We consider the problem of minimizing a finite sum of convex functions subject to the set of mini... more We consider the problem of minimizing a finite sum of convex functions subject to the set of minimizers of a convex differentiable function. In order to solve the problem, an algorithm combining the incremental proximal gradient method with smooth penalization technique is proposed. We show the convergence of the generated sequence of iterates to an optimal solution of the optimization problems, provided that a condition expressed via the Fenchel conjugate of the constraint function is fulfilled. Finally, the functionality of the method is illustrated by some numerical experiments addressing image inpainting problems and generalized Heron problems with least squares constraints.
This paper presents two inertial extragradient algorithms for finding a solution of split pseudom... more This paper presents two inertial extragradient algorithms for finding a solution of split pseudomonotone equilibrium problems in the setting of real Hilbert spaces. The weak and strong convergence theorems of the introduced algorithms are presented under some constraint qualifications of the scalar sequences. The discussions on the numerical experiments are also provided to demonstrate the effectiveness of the proposed algorithms.
We propose a modified extragradient method for solving the variational inequality problem in a Hi... more We propose a modified extragradient method for solving the variational inequality problem in a Hilbert space. The method is a combination of the well-known subgradient extragradient with the Mann’s mean value method in which the updated iterate is picked in the convex hull of all previous iterates. We show weak convergence of the mean value iterate to a solution of the variational inequality problem, provided that a condition on the corresponding averaging matrix is fulfilled. Some numerical experiments are given to show the effectiveness of the obtained theoretical result.
This study provides the important properties of the lexicographic tolerable robust solution for u... more This study provides the important properties of the lexicographic tolerable robust solution for uncertain multi-objective optimization problems which was introduced by Boriwan et al. [Boriwan, P.; Ehrgott, M.; Kuroiwa, D.; Petrot, N. The lexicographic tolerable robustness concept for uncertain multi-objective optimization problems: a study on water resources management. Sustainability. 12 (2020), no. 18, article number 7582.]. Also, the relationship between the lexicographic tolerable solution concept and the well-known robust solution, as the set-based robust efficiency [Ehrgott, M.; Ide, J.; Schöbel, A. Minmax robustness for multi-objective optimization problems. Eur. J. Oper. Res. 239 (2014), no. 1, 17–31.], are provided.
Journal of Nonlinear Sciences and Applications, May 22, 2016
In this paper, we consider the split quasi variational inequality problems over a class of noncon... more In this paper, we consider the split quasi variational inequality problems over a class of nonconvex sets, as uniformly prox-regular sets. The sufficient conditions for the existence of solutions of such a problem are provided. Furthermore, an iterative algorithm for finding a solution is constructed and its convergence analysis are considered. The results in this paper improve and extend the variational inequality problems which have been appeared in literature.
We introduce a generalized forward-backward splitting method with penalty term for solving monoto... more We introduce a generalized forward-backward splitting method with penalty term for solving monotone inclusion problems involving the sum of a finite number of maximally monotone operators and the normal cone to the nonempty set of zeros of another maximally monotone operator. We show weak ergodic convergence of the generated sequence of iterates to a solution of the considered monotone inclusion problem, provided that the condition corresponding to the Fitzpatrick function of the operator describing the set of the normal cone is fulfilled. Under strong monotonicity of an operator, we show strong convergence of the iterates. Furthermore, we utilize the proposed method for minimizing a largescale hierarchical minimization problem concerning the sum of differentiable and nondifferentiable convex functions subject to the set of minima of another differentiable convex function. We illustrate the functionality of the method through numerical experiments addressing constrained elastic net and generalized Heron location problems.
International journal on fixed point theory computation and applications
In this paper, we introduce a new iterative scheme for finding solutions the common element of th... more In this paper, we introduce a new iterative scheme for finding solutions the common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inclusion problem with a multivalued maximal monotone mapping and an α-inverse-strongly monotone mapping. We show that the sequence converges strongly to a common solutions for quasi variational inclusion and fixed point problems under some parameters controlling conditions. This main theorem extends a recent result of Zhang et al. [Algorithms of common solutions to quasi variational inclusion and fixed point problems.
The main objective of this paper is to find a common solution of split variational inclusion prob... more The main objective of this paper is to find a common solution of split variational inclusion problem and fixed point problem of infinite family of nonexpansive operators in a setting of real Hilbert spaces. To reach this goal, the iterative algorithms which combine Moudafi's viscosity approximation method with some fixed point technically proving methods are utilized for solving the problem. We prove that the iterative schemes with some suitable control conditions converge strongly to a common solution of the considered problem. We also show that many interesting problems can be solved by using our presented results. Index Terms—Split variational inclusion problem, fixed point problem, nonexpansive operators, resolvent operators, strong convergence.
Necessary and sufficient conditions under which the Cesàro-Orlicz sequence space ces φ is nontriv... more Necessary and sufficient conditions under which the Cesàro-Orlicz sequence space ces φ is nontrivial are presented. It is proved that for the Luxemburg norm, Cesàro-Orlicz spaces ces φ have the Fatou property. Consequently, the spaces are complete. It is also proved that the subspace of order continuous elements in ces φ can be defined in two ways. Finally, criteria for strict monotonicity, uniform monotonicity and rotundity (= strict convexity) of the spaces ces φ are given.
The main objective of this paper is to introduce a split hierarchical variational inequality prob... more The main objective of this paper is to introduce a split hierarchical variational inequality problem. Several related problems are also considered. We propose an iterative method for finding a solution of our problem. The weak convergence of the sequence generated by the proposed method is studied.
In this paper, some basic properties of the general modular space are proven. Criteria for strict... more In this paper, some basic properties of the general modular space are proven. Criteria for strictly monotone points, extreme points and SU-points in generalized Calderón-Lozanovskiǐ spaces are obtained. Consequently, the sufficient and necessary conditions for the rotundity properties of such spaces are given.
Some existence theorems for the mixed variational-like inequality for fuzzy mappings FMVLIP in a ... more Some existence theorems for the mixed variational-like inequality for fuzzy mappings FMVLIP in a reflexive Banach space are established. Further, the auxiliary principle technique is used to suggest a novel and innovative iterative algorithm for computing the approximate solution. Consequently, not only the existence of solutions of the FMVLIP is shown, but also the convergence of iterative sequences generated by the algorithm is also proven. The results proved in this paper represent an improvement of previously known results.
At the present article, we consider a new class of general nonlinear random Amaximal m-relaxed h-... more At the present article, we consider a new class of general nonlinear random Amaximal m-relaxed h-accretive equations with random relaxed cocoercive mappings and random fuzzy mappings in q-uniformly smooth Banach spaces. By using the resolvent mapping technique for A-maximal m-relaxed h-accretive mappings due to Lan et al. and Chang's lemma, we construct a new iterative algorithm with mixed errors for finding the approximate solutions of this class of nonlinear random equations. We also verify that the approximate solutions obtained by the our proposed algorithm converge to the exact solution of the general nonlinear random A-maximal m-relaxed h-accretive equations with random relaxed cocoercive mappings and random fuzzy mappings in q-uniformly smooth Banach spaces.
We use the Wiener-Hopf equations and the Yosida approximation notions to prove the existence theo... more We use the Wiener-Hopf equations and the Yosida approximation notions to prove the existence theorem of a system of nonlinear mixed implicit equilibrium problems SMIE in Hilbert spaces. The algorithm for finding a solution of the problem SMIE is suggested; the convergence criteria and stability of the iterative algorithm are discussed. The results presented in this paper are more general and are viewed as an extension, refinement, and improvement of the previously known results in the literature.
On the Orlicz- Cesaro sequence spaces ( ces<f> ) which are defined by using Orlicz function... more On the Orlicz- Cesaro sequence spaces ( ces<f> ) which are defined by using Orlicz function <I> , we show that the space ces<f> equipped with both Amemiya and Luxemburg norms possesses uniform Opial property and uniform Kadec-Klee property if <I> satisfy the 52 -condition.
International Journal of Mathematics and Mathematical Sciences, 2004
We consider the generalized Cesàro sequence spaces defined by Suantai (2003) and consider it equi... more We consider the generalized Cesàro sequence spaces defined by Suantai (2003) and consider it equipped with the Amemiya norm. The main purpose of this paper is to show thatces(p)equipped with the Amemiya norm is rotund and has uniform Kadec-Klee property.
In this paper, we introduce and consider a new system of generalized nonlinear mixed variational ... more In this paper, we introduce and consider a new system of generalized nonlinear mixed variational inequalities involving six different nonlinear operators and discuss the existence and uniqueness of solution of the aforesaid system. We use three nearly uniformly Lipschitzian mappings S i (i = 1, 2, 3) to suggest and analyze some new three-step resolvent iterative algorithms with mixed errors for finding an element of the set of fixed points of the nearly uniformly Lipschitzian mapping Q = (S 1 , S 2 , S 3), which is the unique solution of the system of generalized nonlinear mixed variational inequalities. The convergence analysis of the suggested iterative algorithms under suitable conditions is studied. In the final section, an important remark on a class of some relaxed cocoercive mappings is discussed.
In this paper, the existing theorems and methods for finding solutions of systems of general nonl... more In this paper, the existing theorems and methods for finding solutions of systems of general nonlinear set-valued mixed variational inequalities problems in Hilbert spaces are studied. To overcome the difficulties, due to the presence of a proper convex lower semi-continuous function, and a mapping g, which appeared in the considered problem, we have used some applications of the resolvent operator technique. We would like to point out that although many authors have proved results for finding solutions of the systems of nonlinear set-valued (mixed) variational inequalities problems, it is clear that it cannot be directly applied to the problems that we have considered in this paper because of and g.
We consider the problem of minimizing a finite sum of convex functions subject to the set of mini... more We consider the problem of minimizing a finite sum of convex functions subject to the set of minimizers of a convex differentiable function. In order to solve the problem, an algorithm combining the incremental proximal gradient method with smooth penalization technique is proposed. We show the convergence of the generated sequence of iterates to an optimal solution of the optimization problems, provided that a condition expressed via the Fenchel conjugate of the constraint function is fulfilled. Finally, the functionality of the method is illustrated by some numerical experiments addressing image inpainting problems and generalized Heron problems with least squares constraints.
This paper presents two inertial extragradient algorithms for finding a solution of split pseudom... more This paper presents two inertial extragradient algorithms for finding a solution of split pseudomonotone equilibrium problems in the setting of real Hilbert spaces. The weak and strong convergence theorems of the introduced algorithms are presented under some constraint qualifications of the scalar sequences. The discussions on the numerical experiments are also provided to demonstrate the effectiveness of the proposed algorithms.
We propose a modified extragradient method for solving the variational inequality problem in a Hi... more We propose a modified extragradient method for solving the variational inequality problem in a Hilbert space. The method is a combination of the well-known subgradient extragradient with the Mann’s mean value method in which the updated iterate is picked in the convex hull of all previous iterates. We show weak convergence of the mean value iterate to a solution of the variational inequality problem, provided that a condition on the corresponding averaging matrix is fulfilled. Some numerical experiments are given to show the effectiveness of the obtained theoretical result.
This study provides the important properties of the lexicographic tolerable robust solution for u... more This study provides the important properties of the lexicographic tolerable robust solution for uncertain multi-objective optimization problems which was introduced by Boriwan et al. [Boriwan, P.; Ehrgott, M.; Kuroiwa, D.; Petrot, N. The lexicographic tolerable robustness concept for uncertain multi-objective optimization problems: a study on water resources management. Sustainability. 12 (2020), no. 18, article number 7582.]. Also, the relationship between the lexicographic tolerable solution concept and the well-known robust solution, as the set-based robust efficiency [Ehrgott, M.; Ide, J.; Schöbel, A. Minmax robustness for multi-objective optimization problems. Eur. J. Oper. Res. 239 (2014), no. 1, 17–31.], are provided.
Journal of Nonlinear Sciences and Applications, May 22, 2016
In this paper, we consider the split quasi variational inequality problems over a class of noncon... more In this paper, we consider the split quasi variational inequality problems over a class of nonconvex sets, as uniformly prox-regular sets. The sufficient conditions for the existence of solutions of such a problem are provided. Furthermore, an iterative algorithm for finding a solution is constructed and its convergence analysis are considered. The results in this paper improve and extend the variational inequality problems which have been appeared in literature.
We introduce a generalized forward-backward splitting method with penalty term for solving monoto... more We introduce a generalized forward-backward splitting method with penalty term for solving monotone inclusion problems involving the sum of a finite number of maximally monotone operators and the normal cone to the nonempty set of zeros of another maximally monotone operator. We show weak ergodic convergence of the generated sequence of iterates to a solution of the considered monotone inclusion problem, provided that the condition corresponding to the Fitzpatrick function of the operator describing the set of the normal cone is fulfilled. Under strong monotonicity of an operator, we show strong convergence of the iterates. Furthermore, we utilize the proposed method for minimizing a largescale hierarchical minimization problem concerning the sum of differentiable and nondifferentiable convex functions subject to the set of minima of another differentiable convex function. We illustrate the functionality of the method through numerical experiments addressing constrained elastic net and generalized Heron location problems.
International journal on fixed point theory computation and applications
In this paper, we introduce a new iterative scheme for finding solutions the common element of th... more In this paper, we introduce a new iterative scheme for finding solutions the common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inclusion problem with a multivalued maximal monotone mapping and an α-inverse-strongly monotone mapping. We show that the sequence converges strongly to a common solutions for quasi variational inclusion and fixed point problems under some parameters controlling conditions. This main theorem extends a recent result of Zhang et al. [Algorithms of common solutions to quasi variational inclusion and fixed point problems.
The main objective of this paper is to find a common solution of split variational inclusion prob... more The main objective of this paper is to find a common solution of split variational inclusion problem and fixed point problem of infinite family of nonexpansive operators in a setting of real Hilbert spaces. To reach this goal, the iterative algorithms which combine Moudafi's viscosity approximation method with some fixed point technically proving methods are utilized for solving the problem. We prove that the iterative schemes with some suitable control conditions converge strongly to a common solution of the considered problem. We also show that many interesting problems can be solved by using our presented results. Index Terms—Split variational inclusion problem, fixed point problem, nonexpansive operators, resolvent operators, strong convergence.
Necessary and sufficient conditions under which the Cesàro-Orlicz sequence space ces φ is nontriv... more Necessary and sufficient conditions under which the Cesàro-Orlicz sequence space ces φ is nontrivial are presented. It is proved that for the Luxemburg norm, Cesàro-Orlicz spaces ces φ have the Fatou property. Consequently, the spaces are complete. It is also proved that the subspace of order continuous elements in ces φ can be defined in two ways. Finally, criteria for strict monotonicity, uniform monotonicity and rotundity (= strict convexity) of the spaces ces φ are given.
The main objective of this paper is to introduce a split hierarchical variational inequality prob... more The main objective of this paper is to introduce a split hierarchical variational inequality problem. Several related problems are also considered. We propose an iterative method for finding a solution of our problem. The weak convergence of the sequence generated by the proposed method is studied.
In this paper, some basic properties of the general modular space are proven. Criteria for strict... more In this paper, some basic properties of the general modular space are proven. Criteria for strictly monotone points, extreme points and SU-points in generalized Calderón-Lozanovskiǐ spaces are obtained. Consequently, the sufficient and necessary conditions for the rotundity properties of such spaces are given.
Some existence theorems for the mixed variational-like inequality for fuzzy mappings FMVLIP in a ... more Some existence theorems for the mixed variational-like inequality for fuzzy mappings FMVLIP in a reflexive Banach space are established. Further, the auxiliary principle technique is used to suggest a novel and innovative iterative algorithm for computing the approximate solution. Consequently, not only the existence of solutions of the FMVLIP is shown, but also the convergence of iterative sequences generated by the algorithm is also proven. The results proved in this paper represent an improvement of previously known results.
At the present article, we consider a new class of general nonlinear random Amaximal m-relaxed h-... more At the present article, we consider a new class of general nonlinear random Amaximal m-relaxed h-accretive equations with random relaxed cocoercive mappings and random fuzzy mappings in q-uniformly smooth Banach spaces. By using the resolvent mapping technique for A-maximal m-relaxed h-accretive mappings due to Lan et al. and Chang's lemma, we construct a new iterative algorithm with mixed errors for finding the approximate solutions of this class of nonlinear random equations. We also verify that the approximate solutions obtained by the our proposed algorithm converge to the exact solution of the general nonlinear random A-maximal m-relaxed h-accretive equations with random relaxed cocoercive mappings and random fuzzy mappings in q-uniformly smooth Banach spaces.
We use the Wiener-Hopf equations and the Yosida approximation notions to prove the existence theo... more We use the Wiener-Hopf equations and the Yosida approximation notions to prove the existence theorem of a system of nonlinear mixed implicit equilibrium problems SMIE in Hilbert spaces. The algorithm for finding a solution of the problem SMIE is suggested; the convergence criteria and stability of the iterative algorithm are discussed. The results presented in this paper are more general and are viewed as an extension, refinement, and improvement of the previously known results in the literature.
On the Orlicz- Cesaro sequence spaces ( ces<f> ) which are defined by using Orlicz function... more On the Orlicz- Cesaro sequence spaces ( ces<f> ) which are defined by using Orlicz function <I> , we show that the space ces<f> equipped with both Amemiya and Luxemburg norms possesses uniform Opial property and uniform Kadec-Klee property if <I> satisfy the 52 -condition.
International Journal of Mathematics and Mathematical Sciences, 2004
We consider the generalized Cesàro sequence spaces defined by Suantai (2003) and consider it equi... more We consider the generalized Cesàro sequence spaces defined by Suantai (2003) and consider it equipped with the Amemiya norm. The main purpose of this paper is to show thatces(p)equipped with the Amemiya norm is rotund and has uniform Kadec-Klee property.
In this paper, we introduce and consider a new system of generalized nonlinear mixed variational ... more In this paper, we introduce and consider a new system of generalized nonlinear mixed variational inequalities involving six different nonlinear operators and discuss the existence and uniqueness of solution of the aforesaid system. We use three nearly uniformly Lipschitzian mappings S i (i = 1, 2, 3) to suggest and analyze some new three-step resolvent iterative algorithms with mixed errors for finding an element of the set of fixed points of the nearly uniformly Lipschitzian mapping Q = (S 1 , S 2 , S 3), which is the unique solution of the system of generalized nonlinear mixed variational inequalities. The convergence analysis of the suggested iterative algorithms under suitable conditions is studied. In the final section, an important remark on a class of some relaxed cocoercive mappings is discussed.
In this paper, the existing theorems and methods for finding solutions of systems of general nonl... more In this paper, the existing theorems and methods for finding solutions of systems of general nonlinear set-valued mixed variational inequalities problems in Hilbert spaces are studied. To overcome the difficulties, due to the presence of a proper convex lower semi-continuous function, and a mapping g, which appeared in the considered problem, we have used some applications of the resolvent operator technique. We would like to point out that although many authors have proved results for finding solutions of the systems of nonlinear set-valued (mixed) variational inequalities problems, it is clear that it cannot be directly applied to the problems that we have considered in this paper because of and g.
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