An analytic function f in the unit disk D ：= {z ∈ C ： ｜z｜ 〈 1}, standardly normalized, is called close-to-convex with respect to the Koebe function k（z） ：= z/（1-z）2, z ∈ D, if there exists δ ∈ （-π/2,...An analytic function f in the unit disk D ：= {z ∈ C ： ｜z｜ 〈 1}, standardly normalized, is called close-to-convex with respect to the Koebe function k（z） ：= z/（1-z）2, z ∈ D, if there exists δ ∈ （-π/2,π/2） such that Re {eiδ（1-z）2f′（z）} 〉 0, z ∈ D. For the class C（k） of all close-to-convex functions with respect to k, related to the class of functions convex in the positive direction of the imaginary axis, the Fekete-Szego problem is studied.展开更多
In this paper, we introduce and study some new classes of biconvex functions with respect to an arbitrary function and a bifunction, which are called the higher order strongly biconvex functions. These functions are n...In this paper, we introduce and study some new classes of biconvex functions with respect to an arbitrary function and a bifunction, which are called the higher order strongly biconvex functions. These functions are nonconvex functions and include the biconvex function, convex functions, and <i>k</i>-convex as special cases. We study some properties of the higher order strongly biconvex functions. Several parallelogram laws for inner product spaces are obtained as novel applications of the higher order strongly biconvex affine functions. It is shown that the minimum of generalized biconvex functions on the <i>k</i>-biconvex sets can be characterized by a class of equilibrium problems, which is called the higher order strongly biequilibrium problems. Using the auxiliary technique involving the Bregman functions, several new inertial type methods for solving the higher order strongly biequilibrium problem are suggested and investigated. Convergence analysis of the proposed methods is considered under suitable conditions. Several important special cases are obtained as novel applications of the derived results. Some open problems are also suggested for future research.展开更多
Characterizations of differentiability are obtained for continuous convex functions defined on nonempty open convex sets of Banach spaces as a generalization and application of a mumber of mathematicians several years...Characterizations of differentiability are obtained for continuous convex functions defined on nonempty open convex sets of Banach spaces as a generalization and application of a mumber of mathematicians several years effort, and a characteristic theorem is given for Banach spaces which are (weak) Asplund spaces.展开更多
In this paper, the Tφ-convex functions were introduced as a generalizations of convex functions. Then the characteristics of the Tφ-convex functions were discussed. Furthermore, some new inequalities for the Tφ-con...In this paper, the Tφ-convex functions were introduced as a generalizations of convex functions. Then the characteristics of the Tφ-convex functions were discussed. Furthermore, some new inequalities for the Tφ-convex functions were derived.展开更多
In this paper, we first introduce the concept ＂harmonically convex functions＂ in the second sense and establish several Hermite-Hadamard type inequalities for harmonically convex functions in the second sense. Final...In this paper, we first introduce the concept ＂harmonically convex functions＂ in the second sense and establish several Hermite-Hadamard type inequalities for harmonically convex functions in the second sense. Finally, some applications to special mean are shown.展开更多
Schur convexity, Schur geometrical convexity and Schur harmonic convexityof a class of symmetric functions are investigated. As consequences some knowninequalities are generalized. In addition, a class of geometric in...Schur convexity, Schur geometrical convexity and Schur harmonic convexityof a class of symmetric functions are investigated. As consequences some knowninequalities are generalized. In addition, a class of geometric inequalities involvingn-dimensional simplex in n-dimensional Euclidean space En and several matrix inequalitiesare established to show the applications of our results.展开更多
In this paper, we shall establish an inequality for differentiable co-ordinated convex functions on a rectangle from the plane. It is connected with the left side and right side of extended Hermite-Hadamard inequality...In this paper, we shall establish an inequality for differentiable co-ordinated convex functions on a rectangle from the plane. It is connected with the left side and right side of extended Hermite-Hadamard inequality in two variables. In addition, six other inequalities are derived from it for some refinements. Finally, this paper shows some examples that these inequalities are able to be applied to some special means.展开更多
In this paper,by making use of Divergence theorem for multiple integrals,we establish some integral inequalities for Schur convex functions defined on bodies B⊂R^(n)that are symmetric,convex and have nonempty interior...In this paper,by making use of Divergence theorem for multiple integrals,we establish some integral inequalities for Schur convex functions defined on bodies B⊂R^(n)that are symmetric,convex and have nonempty interiors.Examples for three dimensional balls are also provided.展开更多
In this article,some basic and important properties of spherically convex functions,such as the Lipschitz-continuity,are investigated.It is shown that,under a weaker condition,every family of spherically convex functi...In this article,some basic and important properties of spherically convex functions,such as the Lipschitz-continuity,are investigated.It is shown that,under a weaker condition,every family of spherically convex functions is equi-Lipschitzian on each closed spherically convex subset contained in the relative interior of their common domain,and from which a powerful result is derived:the pointwise convergence of a sequence of spherically convex functions implies its uniform convergence on each closed spherically convex subset contained in the relative interior of their common domain.展开更多
Given α∈[0, 1], let hα（z） ：= z/（1 - αz）, z ∈ D ：= {z ∈ C： ｜z｜ 〈 1}. An analytic standardly normalized function f in D is called close-to-convex with respect to hα if there exists δ ∈ （-π/2, π/2）...Given α∈[0, 1], let hα（z） ：= z/（1 - αz）, z ∈ D ：= {z ∈ C： ｜z｜ 〈 1}. An analytic standardly normalized function f in D is called close-to-convex with respect to hα if there exists δ ∈ （-π/2, π/2） such that Re{e^iδ zf′（z）/hα（z）} 〉 0, z ∈ D. For the class l（hα） of all close-to-convex functions with respect to hα, the Fekete-Szego problem is studied.展开更多
A class of functions and a sort of nonlinear programming called respectively E-convex functions and E-convex programming were presented and studied recently by Youness in [1], In this paper, we point out the most resu...A class of functions and a sort of nonlinear programming called respectively E-convex functions and E-convex programming were presented and studied recently by Youness in [1], In this paper, we point out the most results for .E-convex functions and E-convex programming in [1] are not true by six counter examples.展开更多
We are investigating the distributed optimization problem,where a network of nodes works together to minimize a global objective that is a finite sum of their stored local functions.Since nodes exchange optimization p...We are investigating the distributed optimization problem,where a network of nodes works together to minimize a global objective that is a finite sum of their stored local functions.Since nodes exchange optimization parameters through the wireless network,large-scale training models can create communication bottlenecks,resulting in slower training times.To address this issue,CHOCO-SGD was proposed,which allows compressing information with arbitrary precision without reducing the convergence rate for strongly convex objective functions.Nevertheless,most convex functions are not strongly convex(such as logistic regression or Lasso),which raises the question of whether this algorithm can be applied to non-strongly convex functions.In this paper,we provide the first theoretical analysis of the convergence rate of CHOCO-SGD on non-strongly convex objectives.We derive a sufficient condition,which limits the fidelity of compression,to guarantee convergence.Moreover,our analysis demonstrates that within the fidelity threshold,this algorithm can significantly reduce transmission burden while maintaining the same convergence rate order as its no-compression equivalent.Numerical experiments further validate the theoretical findings by demonstrating that CHOCO-SGD improves communication efficiency and keeps the same convergence rate order simultaneously.And experiments also show that the algorithm fails to converge with low compression fidelity and in time-varying topologies.Overall,our study offers valuable insights into the potential applicability of CHOCO-SGD for non-strongly convex objectives.Additionally,we provide practical guidelines for researchers seeking to utilize this algorithm in real-world scenarios.展开更多
This paper develops a quadratic function convex approximation approach to deal with the negative definite problem of the quadratic function induced by stability analysis of linear systems with time-varying delays.By i...This paper develops a quadratic function convex approximation approach to deal with the negative definite problem of the quadratic function induced by stability analysis of linear systems with time-varying delays.By introducing two adjustable parameters and two free variables,a novel convex function greater than or equal to the quadratic function is constructed,regardless of the sign of the coefficient in the quadratic term.The developed lemma can also be degenerated into the existing quadratic function negative-determination(QFND)lemma and relaxed QFND lemma respectively,by setting two adjustable parameters and two free variables as some particular values.Moreover,for a linear system with time-varying delays,a relaxed stability criterion is established via our developed lemma,together with the quivalent reciprocal combination technique and the Bessel-Legendre inequality.As a result,the conservatism can be reduced via the proposed approach in the context of constructing Lyapunov-Krasovskii functionals for the stability analysis of linear time-varying delay systems.Finally,the superiority of our results is illustrated through three numerical examples.展开更多
Let a_0 < a_1 < … < a_n be positive integers with sums Σ_(i=0)~n∈_ia_i(∈_i = 0,1) distinct. P. Erdos conjectured that Σ_(i=0)~n 1/a_i ≤ Σ_(i=0)~n 1/2~i. Thebest known result along this line is that of ...Let a_0 < a_1 < … < a_n be positive integers with sums Σ_(i=0)~n∈_ia_i(∈_i = 0,1) distinct. P. Erdos conjectured that Σ_(i=0)~n 1/a_i ≤ Σ_(i=0)~n 1/2~i. Thebest known result along this line is that of Chen: Let f be any given convex decreasing function on[A, B] with α_0, α_1, …, α_n , β_0, β_1, …, β_n being real numbers in [A, B] with α_0 ≤α_1 ≤ … ≤ α_n, Σ_(i=0)~n α_i ≥ Σ_(i=0)~n β_i, k = 0, …, n. Then Σ_(i=0)~n f(α_i) ≤Σ_(i=0)~n f(β_i). In this paper, we obtain two generalizations of the above result; each is ofspecial interest in itself. We prove:Theorem 1 Let f and g be two given non-negative convex decreasing functions on [A, B], and α_0,α_1, …, α_n , β_0, β_1, …, β_n, α'_0, α'_1, …, α'_n , β'_0, β'_1, …, β'_n be realnumbers in [A, B] with α'_0 ≤ α'_1 ≤ … ≤ α_n. Then Σ_(i=0)~n f(α_i)g(α'_i) ≤ Σ_(i=0)~nf(β_i)g(β'_i), k = 0, …, n. Theorem 2 Let f be any given convex decreasing function on [A, B]with k_0, k_1, …, k_n being nonnegative real numbers and α_0, α_1, …, α_n , β_0, β_1, …,β_n being real numbers in [A, B] with α_0 ≤ α_1 ≤ … ≤ α_n, Σ_(i=0)~t k_i α_i ≥ Σ_(i=0)~tk_iβ_i, t = 0, …, n. Then Σ_(i=0)~t k_if(α_i) ≤ Σ_(i=0)~t k_if_(β_i).展开更多
We discuss the relationship between Lipschitz functions and convex functions. By these relations, we give a sufficient condition for the set of points where Lipschitz functions on a Hilbert space is Frechet differenti...We discuss the relationship between Lipschitz functions and convex functions. By these relations, we give a sufficient condition for the set of points where Lipschitz functions on a Hilbert space is Frechet differentiate to be residual.展开更多
Purpose–The purpose of this paper is to describe imperialist competitive algorithm(ICA),a novel socio-politically inspired optimization strategy for proposing a fuzzy variant of this algorithm.ICA is a meta-heuristic...Purpose–The purpose of this paper is to describe imperialist competitive algorithm(ICA),a novel socio-politically inspired optimization strategy for proposing a fuzzy variant of this algorithm.ICA is a meta-heuristic algorithm for dealing with different optimization tasks.The basis of the algorithm is inspired by imperialistic competition.It attempts to present the social policy of imperialisms(referred to empires)to control more countries(referred to colonies)and use their sources.If one empire loses its power,among the others making a competition to take possession of it.Design/methodology/approach–In fuzzy imperialist competitive algorithm(FICA),the colonies have a degree of belonging to their imperialists and the top imperialist,as in fuzzy logic,rather than belonging completely to just one empire therefore the colonies move toward the superior empire and their relevant empires.Simultaneously for balancing the exploration and exploitation abilities of the ICA.The algorithms are used for optimization have shortcoming to deal with accuracy rate and local optimum trap and they need complex tuning procedures.FICA is proposed a way for optimizing convex function with high accuracy and avoiding to trap in local optima rather than using original ICA algorithm by implementing fuzzy logic on it.Findings–Therefore several solution procedures,including ICA,FICA,genetic algorithm,particle swarm optimization,tabu search and simulated annealing optimization algorithm are considered.Finally numerical experiments are carried out to evaluate the effectiveness of models as well as solution procedures.Test results present the suitability of the proposed fuzzy ICA for convex functions with little fluctuations.Originality/value–The proposed evolutionary algorithm,FICA,can be used in diverse areas of optimization problems where convex functions properties are appeared including,industrial planning,resource allocation,scheduling,decision making,pattern recognition and machine learning(optimization techniques;fuzzy logic;convex functions).展开更多
This paper through discussing subdifferentiability and convexity of convex functions shows that a Banach space admits an equivalent uniformly [locally uniformly, strictly] convex norm if and only if there exists a con...This paper through discussing subdifferentiability and convexity of convex functions shows that a Banach space admits an equivalent uniformly [locally uniformly, strictly] convex norm if and only if there exists a continuous uniformly [locally uniformly, strictly] convex function on some nonempty open convex subset of the space and presents some characterizations of super-reflexive Banach spaces.展开更多
文摘An analytic function f in the unit disk D ：= {z ∈ C ： ｜z｜ 〈 1}, standardly normalized, is called close-to-convex with respect to the Koebe function k（z） ：= z/（1-z）2, z ∈ D, if there exists δ ∈ （-π/2,π/2） such that Re {eiδ（1-z）2f′（z）} 〉 0, z ∈ D. For the class C（k） of all close-to-convex functions with respect to k, related to the class of functions convex in the positive direction of the imaginary axis, the Fekete-Szego problem is studied.
文摘In this paper, we introduce and study some new classes of biconvex functions with respect to an arbitrary function and a bifunction, which are called the higher order strongly biconvex functions. These functions are nonconvex functions and include the biconvex function, convex functions, and <i>k</i>-convex as special cases. We study some properties of the higher order strongly biconvex functions. Several parallelogram laws for inner product spaces are obtained as novel applications of the higher order strongly biconvex affine functions. It is shown that the minimum of generalized biconvex functions on the <i>k</i>-biconvex sets can be characterized by a class of equilibrium problems, which is called the higher order strongly biequilibrium problems. Using the auxiliary technique involving the Bregman functions, several new inertial type methods for solving the higher order strongly biequilibrium problem are suggested and investigated. Convergence analysis of the proposed methods is considered under suitable conditions. Several important special cases are obtained as novel applications of the derived results. Some open problems are also suggested for future research.
文摘Characterizations of differentiability are obtained for continuous convex functions defined on nonempty open convex sets of Banach spaces as a generalization and application of a mumber of mathematicians several years effort, and a characteristic theorem is given for Banach spaces which are (weak) Asplund spaces.
基金Project supported by the National Natural Science Foundation of China（Grant No.10271071）
文摘In this paper, the Tφ-convex functions were introduced as a generalizations of convex functions. Then the characteristics of the Tφ-convex functions were discussed. Furthermore, some new inequalities for the Tφ-convex functions were derived.
基金The Doctoral Programs Foundation(20113401110009)of Education Ministry of ChinaNatural Science Research Project(2012kj11)of Hefei Normal University+1 种基金Universities Natural Science Foundation(KJ2013A220)of Anhui ProvinceResearch Project of Graduates Innovation Fund(2014yjs02)
文摘In this paper, we first introduce the concept ＂harmonically convex functions＂ in the second sense and establish several Hermite-Hadamard type inequalities for harmonically convex functions in the second sense. Finally, some applications to special mean are shown.
基金The Doctoral Programs Foundation(20113401110009) of Education Ministry of Chinathe Natural Science Research Project(2012kj11) of Hefei Normal Universitythe NSF(KJ2013A220) of Anhui Province
文摘Schur convexity, Schur geometrical convexity and Schur harmonic convexityof a class of symmetric functions are investigated. As consequences some knowninequalities are generalized. In addition, a class of geometric inequalities involvingn-dimensional simplex in n-dimensional Euclidean space En and several matrix inequalitiesare established to show the applications of our results.
文摘In this paper, we shall establish an inequality for differentiable co-ordinated convex functions on a rectangle from the plane. It is connected with the left side and right side of extended Hermite-Hadamard inequality in two variables. In addition, six other inequalities are derived from it for some refinements. Finally, this paper shows some examples that these inequalities are able to be applied to some special means.
文摘In this paper,by making use of Divergence theorem for multiple integrals,we establish some integral inequalities for Schur convex functions defined on bodies B⊂R^(n)that are symmetric,convex and have nonempty interiors.Examples for three dimensional balls are also provided.
基金Supported by the National NSF of China(Grant Nos.12071334,11671293)。
文摘In this article,some basic and important properties of spherically convex functions,such as the Lipschitz-continuity,are investigated.It is shown that,under a weaker condition,every family of spherically convex functions is equi-Lipschitzian on each closed spherically convex subset contained in the relative interior of their common domain,and from which a powerful result is derived:the pointwise convergence of a sequence of spherically convex functions implies its uniform convergence on each closed spherically convex subset contained in the relative interior of their common domain.
文摘Given α∈[0, 1], let hα（z） ：= z/（1 - αz）, z ∈ D ：= {z ∈ C： ｜z｜ 〈 1}. An analytic standardly normalized function f in D is called close-to-convex with respect to hα if there exists δ ∈ （-π/2, π/2） such that Re{e^iδ zf′（z）/hα（z）} 〉 0, z ∈ D. For the class l（hα） of all close-to-convex functions with respect to hα, the Fekete-Szego problem is studied.
基金National Natural Science Foundation of China(10261001)Science Foundation of Guangxi(0236001)
文摘A class of functions and a sort of nonlinear programming called respectively E-convex functions and E-convex programming were presented and studied recently by Youness in [1], In this paper, we point out the most results for .E-convex functions and E-convex programming in [1] are not true by six counter examples.
基金supported in part by the Shanghai Natural Science Foundation under the Grant 22ZR1407000.
文摘We are investigating the distributed optimization problem,where a network of nodes works together to minimize a global objective that is a finite sum of their stored local functions.Since nodes exchange optimization parameters through the wireless network,large-scale training models can create communication bottlenecks,resulting in slower training times.To address this issue,CHOCO-SGD was proposed,which allows compressing information with arbitrary precision without reducing the convergence rate for strongly convex objective functions.Nevertheless,most convex functions are not strongly convex(such as logistic regression or Lasso),which raises the question of whether this algorithm can be applied to non-strongly convex functions.In this paper,we provide the first theoretical analysis of the convergence rate of CHOCO-SGD on non-strongly convex objectives.We derive a sufficient condition,which limits the fidelity of compression,to guarantee convergence.Moreover,our analysis demonstrates that within the fidelity threshold,this algorithm can significantly reduce transmission burden while maintaining the same convergence rate order as its no-compression equivalent.Numerical experiments further validate the theoretical findings by demonstrating that CHOCO-SGD improves communication efficiency and keeps the same convergence rate order simultaneously.And experiments also show that the algorithm fails to converge with low compression fidelity and in time-varying topologies.Overall,our study offers valuable insights into the potential applicability of CHOCO-SGD for non-strongly convex objectives.Additionally,we provide practical guidelines for researchers seeking to utilize this algorithm in real-world scenarios.
基金the National Natural Science Foundation of China(62273058,U22A2045)the Key Science and Technology Projects of Jilin Province(20200401075GX)the Youth Science and Technology Innovation and Entrepreneurship Outstanding Talents Project of Jilin Province(20230508043RC)。
文摘This paper develops a quadratic function convex approximation approach to deal with the negative definite problem of the quadratic function induced by stability analysis of linear systems with time-varying delays.By introducing two adjustable parameters and two free variables,a novel convex function greater than or equal to the quadratic function is constructed,regardless of the sign of the coefficient in the quadratic term.The developed lemma can also be degenerated into the existing quadratic function negative-determination(QFND)lemma and relaxed QFND lemma respectively,by setting two adjustable parameters and two free variables as some particular values.Moreover,for a linear system with time-varying delays,a relaxed stability criterion is established via our developed lemma,together with the quivalent reciprocal combination technique and the Bessel-Legendre inequality.As a result,the conservatism can be reduced via the proposed approach in the context of constructing Lyapunov-Krasovskii functionals for the stability analysis of linear time-varying delay systems.Finally,the superiority of our results is illustrated through three numerical examples.
基金supported by the National Natural Science Foundation of China(No.10071016)
文摘Let a_0 < a_1 < … < a_n be positive integers with sums Σ_(i=0)~n∈_ia_i(∈_i = 0,1) distinct. P. Erdos conjectured that Σ_(i=0)~n 1/a_i ≤ Σ_(i=0)~n 1/2~i. Thebest known result along this line is that of Chen: Let f be any given convex decreasing function on[A, B] with α_0, α_1, …, α_n , β_0, β_1, …, β_n being real numbers in [A, B] with α_0 ≤α_1 ≤ … ≤ α_n, Σ_(i=0)~n α_i ≥ Σ_(i=0)~n β_i, k = 0, …, n. Then Σ_(i=0)~n f(α_i) ≤Σ_(i=0)~n f(β_i). In this paper, we obtain two generalizations of the above result; each is ofspecial interest in itself. We prove:Theorem 1 Let f and g be two given non-negative convex decreasing functions on [A, B], and α_0,α_1, …, α_n , β_0, β_1, …, β_n, α'_0, α'_1, …, α'_n , β'_0, β'_1, …, β'_n be realnumbers in [A, B] with α'_0 ≤ α'_1 ≤ … ≤ α_n. Then Σ_(i=0)~n f(α_i)g(α'_i) ≤ Σ_(i=0)~nf(β_i)g(β'_i), k = 0, …, n. Theorem 2 Let f be any given convex decreasing function on [A, B]with k_0, k_1, …, k_n being nonnegative real numbers and α_0, α_1, …, α_n , β_0, β_1, …,β_n being real numbers in [A, B] with α_0 ≤ α_1 ≤ … ≤ α_n, Σ_(i=0)~t k_i α_i ≥ Σ_(i=0)~tk_iβ_i, t = 0, …, n. Then Σ_(i=0)~t k_if(α_i) ≤ Σ_(i=0)~t k_if_(β_i).
文摘We discuss the relationship between Lipschitz functions and convex functions. By these relations, we give a sufficient condition for the set of points where Lipschitz functions on a Hilbert space is Frechet differentiate to be residual.
文摘Purpose–The purpose of this paper is to describe imperialist competitive algorithm(ICA),a novel socio-politically inspired optimization strategy for proposing a fuzzy variant of this algorithm.ICA is a meta-heuristic algorithm for dealing with different optimization tasks.The basis of the algorithm is inspired by imperialistic competition.It attempts to present the social policy of imperialisms(referred to empires)to control more countries(referred to colonies)and use their sources.If one empire loses its power,among the others making a competition to take possession of it.Design/methodology/approach–In fuzzy imperialist competitive algorithm(FICA),the colonies have a degree of belonging to their imperialists and the top imperialist,as in fuzzy logic,rather than belonging completely to just one empire therefore the colonies move toward the superior empire and their relevant empires.Simultaneously for balancing the exploration and exploitation abilities of the ICA.The algorithms are used for optimization have shortcoming to deal with accuracy rate and local optimum trap and they need complex tuning procedures.FICA is proposed a way for optimizing convex function with high accuracy and avoiding to trap in local optima rather than using original ICA algorithm by implementing fuzzy logic on it.Findings–Therefore several solution procedures,including ICA,FICA,genetic algorithm,particle swarm optimization,tabu search and simulated annealing optimization algorithm are considered.Finally numerical experiments are carried out to evaluate the effectiveness of models as well as solution procedures.Test results present the suitability of the proposed fuzzy ICA for convex functions with little fluctuations.Originality/value–The proposed evolutionary algorithm,FICA,can be used in diverse areas of optimization problems where convex functions properties are appeared including,industrial planning,resource allocation,scheduling,decision making,pattern recognition and machine learning(optimization techniques;fuzzy logic;convex functions).
文摘This paper through discussing subdifferentiability and convexity of convex functions shows that a Banach space admits an equivalent uniformly [locally uniformly, strictly] convex norm if and only if there exists a continuous uniformly [locally uniformly, strictly] convex function on some nonempty open convex subset of the space and presents some characterizations of super-reflexive Banach spaces.
