Several equivalent statements of generalized subconvexlike set-valued map are established in ordered linear spaces. Using vector closure, we introduce Benson proper efficient solution of vector optimization problem. U...Several equivalent statements of generalized subconvexlike set-valued map are established in ordered linear spaces. Using vector closure, we introduce Benson proper efficient solution of vector optimization problem. Under the assumption of generalized subconvexlikeness, scalarization, multiplier and saddle point theorems are obtained in the sense of Benson proper efficiency.展开更多
A kind of tangent derivative and the concepts of strong and weak * pseudoconvexity for a set-valued map are introduced. By the standard separation theorems of the convex sets and cones the optimality Fritz John condit...A kind of tangent derivative and the concepts of strong and weak * pseudoconvexity for a set-valued map are introduced. By the standard separation theorems of the convex sets and cones the optimality Fritz John condition of set-valued optimization under Benson proper efficiency is established, its sufficience is discussed. The form of the optimality conditions obtained here completely tally with the classical results when the set-valued map is specialized to be a single-valued map.展开更多
In this paper,the ε-super effcient solution for set-valued map vector optimization in locally convex space is introduced.And under the assumption of the nearly generalized cone-subconvexlikeness for set-valued maps,u...In this paper,the ε-super effcient solution for set-valued map vector optimization in locally convex space is introduced.And under the assumption of the nearly generalized cone-subconvexlikeness for set-valued maps,used new methods,the scalarization theorem and Lagrange multiplier theorem for ε-super effcient solution are established.展开更多
The definitions of cone-subconvexlike set-valued maps and generalized cone-subconvexlike set-valued maps in topological vector spaces are defined by using the relative interiors of ordering cone. The relationships bet...The definitions of cone-subconvexlike set-valued maps and generalized cone-subconvexlike set-valued maps in topological vector spaces are defined by using the relative interiors of ordering cone. The relationships between the two classes of set-valued maps are investigated, and some properties of them are shown. A Gordan type alternative theorem under the assumption of generalized cone-subconvexlikeness of set-valued maps is proved by applying convex separation theorems involving the relative interiors in infinite dimensional spaces. Finally a necessary optimality condition theorem is shown for a general kind of set-valued vector optimization in a sense of weak E-minimizer.展开更多
In locally convex Hausdorff topological vector spaces,ε-strongly efficient solutions for vector optimization with set-valued maps are discussed.Firstly,ε-strongly efficient point of set is introduced.Secondly,under ...In locally convex Hausdorff topological vector spaces,ε-strongly efficient solutions for vector optimization with set-valued maps are discussed.Firstly,ε-strongly efficient point of set is introduced.Secondly,under the nearly cone-subconvexlike set-valued maps,the theorem of scalarization for vector optimization is obtained.Finally,optimality conditions of ε-strongly efficient solutions for vector optimization with generalized inequality constraints and equality constraints are obtained.展开更多
In this paper, we will construct a new class of subadditive set-valued maps and use Cantor theorem to prove that the set-valued map has an unique additive selection map when the set-valued map satisfies some certain c...In this paper, we will construct a new class of subadditive set-valued maps and use Cantor theorem to prove that the set-valued map has an unique additive selection map when the set-valued map satisfies some certain conditions, and then compare the obtained result with the well-known results.展开更多
In this paper, firstly, a new notion of generalized cone convex set-valued map is introduced in real normed spaces. Secondly, a property of the generalized cone convex set-valued map involving the contingent epideriva...In this paper, firstly, a new notion of generalized cone convex set-valued map is introduced in real normed spaces. Secondly, a property of the generalized cone convex set-valued map involving the contingent epiderivative is obtained. Finally, as the applications of this property, we use the contingent epiderivative to establish optimality conditions of the set-valued optimization problem with generalized cone convex set-valued maps in the sense of Henig proper efficiency. The results obtained in this paper generalize and improve some known results in the literature.展开更多
In this paper, we consider the following subadditive set-valued map F ： X→P0（Y） ：where r and s are two natural numbers. And we discuss the existence and unique problem of additive selection maps for the above set...In this paper, we consider the following subadditive set-valued map F ： X→P0（Y） ：where r and s are two natural numbers. And we discuss the existence and unique problem of additive selection maps for the above set-valued map.展开更多
In this paper, we discuss the unique existent problems of a additive selection map for the following map F:X→P0(Y):F(r∑i=1αixi)■r∑i=1αiF(xi),■xi∈K,i=1, ··· , r,where αi and αi (i = 1, ·&#...In this paper, we discuss the unique existent problems of a additive selection map for the following map F:X→P0(Y):F(r∑i=1αixi)■r∑i=1αiF(xi),■xi∈K,i=1, ··· , r,where αi and αi (i = 1, ··· , r) are all non-negative real numbers.展开更多
In this paper,the dynamics(including shadowing property,expansiveness,topological stability and entropy)of several types of upper semi-continuous set-valued maps are mainly considered from differentiable dynamical sys...In this paper,the dynamics(including shadowing property,expansiveness,topological stability and entropy)of several types of upper semi-continuous set-valued maps are mainly considered from differentiable dynamical systems points of view.It is shown that(1)if f is a hyperbolic endomorphism then for eachε>0 there exists a C^(1)-neighborhood U of f such that the induced set-valued map F_(f,U)has theε-shadowing property,and moreover,if f is an expanding endomorphism then there exists a C^(1)-neighborhood U of f such that the induced set-valued map F_(f,U)has the Lipschitz shadowing property;(2)when a set-valued map F is generated by finite expanding endomorphisms,it has the shadowing property,and moreover,if the collection of the generators has no coincidence point then F is expansive and hence is topologically stable;(3)if f is an expanding endomorphism then for eachε>0 there exists a C^(1)-neighborhood U of f such that h(F_(f,U,ε))=h(f);(4)when F is generated by finite expanding endomorphisms with no coincidence point,the entropy formula of F is given.Furthermore,the dynamics of the set-valued maps based on discontinuous maps on the interval are also considered.展开更多
This paper deals with higher-order optimality conditions for Henig effcient solutions of set-valued optimization problems.By virtue of the higher-order tangent sets, necessary and suffcient conditions are obtained for...This paper deals with higher-order optimality conditions for Henig effcient solutions of set-valued optimization problems.By virtue of the higher-order tangent sets, necessary and suffcient conditions are obtained for Henig effcient solutions of set-valued optimization problems whose constraint condition is determined by a fixed set.展开更多
In complete metric spaces, the common fixed point theorems for sequences of φ-type contraction set-valued mappings are established, and the corresponding random com- mon fixed point theorems for set-valued mappings a...In complete metric spaces, the common fixed point theorems for sequences of φ-type contraction set-valued mappings are established, and the corresponding random com- mon fixed point theorems for set-valued mappings are also obtained.展开更多
Image-maps,a hybrid design with satellite images as background and map symbols uploaded,aim to combine the advantages of maps’high interpretation efficiency and satellite images’realism.The usability of image-maps i...Image-maps,a hybrid design with satellite images as background and map symbols uploaded,aim to combine the advantages of maps’high interpretation efficiency and satellite images’realism.The usability of image-maps is influenced by the representations of background images and map symbols.Many researchers explored the optimizations for background images and symbolization techniques for symbols to reduce the complexity of image-maps and improve the usability.However,little literature was found for the optimum amount of symbol loading.This study focuses on the effects of background image complexity and map symbol load on the usability(i.e.,effectiveness and efficiency)of image-maps.Experiments were conducted by user studies via eye-tracking equipment and an online questionnaire survey.Experimental data sets included image-maps with ten levels of map symbol load in ten areas.Forty volunteers took part in the target searching experiments.It has been found that the usability,i.e.,average time viewed(efficiency)and average revisits(effectiveness)of targets recorded,is influenced by the complexity of background images,a peak exists for optimum symbol load for an image-map.The optimum levels for symbol load for different image-maps also have a peak when the complexity of the background image/image map increases.The complexity of background images serves as a guideline for optimum map symbol load in image-map design.This study enhanced user experience by optimizing visual clarity and managing cognitive load.Understanding how these factors interact can help create adaptive maps that maintain clarity and usability,guiding AI algorithms to adjust symbol density based on user context.This research establishes the practices for map design,making cartographic tools more innovative and more user-centric.展开更多
Topographic maps,as essential tools and sources of information for geographic research,contain precise spatial locations and rich map features,and they illustrate spatio-temporal information on the distribution and di...Topographic maps,as essential tools and sources of information for geographic research,contain precise spatial locations and rich map features,and they illustrate spatio-temporal information on the distribution and differences of various surface features.Currently,topographic maps are mainly stored in raster and vector formats.Extraction of the spatio-temporal knowledge in the maps—such as spatial distribution patterns,feature relationships,and dynamic evolution—still primarily relies on manual interpretation.However,manual interpretation is time-consuming and laborious,especially for large-scale,long-term map knowledge extraction and application.With the development of artificial intelligence technology,it is possible to improve the automation level of map knowledge interpretation.Therefore,the present study proposes an automatic interpretation method for raster topographic map knowledge based on deep learning.To address the limitations of current data-driven intelligent technology in learning map spatial relations and cognitive logic,we establish a formal description of map knowledge by mapping the relationship between map knowledge and features,thereby ensuring interpretation accuracy.Subsequently,deep learning techniques are employed to extract map features automatically,and the spatio-temporal knowledge is constructed by combining formal descriptions of geographic feature knowledge.Validation experiments demonstrate that the proposed method effectively achieves automatic interpretation of spatio-temporal knowledge of geographic features in maps,with an accuracy exceeding 80%.The findings of the present study contribute to machine understanding of spatio-temporal differences in map knowledge and advances the intelligent interpretation and utilization of cartographic information.展开更多
We introduced a new class of fuzzy set-valued variational inclusions with (H,η)-monotone mappings. Using the resolvent operator method in Hilbert spaces, we suggested a new proximal point algorithm for finding approx...We introduced a new class of fuzzy set-valued variational inclusions with (H,η)-monotone mappings. Using the resolvent operator method in Hilbert spaces, we suggested a new proximal point algorithm for finding approximate solutions, which strongly converge to the exact solution of a fuzzy set-valued variational inclusion with (H,η)-monotone. The results improved and generalized the general quasi-variational inclusions with fuzzy set-valued mappings proposed by Jin and Tian Jin MM, Perturbed proximal point algorithm for general quasi-variational inclusions with fuzzy set-valued mappings, OR Transactions, 2005, 9(3): 31-38, (In Chinese); Tian YX, Generalized nonlinear implicit quasi-variational inclusions with fuzzy mappings, Computers & Mathematics with Applications, 2001, 42: 101-108.展开更多
Formal concept analysis （FCA） is a discipline that studied the hierarchical structures induced by a binary relation between a pair of sets, and applies in data analysis, information retrieval, knowledge discovery, e...Formal concept analysis （FCA） is a discipline that studied the hierarchical structures induced by a binary relation between a pair of sets, and applies in data analysis, information retrieval, knowledge discovery, etc. In this paper, it is shown that a formal context T is equivalent to a set-valued mapping S ： G → P（М）, and formal concepts could be defined in the set-valued mapping S. It is known that the topology and set-valued mapping are linked. Hence, the advantage of this paper is that the conclusion make us to construct formal concept lattice based on the topology.展开更多
In this paper,we discuss the continuities of some natural mappings on pointcompact continuous set-valued mapping spaces with compact-open topology and obtain the properties of set-valued injective mappings,set-valued ...In this paper,we discuss the continuities of some natural mappings on pointcompact continuous set-valued mapping spaces with compact-open topology and obtain the properties of set-valued injective mappings,set-valued diagonal mappings,induced mappings,set-valued evaluation mappings,set-valued topological sum mappings and set-valued topological product mappings.展开更多
Under the conditions of compatility or sub -c ompatility between a sigle-valued mapping and set-valued mapping, this paper d iscusses the existence of common fixed points for two set-valued mappings and a single-value...Under the conditions of compatility or sub -c ompatility between a sigle-valued mapping and set-valued mapping, this paper d iscusses the existence of common fixed points for two set-valued mappings and a single-valued mapping in complete, convex matric spaces. We extend and develop the main results.展开更多
A new conception of generalized set-valued Ф-hemi-contractive mapping in Banach spaces is presented. Some strong convergence theorems of Ishikawa and Mann iterative approximation with errors is proved. The results in...A new conception of generalized set-valued Ф-hemi-contractive mapping in Banach spaces is presented. Some strong convergence theorems of Ishikawa and Mann iterative approximation with errors is proved. The results in this paper improve and extend the earlier results.展开更多
文摘Several equivalent statements of generalized subconvexlike set-valued map are established in ordered linear spaces. Using vector closure, we introduce Benson proper efficient solution of vector optimization problem. Under the assumption of generalized subconvexlikeness, scalarization, multiplier and saddle point theorems are obtained in the sense of Benson proper efficiency.
文摘A kind of tangent derivative and the concepts of strong and weak * pseudoconvexity for a set-valued map are introduced. By the standard separation theorems of the convex sets and cones the optimality Fritz John condition of set-valued optimization under Benson proper efficiency is established, its sufficience is discussed. The form of the optimality conditions obtained here completely tally with the classical results when the set-valued map is specialized to be a single-valued map.
基金Supported by the Natural Science Foundation of the Education Department of Henan Province（2004110008）
文摘In this paper,the ε-super effcient solution for set-valued map vector optimization in locally convex space is introduced.And under the assumption of the nearly generalized cone-subconvexlikeness for set-valued maps,used new methods,the scalarization theorem and Lagrange multiplier theorem for ε-super effcient solution are established.
文摘The definitions of cone-subconvexlike set-valued maps and generalized cone-subconvexlike set-valued maps in topological vector spaces are defined by using the relative interiors of ordering cone. The relationships between the two classes of set-valued maps are investigated, and some properties of them are shown. A Gordan type alternative theorem under the assumption of generalized cone-subconvexlikeness of set-valued maps is proved by applying convex separation theorems involving the relative interiors in infinite dimensional spaces. Finally a necessary optimality condition theorem is shown for a general kind of set-valued vector optimization in a sense of weak E-minimizer.
基金Foundation item： Supported by the Natural Science Foundation of China（10871216） Supported by the Natural Science Foundation Project of CQ CSTC（2008BB0346, 2007BB0441） Supported by the Excellent Young Teachers Program of Chongqing Jiaotong University（EYT08-016） Acknowledgement The author would like to thank the anonymous referee for the valuable remarks that helped considerably to correct and to improve the presentation.
文摘In locally convex Hausdorff topological vector spaces,ε-strongly efficient solutions for vector optimization with set-valued maps are discussed.Firstly,ε-strongly efficient point of set is introduced.Secondly,under the nearly cone-subconvexlike set-valued maps,the theorem of scalarization for vector optimization is obtained.Finally,optimality conditions of ε-strongly efficient solutions for vector optimization with generalized inequality constraints and equality constraints are obtained.
基金Supported by the National Natural Science Foundation of China（11361064）
文摘In this paper, we will construct a new class of subadditive set-valued maps and use Cantor theorem to prove that the set-valued map has an unique additive selection map when the set-valued map satisfies some certain conditions, and then compare the obtained result with the well-known results.
基金supported by the National Nature Science Foundation of China(11431004,11471291)the General Project of Chongqing Frontier and Applied Foundation Research(cstc2015jcyj A00050)the Key Project of Chongqing Frontier and Applied Foundation Research(cstc2017jcyj BX0055,cstc2015jcyj BX0113)
文摘In this paper, firstly, a new notion of generalized cone convex set-valued map is introduced in real normed spaces. Secondly, a property of the generalized cone convex set-valued map involving the contingent epiderivative is obtained. Finally, as the applications of this property, we use the contingent epiderivative to establish optimality conditions of the set-valued optimization problem with generalized cone convex set-valued maps in the sense of Henig proper efficiency. The results obtained in this paper generalize and improve some known results in the literature.
基金Supported by Science Foundation of Education Committee of Jilin Province of China([2011]No.434)
文摘In this paper, we consider the following subadditive set-valued map F ： X→P0（Y） ：where r and s are two natural numbers. And we discuss the existence and unique problem of additive selection maps for the above set-valued map.
基金Foundation item： Supported by the Science Foundation of Education Committee of Jilin Province（[2011] No434）
文摘In this paper, we discuss the unique existent problems of a additive selection map for the following map F:X→P0(Y):F(r∑i=1αixi)■r∑i=1αiF(xi),■xi∈K,i=1, ··· , r,where αi and αi (i = 1, ··· , r) are all non-negative real numbers.
文摘In this paper,the dynamics(including shadowing property,expansiveness,topological stability and entropy)of several types of upper semi-continuous set-valued maps are mainly considered from differentiable dynamical systems points of view.It is shown that(1)if f is a hyperbolic endomorphism then for eachε>0 there exists a C^(1)-neighborhood U of f such that the induced set-valued map F_(f,U)has theε-shadowing property,and moreover,if f is an expanding endomorphism then there exists a C^(1)-neighborhood U of f such that the induced set-valued map F_(f,U)has the Lipschitz shadowing property;(2)when a set-valued map F is generated by finite expanding endomorphisms,it has the shadowing property,and moreover,if the collection of the generators has no coincidence point then F is expansive and hence is topologically stable;(3)if f is an expanding endomorphism then for eachε>0 there exists a C^(1)-neighborhood U of f such that h(F_(f,U,ε))=h(f);(4)when F is generated by finite expanding endomorphisms with no coincidence point,the entropy formula of F is given.Furthermore,the dynamics of the set-valued maps based on discontinuous maps on the interval are also considered.
基金Supported by the National Natural Science Foundation of China（10871216） Supported by the Science and Technology Research Project of Chongqing Municipal Education Commission（KJ100419） Supported by the Natural Science Foundation Project of CQ CSTC（cstcjjA00019）
文摘This paper deals with higher-order optimality conditions for Henig effcient solutions of set-valued optimization problems.By virtue of the higher-order tangent sets, necessary and suffcient conditions are obtained for Henig effcient solutions of set-valued optimization problems whose constraint condition is determined by a fixed set.
基金Foundation item： Supported by the Science Foundation from the Ministry of Education of Jiangsu Province（04KJD110168, 06KJBll0107）
文摘In complete metric spaces, the common fixed point theorems for sequences of φ-type contraction set-valued mappings are established, and the corresponding random com- mon fixed point theorems for set-valued mappings are also obtained.
基金National Natural Science Foundation of China(No.42301518)Hubei Key Laboratory of Regional Development and Environmental Response(No.2023(A)002)Key Laboratory of the Evaluation and Monitoring of Southwest Land Resources(Ministry of Education)(No.TDSYS202304).
文摘Image-maps,a hybrid design with satellite images as background and map symbols uploaded,aim to combine the advantages of maps’high interpretation efficiency and satellite images’realism.The usability of image-maps is influenced by the representations of background images and map symbols.Many researchers explored the optimizations for background images and symbolization techniques for symbols to reduce the complexity of image-maps and improve the usability.However,little literature was found for the optimum amount of symbol loading.This study focuses on the effects of background image complexity and map symbol load on the usability(i.e.,effectiveness and efficiency)of image-maps.Experiments were conducted by user studies via eye-tracking equipment and an online questionnaire survey.Experimental data sets included image-maps with ten levels of map symbol load in ten areas.Forty volunteers took part in the target searching experiments.It has been found that the usability,i.e.,average time viewed(efficiency)and average revisits(effectiveness)of targets recorded,is influenced by the complexity of background images,a peak exists for optimum symbol load for an image-map.The optimum levels for symbol load for different image-maps also have a peak when the complexity of the background image/image map increases.The complexity of background images serves as a guideline for optimum map symbol load in image-map design.This study enhanced user experience by optimizing visual clarity and managing cognitive load.Understanding how these factors interact can help create adaptive maps that maintain clarity and usability,guiding AI algorithms to adjust symbol density based on user context.This research establishes the practices for map design,making cartographic tools more innovative and more user-centric.
基金Deep-time Digital Earth(DDE)Big Science Program(No.GJ-C03-SGF-2025-004)National Natural Science Foundation of China(No.42394063)Sichuan Science and Technology Program(No.2025ZNSFSC0325).
文摘Topographic maps,as essential tools and sources of information for geographic research,contain precise spatial locations and rich map features,and they illustrate spatio-temporal information on the distribution and differences of various surface features.Currently,topographic maps are mainly stored in raster and vector formats.Extraction of the spatio-temporal knowledge in the maps—such as spatial distribution patterns,feature relationships,and dynamic evolution—still primarily relies on manual interpretation.However,manual interpretation is time-consuming and laborious,especially for large-scale,long-term map knowledge extraction and application.With the development of artificial intelligence technology,it is possible to improve the automation level of map knowledge interpretation.Therefore,the present study proposes an automatic interpretation method for raster topographic map knowledge based on deep learning.To address the limitations of current data-driven intelligent technology in learning map spatial relations and cognitive logic,we establish a formal description of map knowledge by mapping the relationship between map knowledge and features,thereby ensuring interpretation accuracy.Subsequently,deep learning techniques are employed to extract map features automatically,and the spatio-temporal knowledge is constructed by combining formal descriptions of geographic feature knowledge.Validation experiments demonstrate that the proposed method effectively achieves automatic interpretation of spatio-temporal knowledge of geographic features in maps,with an accuracy exceeding 80%.The findings of the present study contribute to machine understanding of spatio-temporal differences in map knowledge and advances the intelligent interpretation and utilization of cartographic information.
基金the Natural Science Foundation of China (No. 10471151)the Educational Science Foundation of Chongqing (KJ051307).
文摘We introduced a new class of fuzzy set-valued variational inclusions with (H,η)-monotone mappings. Using the resolvent operator method in Hilbert spaces, we suggested a new proximal point algorithm for finding approximate solutions, which strongly converge to the exact solution of a fuzzy set-valued variational inclusion with (H,η)-monotone. The results improved and generalized the general quasi-variational inclusions with fuzzy set-valued mappings proposed by Jin and Tian Jin MM, Perturbed proximal point algorithm for general quasi-variational inclusions with fuzzy set-valued mappings, OR Transactions, 2005, 9(3): 31-38, (In Chinese); Tian YX, Generalized nonlinear implicit quasi-variational inclusions with fuzzy mappings, Computers & Mathematics with Applications, 2001, 42: 101-108.
基金the Young Foundation of Sichuan Province(06ZQ026-037) the Education Department Foundation of Sichuan Province(2005A1212006A084)
文摘Formal concept analysis （FCA） is a discipline that studied the hierarchical structures induced by a binary relation between a pair of sets, and applies in data analysis, information retrieval, knowledge discovery, etc. In this paper, it is shown that a formal context T is equivalent to a set-valued mapping S ： G → P（М）, and formal concepts could be defined in the set-valued mapping S. It is known that the topology and set-valued mapping are linked. Hence, the advantage of this paper is that the conclusion make us to construct formal concept lattice based on the topology.
基金Supported by the Science Foundation of Hangzhou Normal University（02010180）
文摘In this paper,we discuss the continuities of some natural mappings on pointcompact continuous set-valued mapping spaces with compact-open topology and obtain the properties of set-valued injective mappings,set-valued diagonal mappings,induced mappings,set-valued evaluation mappings,set-valued topological sum mappings and set-valued topological product mappings.
文摘Under the conditions of compatility or sub -c ompatility between a sigle-valued mapping and set-valued mapping, this paper d iscusses the existence of common fixed points for two set-valued mappings and a single-valued mapping in complete, convex matric spaces. We extend and develop the main results.
文摘A new conception of generalized set-valued Ф-hemi-contractive mapping in Banach spaces is presented. Some strong convergence theorems of Ishikawa and Mann iterative approximation with errors is proved. The results in this paper improve and extend the earlier results.
