This paper studies a strongly convergent inertial forward-backward-forward algorithm for the variational inequality problem in Hilbert spaces.In our convergence analysis,we do not assume the on-line rule of the inerti...This paper studies a strongly convergent inertial forward-backward-forward algorithm for the variational inequality problem in Hilbert spaces.In our convergence analysis,we do not assume the on-line rule of the inertial parameters and the iterates,which have been assumed by several authors whenever a strongly convergent algorithm with an inertial extrapolation step is proposed for a variational inequality problem.Consequently,our proof arguments are different from what is obtainable in the relevant literature.Finally,we give numerical tests to confirm the theoretical analysis and show that our proposed algorithm is superior to related ones in the literature.展开更多
By introducing a smooth merit function for the median function, a new smooth merit function for box constrained variational inequalities (BVIs) was constructed. The function is simple and has some good differential pr...By introducing a smooth merit function for the median function, a new smooth merit function for box constrained variational inequalities (BVIs) was constructed. The function is simple and has some good differential properties. A damped Newton type method was presented based on it.Global and local superlinear/quadratic convergence results were obtained under mild conditions, and the finite termination property was also shown for the linear BVIs. Numerical results suggest that the method is efficient and promising.展开更多
This paper proposes two kinds of approximate proximal point algorithms （APPA） for monotone variational inequalities, both of which can be viewed as two extended versions of Solodov and Svaiter＇s APPA in the paper ...This paper proposes two kinds of approximate proximal point algorithms （APPA） for monotone variational inequalities, both of which can be viewed as two extended versions of Solodov and Svaiter＇s APPA in the paper ＂Error bounds for proximal point subproblems and associated inexact proximal point algorithms＂ published in 2000. They are both prediction- correction methods which use the same inexactness restriction; the only difference is that they use different search directions in the correction steps. This paper also chooses an optimal step size in the two versions of the APPA to improve the profit at each iteration. Analysis also shows that the two APPAs are globally convergent under appropriate assumptions, and we can expect algorithm 2 to get more progress in every iteration than algorithm 1. Numerical experiments indicate that algorithm 2 is more efficient than algorithm 1 with the same correction step size,展开更多
Many solutions of variational inequalities have been proposed,among which the subgradient extragradient method has obvious advantages.Two different algorithms are given for solving variational inequality problem in th...Many solutions of variational inequalities have been proposed,among which the subgradient extragradient method has obvious advantages.Two different algorithms are given for solving variational inequality problem in this paper.The problem we study is defined in a real Hilbert space and has L-Lipschitz and pseudomonotone condition.Two new algorithms adopt inertial technology and non-monotonic step size rule,and their convergence can still be proved when the value of L is not given in advance.Finally,some numerical results are designed to demonstrate the computational efficiency of our two new algorithms.展开更多
In this paper,we consider a new differential variational inequality（DVI,for short）which is composed of an evolution equation and a variational inequality in infinite Banach spaces.This kind of problems may be regard...In this paper,we consider a new differential variational inequality（DVI,for short）which is composed of an evolution equation and a variational inequality in infinite Banach spaces.This kind of problems may be regarded as a special feedback control problem.Based on the Browder＇s theorem and the optimal control theory,we show the existence of solutions to the mentioned problem.展开更多
The goal of the present paper is to investigate an abstract system, called fractional differential variational inequality, which consists of a mixed variational inequality combined with a fractional evolution equation...The goal of the present paper is to investigate an abstract system, called fractional differential variational inequality, which consists of a mixed variational inequality combined with a fractional evolution equation in the framework of Banach spaces. Using discrete approximation approach, an existence theorem of solutions for the inequality is established under some suitable assumptions.展开更多
In this article, a new differential inverse variational inequality is introduced and studied in finite dimensional Euclidean spaces. Some results concerned with the linear growth of the solution set for the differenti...In this article, a new differential inverse variational inequality is introduced and studied in finite dimensional Euclidean spaces. Some results concerned with the linear growth of the solution set for the differential inverse variational inequalities are obtained under different conditions. Some existence theorems of Caratheodory weak solutions for the differential inverse variational inequality are also established under suitable conditions. An application to the time-dependent spatial price equilibrium control problem is also given.展开更多
In this paper, a convex feasibility problem is considered. We construct an iterative method to approximate a common element of the solution set of classical variational inequalities and of the fixed point set of a str...In this paper, a convex feasibility problem is considered. We construct an iterative method to approximate a common element of the solution set of classical variational inequalities and of the fixed point set of a strict pseudocontraction. Strong convergence theorems for the common element are established in the framework of Hilbert spaces.展开更多
The proximal-based decomposition method was originally proposed by Chen and Teboulle （Math. Programming, 1994, 64：81-101 for solving corrvex minimization problems. This paper extends it to solving monotone variation...The proximal-based decomposition method was originally proposed by Chen and Teboulle （Math. Programming, 1994, 64：81-101 for solving corrvex minimization problems. This paper extends it to solving monotone variational inequalities associated with separable structures with the improvements that the restrictive assumptions on the involved parameters are much relaxed, and thus makes it practical to solve the subproblems easily. Without additional assumptions, global convergence of the new method is proved under the same mild assumptions on the problem＇s data as the original method.展开更多
We propose a projection-type algorithm for generalized mixed variational in- equality problem in Euclidean space Rn. We establish the convergence theorem for the pro- posed algorithm, provided the multi-valued mapping...We propose a projection-type algorithm for generalized mixed variational in- equality problem in Euclidean space Rn. We establish the convergence theorem for the pro- posed algorithm, provided the multi-valued mapping is continuous and f-pseudomonotone with nonempty compact convex values on dom（f）, where f ： Rn --RU{＋∞} is a proper func- tion. The algorithm presented in this paper generalize and improve some known algorithms in literatures. Preliminary computational experience is also reported.展开更多
By introducing a smooth merit function for the median function, a new smooth merit function for box constrained variational inequalities （BVIs） was constructed. The function is simple and has some good differential ...By introducing a smooth merit function for the median function, a new smooth merit function for box constrained variational inequalities （BVIs） was constructed. The function is simple and has some good differential properties. A damped Newton type method was presented based on it. Global and local superlinear/ quadratic convergence results were obtained under mild conditions, and the finite termination property was also shown for the linear BVIs. Numerical results suggest that the method is efficient and promising.展开更多
In this paper, the authors introduce and study system of generalized vector variational inequalities. Under suitable conditions, the existence of solutions for system of generalized vector variational inequalities is ...In this paper, the authors introduce and study system of generalized vector variational inequalities. Under suitable conditions, the existence of solutions for system of generalized vector variational inequalities is presented by Kakutani-Fan-Glicksberg fixed point theorem.展开更多
Based on the nonlinear characiers of the discrete problems of some ellipticalvariational inequalities, this paper presents a numerical iterative method, the schemesof which are pithy and converge rapidly The new metho...Based on the nonlinear characiers of the discrete problems of some ellipticalvariational inequalities, this paper presents a numerical iterative method, the schemesof which are pithy and converge rapidly The new method possesses a high efficiency. insolving such applied engineering problems as obstacle problems and .free boundary.problems arising in fluid lubrications.展开更多
The approximation solvability of a generalized system for strongly g-r- pseudomonotonic nonlinear variational inequalities in Hilbert spaces is studied based on the convergence of the projection method. The results pr...The approximation solvability of a generalized system for strongly g-r- pseudomonotonic nonlinear variational inequalities in Hilbert spaces is studied based on the convergence of the projection method. The results presented in this paper improve, generalize and unify some recent results in the literature.展开更多
A new concept of finitely continuous topological spaces （in short, FC-space） without convexity structure and linear structure was introduced. Some KKM type theorems in noncompact FC-spaces were obtained, and from wh...A new concept of finitely continuous topological spaces （in short, FC-space） without convexity structure and linear structure was introduced. Some KKM type theorems in noncompact FC-spaces were obtained, and from which, some section theorems and wriational inequality theorems were proved under much weak assumptions. Our results improve and generalize the corresponding conclusions in recent literature.展开更多
We study the single projection algorithm of Tseng for solving a variational inequality problem in a 2-uniformly convex Banach space.The underline cost function of the variational inequality is assumed to be monotone a...We study the single projection algorithm of Tseng for solving a variational inequality problem in a 2-uniformly convex Banach space.The underline cost function of the variational inequality is assumed to be monotone and Lipschitz continuous.A weak convergence result is obtained under reasonable assumptions on the variable step-sizes.We also give the strong convergence result for when the underline cost function is strongly monotone and Lipchitz continuous.For this strong convergence case,the proposed method does not require prior knowledge of the modulus of strong monotonicity and the Lipschitz constant of the cost function as input parameters,rather,the variable step-sizes are diminishing and non-summable.The asymptotic estimate of the convergence rate for the strong convergence case is also given.For completeness,we give another strong convergence result using the idea of Halpern iteration when the cost function is monotone and Lipschitz continuous and the variable step-sizes are bounded by the inverse of the Lipschitz constant of the cost function.Finally,we give an example of a contact problem where our proposed method can be applied.展开更多
In this paper,we investigate pseudomonotone and Lipschitz continuous variational inequalities in real Hilbert spaces.For solving this problem,we propose a new method that combines the advantages of the subgradient ext...In this paper,we investigate pseudomonotone and Lipschitz continuous variational inequalities in real Hilbert spaces.For solving this problem,we propose a new method that combines the advantages of the subgradient extragradient method and the projection contraction method.Some very recent papers have considered different inertial algorithms which allowed the inertial factor is chosen in[0;1].The purpose of this work is to continue working in this direction,we propose another inertial subgradient extragradient method that the inertial factor can be chosen in a special case to be 1.Under suitable mild conditions,we establish the weak convergence of the proposed algorithm.Moreover,linear convergence is obtained under strong pseudomonotonicity and Lipschitz continuity assumptions.Finally,some numerical illustrations are given to confirm the theoretical analysis.展开更多
A quasi-variational inequality is proved in paracompact setting which generalizes the results of Zhou Chen andAubin. As applications, two existence theorems on the solutions of optimization problems and social equilib...A quasi-variational inequality is proved in paracompact setting which generalizes the results of Zhou Chen andAubin. As applications, two existence theorems on the solutions of optimization problems and social equilibria ofmetagames are showed which improve and extend the recent results of Kaczynski-Zeidan and Aubin.展开更多
In this work,we investigate a classical pseudomonotone and Lipschitz continuous variational inequality in the setting of Hilbert space,and present a projection-type approximation method for solving this problem.Our me...In this work,we investigate a classical pseudomonotone and Lipschitz continuous variational inequality in the setting of Hilbert space,and present a projection-type approximation method for solving this problem.Our method requires only to compute one projection onto the feasible set per iteration and without any linesearch procedure or additional projections as well as does not need to the prior knowledge of the Lipschitz constant and the sequentially weakly continuity of the variational inequality mapping.A strong convergence is established for the proposed method to a solution of a variational inequality problem under certain mild assumptions.Finally,we give some numerical experiments illustrating the performance of the proposed method for variational inequality problems.展开更多
文摘This paper studies a strongly convergent inertial forward-backward-forward algorithm for the variational inequality problem in Hilbert spaces.In our convergence analysis,we do not assume the on-line rule of the inertial parameters and the iterates,which have been assumed by several authors whenever a strongly convergent algorithm with an inertial extrapolation step is proposed for a variational inequality problem.Consequently,our proof arguments are different from what is obtainable in the relevant literature.Finally,we give numerical tests to confirm the theoretical analysis and show that our proposed algorithm is superior to related ones in the literature.
基金Project supported by the Teaching and Research Award Program for the Outstanding YoungTeachers in Higher Education Institutes of Munistry of Education, P.R.China
文摘By introducing a smooth merit function for the median function, a new smooth merit function for box constrained variational inequalities (BVIs) was constructed. The function is simple and has some good differential properties. A damped Newton type method was presented based on it.Global and local superlinear/quadratic convergence results were obtained under mild conditions, and the finite termination property was also shown for the linear BVIs. Numerical results suggest that the method is efficient and promising.
文摘This paper proposes two kinds of approximate proximal point algorithms （APPA） for monotone variational inequalities, both of which can be viewed as two extended versions of Solodov and Svaiter＇s APPA in the paper ＂Error bounds for proximal point subproblems and associated inexact proximal point algorithms＂ published in 2000. They are both prediction- correction methods which use the same inexactness restriction; the only difference is that they use different search directions in the correction steps. This paper also chooses an optimal step size in the two versions of the APPA to improve the profit at each iteration. Analysis also shows that the two APPAs are globally convergent under appropriate assumptions, and we can expect algorithm 2 to get more progress in every iteration than algorithm 1. Numerical experiments indicate that algorithm 2 is more efficient than algorithm 1 with the same correction step size,
文摘Many solutions of variational inequalities have been proposed,among which the subgradient extragradient method has obvious advantages.Two different algorithms are given for solving variational inequality problem in this paper.The problem we study is defined in a real Hilbert space and has L-Lipschitz and pseudomonotone condition.Two new algorithms adopt inertial technology and non-monotonic step size rule,and their convergence can still be proved when the value of L is not given in advance.Finally,some numerical results are designed to demonstrate the computational efficiency of our two new algorithms.
基金supported by NNSF of China(11671101)the National Science Center of Poland Under Maestro Advanced Project(UMO-2012/06/A/ST1/00262)Special Funds of Guangxi Distinguished Experts Construction Engineering
文摘In this paper,we consider a new differential variational inequality（DVI,for short）which is composed of an evolution equation and a variational inequality in infinite Banach spaces.This kind of problems may be regarded as a special feedback control problem.Based on the Browder＇s theorem and the optimal control theory,we show the existence of solutions to the mentioned problem.
基金received funding from the European Union's Horizon 2020 Research and Innovation Programme under the Marie Sklodowska-Curie grant agreement(823731-CONMECH)supported by the National Science Center of Poland under Maestro Project(UMO-2012/06/A/ST1/00262)+3 种基金National Science Center of Poland under Preludium Project(2017/25/N/ST1/00611)supported by the International Project co-financed by the Ministry of Science and Higher Education of Republic of Poland(3792/GGPJ/H2020/2017/0)Qinzhou University Project(2018KYQD06)National Natural Sciences Foundation of Guangxi(2018JJA110006)
文摘The goal of the present paper is to investigate an abstract system, called fractional differential variational inequality, which consists of a mixed variational inequality combined with a fractional evolution equation in the framework of Banach spaces. Using discrete approximation approach, an existence theorem of solutions for the inequality is established under some suitable assumptions.
基金supported by the National Natural Science Foundation of China(11301359,11171237)the Key Program of NSFC(70831005)
文摘In this article, a new differential inverse variational inequality is introduced and studied in finite dimensional Euclidean spaces. Some results concerned with the linear growth of the solution set for the differential inverse variational inequalities are obtained under different conditions. Some existence theorems of Caratheodory weak solutions for the differential inverse variational inequality are also established under suitable conditions. An application to the time-dependent spatial price equilibrium control problem is also given.
文摘In this paper, a convex feasibility problem is considered. We construct an iterative method to approximate a common element of the solution set of classical variational inequalities and of the fixed point set of a strict pseudocontraction. Strong convergence theorems for the common element are established in the framework of Hilbert spaces.
基金the National Natural Science Foundation of China(No.70671024)the Na-tional High-Tech Research and Development Program of China(863 Program)(No.2006AA11Z209)
文摘The proximal-based decomposition method was originally proposed by Chen and Teboulle （Math. Programming, 1994, 64：81-101 for solving corrvex minimization problems. This paper extends it to solving monotone variational inequalities associated with separable structures with the improvements that the restrictive assumptions on the involved parameters are much relaxed, and thus makes it practical to solve the subproblems easily. Without additional assumptions, global convergence of the new method is proved under the same mild assumptions on the problem＇s data as the original method.
基金supported by the Scientific Research Foundation of Sichuan Normal University(20151602)National Natural Science Foundation of China(10671135,61179033)and the Key Project of Chinese Ministry of Education(212147)
文摘We propose a projection-type algorithm for generalized mixed variational in- equality problem in Euclidean space Rn. We establish the convergence theorem for the pro- posed algorithm, provided the multi-valued mapping is continuous and f-pseudomonotone with nonempty compact convex values on dom（f）, where f ： Rn --RU{＋∞} is a proper func- tion. The algorithm presented in this paper generalize and improve some known algorithms in literatures. Preliminary computational experience is also reported.
文摘By introducing a smooth merit function for the median function, a new smooth merit function for box constrained variational inequalities （BVIs） was constructed. The function is simple and has some good differential properties. A damped Newton type method was presented based on it. Global and local superlinear/ quadratic convergence results were obtained under mild conditions, and the finite termination property was also shown for the linear BVIs. Numerical results suggest that the method is efficient and promising.
文摘In this paper, the authors introduce and study system of generalized vector variational inequalities. Under suitable conditions, the existence of solutions for system of generalized vector variational inequalities is presented by Kakutani-Fan-Glicksberg fixed point theorem.
文摘Based on the nonlinear characiers of the discrete problems of some ellipticalvariational inequalities, this paper presents a numerical iterative method, the schemesof which are pithy and converge rapidly The new method possesses a high efficiency. insolving such applied engineering problems as obstacle problems and .free boundary.problems arising in fluid lubrications.
基金Supported by the Science and Technology Research Project of Chinese Ministry of Education (206123)
文摘The approximation solvability of a generalized system for strongly g-r- pseudomonotonic nonlinear variational inequalities in Hilbert spaces is studied based on the convergence of the projection method. The results presented in this paper improve, generalize and unify some recent results in the literature.
文摘A new concept of finitely continuous topological spaces （in short, FC-space） without convexity structure and linear structure was introduced. Some KKM type theorems in noncompact FC-spaces were obtained, and from which, some section theorems and wriational inequality theorems were proved under much weak assumptions. Our results improve and generalize the corresponding conclusions in recent literature.
文摘We study the single projection algorithm of Tseng for solving a variational inequality problem in a 2-uniformly convex Banach space.The underline cost function of the variational inequality is assumed to be monotone and Lipschitz continuous.A weak convergence result is obtained under reasonable assumptions on the variable step-sizes.We also give the strong convergence result for when the underline cost function is strongly monotone and Lipchitz continuous.For this strong convergence case,the proposed method does not require prior knowledge of the modulus of strong monotonicity and the Lipschitz constant of the cost function as input parameters,rather,the variable step-sizes are diminishing and non-summable.The asymptotic estimate of the convergence rate for the strong convergence case is also given.For completeness,we give another strong convergence result using the idea of Halpern iteration when the cost function is monotone and Lipschitz continuous and the variable step-sizes are bounded by the inverse of the Lipschitz constant of the cost function.Finally,we give an example of a contact problem where our proposed method can be applied.
基金funded by the University of Science,Vietnam National University,Hanoi under project number TN.21.01。
文摘In this paper,we investigate pseudomonotone and Lipschitz continuous variational inequalities in real Hilbert spaces.For solving this problem,we propose a new method that combines the advantages of the subgradient extragradient method and the projection contraction method.Some very recent papers have considered different inertial algorithms which allowed the inertial factor is chosen in[0;1].The purpose of this work is to continue working in this direction,we propose another inertial subgradient extragradient method that the inertial factor can be chosen in a special case to be 1.Under suitable mild conditions,we establish the weak convergence of the proposed algorithm.Moreover,linear convergence is obtained under strong pseudomonotonicity and Lipschitz continuity assumptions.Finally,some numerical illustrations are given to confirm the theoretical analysis.
文摘A quasi-variational inequality is proved in paracompact setting which generalizes the results of Zhou Chen andAubin. As applications, two existence theorems on the solutions of optimization problems and social equilibria ofmetagames are showed which improve and extend the recent results of Kaczynski-Zeidan and Aubin.
基金funded by National University ofCivil Engineering(NUCE)under grant number 15-2020/KHXD-TD。
文摘In this work,we investigate a classical pseudomonotone and Lipschitz continuous variational inequality in the setting of Hilbert space,and present a projection-type approximation method for solving this problem.Our method requires only to compute one projection onto the feasible set per iteration and without any linesearch procedure or additional projections as well as does not need to the prior knowledge of the Lipschitz constant and the sequentially weakly continuity of the variational inequality mapping.A strong convergence is established for the proposed method to a solution of a variational inequality problem under certain mild assumptions.Finally,we give some numerical experiments illustrating the performance of the proposed method for variational inequality problems.
