image Space Analysis","authors":"Yingrang Xu, S. Li","doi":"10.1080/01630563.2023.2208867","DOIUrl":"https://doi.org/10.1080/01630563.2023.2208867","url":null,"abstract":"Abstract In this article, optimality conditions on (weak) efficient solutions in multiobjective optimization problems are investigated by using the image space analysis. A class of strong separation functions is constructed by oriented distance functions. Simultaneously, a generalized Lagrange function is introduced by the class of strong separation functions. Then, generalized Karush-Kuhn-Tucker (KKT for short) necessary optimality conditions are established without constraint qualifications or regularity conditions. Under the suitable assumptions, Lagrangian-type sufficient optimality conditions are also characterized. Moreover, the difference between strong separation and weak separation methods is explained.","PeriodicalId":54707,"journal":{"name":"Numerical Functional Analysis and Optimization","volume":"44 1","pages":"708 - 723"},"PeriodicalIF":1.2,"publicationDate":"2023-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44295537","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-05-12DOI: 10.1080/01630563.2023.2209147
J. Balooee, Shih-sen Chang, Min Liu, J. Yao
Abstract In this article, we pursue two goals. First, a new iterative scheme based on the resolvent operator method for finding a common element of the set of solutions of a generalized variational inclusion problem and the set of fixed points of a total asymptotically nonexpansive mapping in a real Banach space is constructed. Under some parameters controlling conditions, the strong convergence of the sequence generated by our proposed iterative algorithm to a common element of the above-mentioned two sets is proved. Our second purpose is to investigate and analyze the concept of H(.,.)-accretive operator that appeared in the literature and to point out some comments concerning it. Several new examples are also provided.
{"title":"Total Asymptotically Nonexpansive Mappings and Generalized Variational Inclusion Problems: Algorithmic and Analytical Approach","authors":"J. Balooee, Shih-sen Chang, Min Liu, J. Yao","doi":"10.1080/01630563.2023.2209147","DOIUrl":"https://doi.org/10.1080/01630563.2023.2209147","url":null,"abstract":"Abstract In this article, we pursue two goals. First, a new iterative scheme based on the resolvent operator method for finding a common element of the set of solutions of a generalized variational inclusion problem and the set of fixed points of a total asymptotically nonexpansive mapping in a real Banach space is constructed. Under some parameters controlling conditions, the strong convergence of the sequence generated by our proposed iterative algorithm to a common element of the above-mentioned two sets is proved. Our second purpose is to investigate and analyze the concept of H(.,.)-accretive operator that appeared in the literature and to point out some comments concerning it. Several new examples are also provided.","PeriodicalId":54707,"journal":{"name":"Numerical Functional Analysis and Optimization","volume":"44 1","pages":"906 - 953"},"PeriodicalIF":1.2,"publicationDate":"2023-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44908735","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-04-27DOI: 10.1080/01630563.2023.2178009
Yuya Yamakawa
Abstract In this paper, we propose a stabilized sequential quadratic programming (SQP) method for optimization problems in function spaces. A form of the problem considered in this paper can widely formulate many types of applications, such as obstacle problems, optimal control problems, and so on. Moreover, the proposed method is based on the existing stabilized SQP method and can find a point satisfying the Karush-Kuhn-Tucker (KKT) or asymptotic KKT conditions. One of the remarkable points is that we prove its global convergence to such a point under some assumptions without any constraint qualifications. In addition, we guarantee that an arbitrary accumulation point generated by the proposed method satisfies the KKT conditions under several additional assumptions. Finally, we report some numerical experiments to examine the effectiveness of the proposed method.
{"title":"A Stabilized Sequential Quadratic Programming Method for Optimization Problems in Function Spaces","authors":"Yuya Yamakawa","doi":"10.1080/01630563.2023.2178009","DOIUrl":"https://doi.org/10.1080/01630563.2023.2178009","url":null,"abstract":"Abstract In this paper, we propose a stabilized sequential quadratic programming (SQP) method for optimization problems in function spaces. A form of the problem considered in this paper can widely formulate many types of applications, such as obstacle problems, optimal control problems, and so on. Moreover, the proposed method is based on the existing stabilized SQP method and can find a point satisfying the Karush-Kuhn-Tucker (KKT) or asymptotic KKT conditions. One of the remarkable points is that we prove its global convergence to such a point under some assumptions without any constraint qualifications. In addition, we guarantee that an arbitrary accumulation point generated by the proposed method satisfies the KKT conditions under several additional assumptions. Finally, we report some numerical experiments to examine the effectiveness of the proposed method.","PeriodicalId":54707,"journal":{"name":"Numerical Functional Analysis and Optimization","volume":"44 1","pages":"867 - 905"},"PeriodicalIF":1.2,"publicationDate":"2023-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43133804","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-04-26DOI: 10.1080/01630563.2023.2172034
M. H. Alizadeh, Javad Hosseinabadi
Abstract The notion of -monotone polarity for -subdifferential is introduced and studied. Also, the concept of Fréchet -subdifferential is introduced and then some results regarding this concept are obtained. In addition, some particular relationships between the -subdifferential and Fréchet -subdifferential are presented.
{"title":"On σ-Subdifferential Polarity and Fréchet σ-Subdifferential","authors":"M. H. Alizadeh, Javad Hosseinabadi","doi":"10.1080/01630563.2023.2172034","DOIUrl":"https://doi.org/10.1080/01630563.2023.2172034","url":null,"abstract":"Abstract The notion of -monotone polarity for -subdifferential is introduced and studied. Also, the concept of Fréchet -subdifferential is introduced and then some results regarding this concept are obtained. In addition, some particular relationships between the -subdifferential and Fréchet -subdifferential are presented.","PeriodicalId":54707,"journal":{"name":"Numerical Functional Analysis and Optimization","volume":"44 1","pages":"603 - 618"},"PeriodicalIF":1.2,"publicationDate":"2023-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49539782","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-04-21DOI: 10.1080/01630563.2023.2183510
S. K. Dixit
Abstract Lanweber-type methods are a well-known regularization methods to solve linear and nonlinear ill-posed problems. In this article, we consider a simplified form of Landweber method, say, an iteratively-regularized simplified Landweber iteration for solving nonlinear ill-posed problems. We study a detailed convergence analysis of the method under standard conditions on the nonlinearity and the rate of convergence under a Hölder-type source condition. Finally, numerical simulations are performed to validate the performance of the method.
{"title":"A Study of an Iteratively-Regularized Simplified Landweber Iteration for Nonlinear Inverse Problems in Hilbert Spaces","authors":"S. K. Dixit","doi":"10.1080/01630563.2023.2183510","DOIUrl":"https://doi.org/10.1080/01630563.2023.2183510","url":null,"abstract":"Abstract Lanweber-type methods are a well-known regularization methods to solve linear and nonlinear ill-posed problems. In this article, we consider a simplified form of Landweber method, say, an iteratively-regularized simplified Landweber iteration for solving nonlinear ill-posed problems. We study a detailed convergence analysis of the method under standard conditions on the nonlinearity and the rate of convergence under a Hölder-type source condition. Finally, numerical simulations are performed to validate the performance of the method.","PeriodicalId":54707,"journal":{"name":"Numerical Functional Analysis and Optimization","volume":"44 1","pages":"619 - 652"},"PeriodicalIF":1.2,"publicationDate":"2023-04-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49305527","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-04-05DOI: 10.1080/01630563.2023.2192593
Xiong-jie Shao, Yimin Wei, J. Yuan
Abstract The nonsymmetric algebraic Riccati equation is proposed by using the tensor product. The existence of its minimal nonnegative solution is studied. The sufficient condition of the existence and the uniqueness of the minimal nonnegative solution is given by -tensor as well. The solution can be obtained by the fast Fourier transform which save computational cost of computing the required solution. Some numerical experiments are performed.
{"title":"Nonsymmetric Algebraic Riccati Equations under the Tensor Product","authors":"Xiong-jie Shao, Yimin Wei, J. Yuan","doi":"10.1080/01630563.2023.2192593","DOIUrl":"https://doi.org/10.1080/01630563.2023.2192593","url":null,"abstract":"Abstract The nonsymmetric algebraic Riccati equation is proposed by using the tensor product. The existence of its minimal nonnegative solution is studied. The sufficient condition of the existence and the uniqueness of the minimal nonnegative solution is given by -tensor as well. The solution can be obtained by the fast Fourier transform which save computational cost of computing the required solution. Some numerical experiments are performed.","PeriodicalId":54707,"journal":{"name":"Numerical Functional Analysis and Optimization","volume":"44 1","pages":"545 - 563"},"PeriodicalIF":1.2,"publicationDate":"2023-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44708721","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-04-05DOI: 10.1080/01630563.2023.2197991
Katarina S. Stojanović
Abstract In this paper, we will introduce two new classes of generalized Drazin invertible operators on Hilbert space, which are called core–EP star and star core–EP operators, using the core–EP inverse and the adjoint of a given operator. We also represent here a few characterizations of these new operators from two points of view, algebraic and geometrical, and make relations to some familiar inverses, which are studied before. Next, we will consider two decompositions of Hilbert space and give the matrix representations of these new operators, following the decompositions. As the special section, there is the case when the operator is Drazin invertible. By using a *core–EP inverse, instead of the core–EP inverse, we get another new classes called *core–EP star and star *core–EP operators.
{"title":"Core–EP Star and Star Core–EP Operators","authors":"Katarina S. Stojanović","doi":"10.1080/01630563.2023.2197991","DOIUrl":"https://doi.org/10.1080/01630563.2023.2197991","url":null,"abstract":"Abstract In this paper, we will introduce two new classes of generalized Drazin invertible operators on Hilbert space, which are called core–EP star and star core–EP operators, using the core–EP inverse and the adjoint of a given operator. We also represent here a few characterizations of these new operators from two points of view, algebraic and geometrical, and make relations to some familiar inverses, which are studied before. Next, we will consider two decompositions of Hilbert space and give the matrix representations of these new operators, following the decompositions. As the special section, there is the case when the operator is Drazin invertible. By using a *core–EP inverse, instead of the core–EP inverse, we get another new classes called *core–EP star and star *core–EP operators.","PeriodicalId":54707,"journal":{"name":"Numerical Functional Analysis and Optimization","volume":"44 1","pages":"687 - 707"},"PeriodicalIF":1.2,"publicationDate":"2023-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45083825","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-03-31DOI: 10.1080/01630563.2023.2192776
Jacob Körner, A. Borzì
Abstract First- and second-order accuracy estimates for an optimal control problem governed by a system of ordinary differential equations with a bilinear control mechanism are presented. The numerical time discretization scheme under consideration is the finite element method with continuous piecewise linear functions. Central to this work is a first- and second-order analysis of optimality of the continuous and approximated optimal control problems. In the case of box constraints on the control, first-order error estimates for the control function are obtained assuming a piecewise constant approximation of the control, whereas second-order accuracy can be obtained in the case of a continuous, piecewise polynomial approximation. Numerical evidence is presented that supports the theoretical findings.
{"title":"Accuracy Estimates for Bilinear Optimal Control Problems Governed by Ordinary Differential Equations","authors":"Jacob Körner, A. Borzì","doi":"10.1080/01630563.2023.2192776","DOIUrl":"https://doi.org/10.1080/01630563.2023.2192776","url":null,"abstract":"Abstract First- and second-order accuracy estimates for an optimal control problem governed by a system of ordinary differential equations with a bilinear control mechanism are presented. The numerical time discretization scheme under consideration is the finite element method with continuous piecewise linear functions. Central to this work is a first- and second-order analysis of optimality of the continuous and approximated optimal control problems. In the case of box constraints on the control, first-order error estimates for the control function are obtained assuming a piecewise constant approximation of the control, whereas second-order accuracy can be obtained in the case of a continuous, piecewise polynomial approximation. Numerical evidence is presented that supports the theoretical findings.","PeriodicalId":54707,"journal":{"name":"Numerical Functional Analysis and Optimization","volume":"44 1","pages":"564 - 602"},"PeriodicalIF":1.2,"publicationDate":"2023-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47612458","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-03-16DOI: 10.1080/01630563.2023.2185896
Km. Lipi, N. Deo
Abstract In this manuscript, we propose a Pólya distribution-based generalization of -Bernstein operators. We establish some fundamental results for convergence as well as order of approximation of the proposed operators. We present theoretical result and graph to demonstrate the proposed operator’s intriguing ability to interpolate at the interval’s end points. In order to illustrate the convergence of proposed operators as well as the effect of changing the parameter “ ” we provide a variety of results and graphs as our paper’s conclusion.
{"title":"λ-Bernstein Operators Based on Pólya Distribution","authors":"Km. Lipi, N. Deo","doi":"10.1080/01630563.2023.2185896","DOIUrl":"https://doi.org/10.1080/01630563.2023.2185896","url":null,"abstract":"Abstract In this manuscript, we propose a Pólya distribution-based generalization of -Bernstein operators. We establish some fundamental results for convergence as well as order of approximation of the proposed operators. We present theoretical result and graph to demonstrate the proposed operator’s intriguing ability to interpolate at the interval’s end points. In order to illustrate the convergence of proposed operators as well as the effect of changing the parameter “ ” we provide a variety of results and graphs as our paper’s conclusion.","PeriodicalId":54707,"journal":{"name":"Numerical Functional Analysis and Optimization","volume":"44 1","pages":"529 - 544"},"PeriodicalIF":1.2,"publicationDate":"2023-03-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45620370","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-03-03DOI: 10.1080/01630563.2023.2180753
Jinlu Li
Abstract Let (X, ) be a topological vector space equipped with a partial order which is induced by a nonempty pointed closed and convex cone K in X. Let (X) = 2 X be the power set of X and (X) = 2 X { }. In this paper, we consider the so called upward preordering on (X), which has been used by many authors (see [4 7, 9, 11 14, 18 19]). We first prove some properties of this ordering relation Let (X) denote the collection of all -closed subsets of X and (X) = (X){ }, which is equipped with the Fell topology Let C be a nonempty subset of X and let F: C (X) be a closed set-valued mapping. By applying the Fell topology on (X) and the Fan-KKM Theorem, we prove some existence theorems for some -minimization and -maximization problems with respect to F subject to the subset C for first countable topological vector spaces. These results will be applied to solve some closed ball-valued optimization problems in partially ordered Hilbert spaces. Some examples will be provided in
{"title":"Applications of Fell Topology to Closed Set-Valued Optimizations in Partially Ordered First Countable Topological Vector Spaces","authors":"Jinlu Li","doi":"10.1080/01630563.2023.2180753","DOIUrl":"https://doi.org/10.1080/01630563.2023.2180753","url":null,"abstract":"Abstract Let (X, ) be a topological vector space equipped with a partial order which is induced by a nonempty pointed closed and convex cone K in X. Let (X) = 2 X be the power set of X and (X) = 2 X { }. In this paper, we consider the so called upward preordering on (X), which has been used by many authors (see [4 7, 9, 11 14, 18 19]). We first prove some properties of this ordering relation Let (X) denote the collection of all -closed subsets of X and (X) = (X){ }, which is equipped with the Fell topology Let C be a nonempty subset of X and let F: C (X) be a closed set-valued mapping. By applying the Fell topology on (X) and the Fan-KKM Theorem, we prove some existence theorems for some -minimization and -maximization problems with respect to F subject to the subset C for first countable topological vector spaces. These results will be applied to solve some closed ball-valued optimization problems in partially ordered Hilbert spaces. Some examples will be provided in","PeriodicalId":54707,"journal":{"name":"Numerical Functional Analysis and Optimization","volume":"44 1","pages":"461 - 489"},"PeriodicalIF":1.2,"publicationDate":"2023-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44599902","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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