Using a modified subgradient extragradient algorithm, this paper proposed a novel approach to solving a supply chain network equilibrium model. The method extends the scope of optimisation and improves the accuracy at...Using a modified subgradient extragradient algorithm, this paper proposed a novel approach to solving a supply chain network equilibrium model. The method extends the scope of optimisation and improves the accuracy at each iteration by incorporating adaptive parameter selection and a more general subgradient projection operator. The advantages of the proposed method are highlighted by the proof of strong convergence presented in the paper. Several concrete examples are given to demonstrate the effectiveness of the algorithm, with comparisons illustrating its superior CPU running time compared to alternative techniques. The practical applicability of the algorithm is also demonstrated by applying it to a realistic supply chain network model.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
In this paper,we propose a modified two-subgradient extragradient algorithm(MTSEGA)for solving monotone and Lipschitz continuous variational inequalities with the feasible set being a level set of a smooth convex func...In this paper,we propose a modified two-subgradient extragradient algorithm(MTSEGA)for solving monotone and Lipschitz continuous variational inequalities with the feasible set being a level set of a smooth convex function in Hilbert space.The advantage of MTSEGA is that all the projections are computed onto a half-space per iteration.Moreover,MTSEGA only needs one computation of the underlying mapping per iteration.Under the same assumptions with the known algorithm,we show that the sequence generated by this algorithm is weakly convergent to a solution of the concerned problem.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
The main purpose of this paper is to introduce and deal with a self adaptive inertial subgradient extragradient iterative algorithm with a new and interesting stepsize rule in real Hilbert spaces.Under some proper con...The main purpose of this paper is to introduce and deal with a self adaptive inertial subgradient extragradient iterative algorithm with a new and interesting stepsize rule in real Hilbert spaces.Under some proper control conditions imposed on the coefficients and operators,we prove a new strong convergence result for solving variational inequalities with regard to pseudomonotone and Lipschitzian operators.Moreover,some numerical simulation results are given to show the rationality and validity of our algorithm.展开更多
The purpose of this paper is to investigate the problem of finding the common element of the set of common fixed points of a countable family of nonexpansive mappings, the set of an equilibrium problem and the set of ...The purpose of this paper is to investigate the problem of finding the common element of the set of common fixed points of a countable family of nonexpansive mappings, the set of an equilibrium problem and the set of solutions of the variational inequality prob- lem for a relaxed cocoercive and Lipschitz continuous mapping in Hilbert spaces. Then, we show that the sequence converges strongly to a common element of the above three sets under some parameter controlling conditions, which are connected with Yao, Liou, Yao[17], Takahashi[12] and many others.展开更多
In order to solve variational inequality problems of pseudomonotonicity and Lipschitz continuity in Hilbert spaces, an inertial subgradient extragradient algorithm is proposed by virtue of non-monotone stepsizes. More...In order to solve variational inequality problems of pseudomonotonicity and Lipschitz continuity in Hilbert spaces, an inertial subgradient extragradient algorithm is proposed by virtue of non-monotone stepsizes. Moreover, weak convergence and R-linear convergence analyses of the algorithm are constructed under appropriate assumptions. Finally, the efficiency of the proposed algorithm is demonstrated through numerical implementations.展开更多
Many approaches inquiring into variational inequality problems have been put forward,among which subgradient extragradient method is of great significance.A novel algorithm is presented in this article for resolving q...Many approaches inquiring into variational inequality problems have been put forward,among which subgradient extragradient method is of great significance.A novel algorithm is presented in this article for resolving quasi-nonexpansive fixed point problem and pseudomonotone variational inequality problem in a real Hilbert interspace.In order to decrease the execution time and quicken the velocity of convergence,the proposed algorithm adopts an inertial technology.Moreover,the algorithm is by virtue of a non-monotonic step size rule to acquire strong convergence theorem without estimating the value of Lipschitz constant.Finally,numerical results on some problems authenticate that the algorithm has preferable efficiency than other algorithms.展开更多
This paper proposes a new hybrid variant of extragradient methods for finding a common solution of an equilibrium problem and a family of strict pseudo-contraction mappings. We present an algorithmic scheme that combi...This paper proposes a new hybrid variant of extragradient methods for finding a common solution of an equilibrium problem and a family of strict pseudo-contraction mappings. We present an algorithmic scheme that combine the idea of an extragradient method and a successive iteration method as a hybrid variant. Then, this algorithm is modified by projecting on a suitable convex set to get a better convergence property. The convergence of two these algorithms are investigated under certain assumptions.展开更多
Inspired by inertial methods and extragradient algorithms,two algorithms were proposed to investigate fixed point problem of quasinonexpansive mapping and pseudomonotone equilibrium problem in this study.In order to e...Inspired by inertial methods and extragradient algorithms,two algorithms were proposed to investigate fixed point problem of quasinonexpansive mapping and pseudomonotone equilibrium problem in this study.In order to enhance the speed of the convergence and reduce computational cost,the algorithms used a new step size and a cutting hyperplane.The first algorithm was proved to be weak convergence,while the second algorithm used a modified version of Halpern iteration to obtain strong convergence.Finally,numerical experiments on several specific problems and comparisons with other algorithms verified the superiority of the proposed algorithms.展开更多
Many approaches have been put forward to resolve the variational inequality problem. The subgradient extragradient method is one of the most effective. This paper proposes a modified subgradient extragradient method a...Many approaches have been put forward to resolve the variational inequality problem. The subgradient extragradient method is one of the most effective. This paper proposes a modified subgradient extragradient method about classical variational inequality in a real Hilbert interspace. By analyzing the operator’s partial message, the proposed method designs a non-monotonic step length strategy which requires no line search and is independent of the value of Lipschitz constant, and is extended to solve the problem of pseudomonotone variational inequality. Meanwhile, the method requires merely one map value and a projective transformation to the practicable set at every iteration. In addition, without knowing the Lipschitz constant for interrelated mapping, weak convergence is given and R-linear convergence rate is established concerning algorithm. Several numerical results further illustrate that the method is superior to other algorithms.展开更多
Gibali[J.Nonlinear Anal.Optim.,2015,6(1):41‒51]presented a self-adaptive subgradient extragradient projection method for solving variational inequalities without Lipschitz continuity,where its next iterative point was...Gibali[J.Nonlinear Anal.Optim.,2015,6(1):41‒51]presented a self-adaptive subgradient extragradient projection method for solving variational inequalities without Lipschitz continuity,where its next iterative point was obtained by projecting a vector onto a specific half-space.In this paper,we present new kinds of self-adaptive subgradient extragradient projection methods by using a new descent direction.With the help of the techniques in the method of He and Liao[J.Optim.Theory Appl,2002,112(1):111‒128],we get a longer step-size for these kinds of algorithms,which proves the global convergence of the generated sequence.Numerical results show that these kinds of extragradient subgradient projection methods are less dependent on the choice of the initial point,the dimension of the variational inequalities,and the tolerance of accuracy than the known methods.Moreover,the new methods proposed in this paper outperform(with respect to the number of iterations and cpu-time)the method presented by Gibali.展开更多
In this paper, we consider an extragradient thresholding algorithm for finding the sparse solution of mixed complementarity problems (MCPs). We establish a relaxation l1 regularized projection minimization model for t...In this paper, we consider an extragradient thresholding algorithm for finding the sparse solution of mixed complementarity problems (MCPs). We establish a relaxation l1 regularized projection minimization model for the original problem and design an extragradient thresholding algorithm (ETA) to solve the regularized model. Furthermore, we prove that any cluster point of the sequence generated by ETA is a solution of MCP. Finally, numerical experiments show that the ETA algorithm can effectively solve the l1 regularized projection minimization model and obtain the sparse solution of the mixed complementarity problem.展开更多
In this paper,we study an extragradient algorithm for approximating solutions of quasi-equilibrium problems in Banach spaces.We prove strong convergence of the sequence generated by the extragradient method to a solut...In this paper,we study an extragradient algorithm for approximating solutions of quasi-equilibrium problems in Banach spaces.We prove strong convergence of the sequence generated by the extragradient method to a solution of the quasi-equilibrium problem.展开更多
Nonconvex–nonconcave saddle-point optimization in machine learning has triggered lots of research for studying non-monotone variational inequalities(VIs).In this work,we introduce two mirror frameworks,called mirror ...Nonconvex–nonconcave saddle-point optimization in machine learning has triggered lots of research for studying non-monotone variational inequalities(VIs).In this work,we introduce two mirror frameworks,called mirror extragradient method and mirror extrapolation method,for approximating solutions to relatively Lipschitz and monotone-like VIs.The former covers the well-known Nemirovski’s mirror prox method and Nesterov’s dual extrapolation method,and the recently proposed Bregman extragradient method;all of them can be reformulated into a scheme that is very similar to the original form of extragradient method.The latter includes the operator extrapolation method and the Bregman extrapolation method as its special cases.The proposed mirror frameworks allow us to present a unified and improved convergence analysis for all these existing methods under relative Lipschitzness and monotone-like conditions that may be the currently weakest assumptions guaranteeing(sub)linear convergence.展开更多
In this paper,we investigate a new inertial viscosity extragradient algorithm for solving variational inequality problems for pseudo-monotone and Lipschitz continuous operator and fixed point problems for quasi-nonexp...In this paper,we investigate a new inertial viscosity extragradient algorithm for solving variational inequality problems for pseudo-monotone and Lipschitz continuous operator and fixed point problems for quasi-nonexpansive mappings in real Hilbert spaces.Strong convergence theorems are obtained under some appropriate conditions on the parameters.Finally,we give some numerical experiments to show the advantages of our proposed algorithms.The results obtained in this paper extend and improve some recent works in the literature.展开更多
文摘Using a modified subgradient extragradient algorithm, this paper proposed a novel approach to solving a supply chain network equilibrium model. The method extends the scope of optimisation and improves the accuracy at each iteration by incorporating adaptive parameter selection and a more general subgradient projection operator. The advantages of the proposed method are highlighted by the proof of strong convergence presented in the paper. Several concrete examples are given to demonstrate the effectiveness of the algorithm, with comparisons illustrating its superior CPU running time compared to alternative techniques. The practical applicability of the algorithm is also demonstrated by applying it to a realistic supply chain network model.
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.
基金Supported by the National Natural Science Foundation of China(Grant Nos.1187105911801455)+1 种基金Sichuan Science and Technology Program(Grant No.2019YFG0299)General Cultivation Program of China West Normal University(Grant No.20A024)。
文摘In this paper,we propose a modified two-subgradient extragradient algorithm(MTSEGA)for solving monotone and Lipschitz continuous variational inequalities with the feasible set being a level set of a smooth convex function in Hilbert space.The advantage of MTSEGA is that all the projections are computed onto a half-space per iteration.Moreover,MTSEGA only needs one computation of the underlying mapping per iteration.Under the same assumptions with the known algorithm,we show that the sequence generated by this algorithm is weakly convergent to a solution of the concerned problem.
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.
文摘The main purpose of this paper is to introduce and deal with a self adaptive inertial subgradient extragradient iterative algorithm with a new and interesting stepsize rule in real Hilbert spaces.Under some proper control conditions imposed on the coefficients and operators,we prove a new strong convergence result for solving variational inequalities with regard to pseudomonotone and Lipschitzian operators.Moreover,some numerical simulation results are given to show the rationality and validity of our algorithm.
基金Supported by King Mongkut's University of Technology Thonburi.KMUTT,(CSEC Project No.E01008)supported by the Faculty of Applied Liberal Arts RMUTR Research Fund and King Mongkut's Diamond scholarship for fostering special academic skills by KMUTT
文摘The purpose of this paper is to investigate the problem of finding the common element of the set of common fixed points of a countable family of nonexpansive mappings, the set of an equilibrium problem and the set of solutions of the variational inequality prob- lem for a relaxed cocoercive and Lipschitz continuous mapping in Hilbert spaces. Then, we show that the sequence converges strongly to a common element of the above three sets under some parameter controlling conditions, which are connected with Yao, Liou, Yao[17], Takahashi[12] and many others.
文摘In order to solve variational inequality problems of pseudomonotonicity and Lipschitz continuity in Hilbert spaces, an inertial subgradient extragradient algorithm is proposed by virtue of non-monotone stepsizes. Moreover, weak convergence and R-linear convergence analyses of the algorithm are constructed under appropriate assumptions. Finally, the efficiency of the proposed algorithm is demonstrated through numerical implementations.
文摘Many approaches inquiring into variational inequality problems have been put forward,among which subgradient extragradient method is of great significance.A novel algorithm is presented in this article for resolving quasi-nonexpansive fixed point problem and pseudomonotone variational inequality problem in a real Hilbert interspace.In order to decrease the execution time and quicken the velocity of convergence,the proposed algorithm adopts an inertial technology.Moreover,the algorithm is by virtue of a non-monotonic step size rule to acquire strong convergence theorem without estimating the value of Lipschitz constant.Finally,numerical results on some problems authenticate that the algorithm has preferable efficiency than other algorithms.
文摘This paper proposes a new hybrid variant of extragradient methods for finding a common solution of an equilibrium problem and a family of strict pseudo-contraction mappings. We present an algorithmic scheme that combine the idea of an extragradient method and a successive iteration method as a hybrid variant. Then, this algorithm is modified by projecting on a suitable convex set to get a better convergence property. The convergence of two these algorithms are investigated under certain assumptions.
文摘Inspired by inertial methods and extragradient algorithms,two algorithms were proposed to investigate fixed point problem of quasinonexpansive mapping and pseudomonotone equilibrium problem in this study.In order to enhance the speed of the convergence and reduce computational cost,the algorithms used a new step size and a cutting hyperplane.The first algorithm was proved to be weak convergence,while the second algorithm used a modified version of Halpern iteration to obtain strong convergence.Finally,numerical experiments on several specific problems and comparisons with other algorithms verified the superiority of the proposed algorithms.
文摘Many approaches have been put forward to resolve the variational inequality problem. The subgradient extragradient method is one of the most effective. This paper proposes a modified subgradient extragradient method about classical variational inequality in a real Hilbert interspace. By analyzing the operator’s partial message, the proposed method designs a non-monotonic step length strategy which requires no line search and is independent of the value of Lipschitz constant, and is extended to solve the problem of pseudomonotone variational inequality. Meanwhile, the method requires merely one map value and a projective transformation to the practicable set at every iteration. In addition, without knowing the Lipschitz constant for interrelated mapping, weak convergence is given and R-linear convergence rate is established concerning algorithm. Several numerical results further illustrate that the method is superior to other algorithms.
文摘Gibali[J.Nonlinear Anal.Optim.,2015,6(1):41‒51]presented a self-adaptive subgradient extragradient projection method for solving variational inequalities without Lipschitz continuity,where its next iterative point was obtained by projecting a vector onto a specific half-space.In this paper,we present new kinds of self-adaptive subgradient extragradient projection methods by using a new descent direction.With the help of the techniques in the method of He and Liao[J.Optim.Theory Appl,2002,112(1):111‒128],we get a longer step-size for these kinds of algorithms,which proves the global convergence of the generated sequence.Numerical results show that these kinds of extragradient subgradient projection methods are less dependent on the choice of the initial point,the dimension of the variational inequalities,and the tolerance of accuracy than the known methods.Moreover,the new methods proposed in this paper outperform(with respect to the number of iterations and cpu-time)the method presented by Gibali.
文摘In this paper, we consider an extragradient thresholding algorithm for finding the sparse solution of mixed complementarity problems (MCPs). We establish a relaxation l1 regularized projection minimization model for the original problem and design an extragradient thresholding algorithm (ETA) to solve the regularized model. Furthermore, we prove that any cluster point of the sequence generated by ETA is a solution of MCP. Finally, numerical experiments show that the ETA algorithm can effectively solve the l1 regularized projection minimization model and obtain the sparse solution of the mixed complementarity problem.
文摘In this paper,we study an extragradient algorithm for approximating solutions of quasi-equilibrium problems in Banach spaces.We prove strong convergence of the sequence generated by the extragradient method to a solution of the quasi-equilibrium problem.
基金supported by the National Natural Science Foundation of China(No.11971480)the Natural Science Fund of Hunan for Excellent Youth(No.2020JJ3038)+2 种基金the Fund for NUDT Young Innovator Awards(No.20190105)supported by the National Natural Science Foundation of China(Nos.11991020,11631013,12021001,11971372 and 11991021)the Strategic Priority Research Program of Chinese Academy of Sciences(No.XDA27000000).
文摘Nonconvex–nonconcave saddle-point optimization in machine learning has triggered lots of research for studying non-monotone variational inequalities(VIs).In this work,we introduce two mirror frameworks,called mirror extragradient method and mirror extrapolation method,for approximating solutions to relatively Lipschitz and monotone-like VIs.The former covers the well-known Nemirovski’s mirror prox method and Nesterov’s dual extrapolation method,and the recently proposed Bregman extragradient method;all of them can be reformulated into a scheme that is very similar to the original form of extragradient method.The latter includes the operator extrapolation method and the Bregman extrapolation method as its special cases.The proposed mirror frameworks allow us to present a unified and improved convergence analysis for all these existing methods under relative Lipschitzness and monotone-like conditions that may be the currently weakest assumptions guaranteeing(sub)linear convergence.
基金Supported by the NSF of China(Grant Nos.11771063,11971082 and 12171062)the Natural Science Foundation of Chongqing(Grant No.cstc2020jcyj-msxm X0455)+2 种基金Science and Technology Project of Chongqing Education Committee(Grant No.KJZD-K201900504)the Program of Chongqing Innovation Research Group Project in University(Grant No.CXQT19018)Open Fund of Tianjin Key Lab for Advanced Signal Processing(Grant No.2019ASP-TJ03)。
文摘In this paper,we investigate a new inertial viscosity extragradient algorithm for solving variational inequality problems for pseudo-monotone and Lipschitz continuous operator and fixed point problems for quasi-nonexpansive mappings in real Hilbert spaces.Strong convergence theorems are obtained under some appropriate conditions on the parameters.Finally,we give some numerical experiments to show the advantages of our proposed algorithms.The results obtained in this paper extend and improve some recent works in the literature.
