Subgradient Extragradient Algorithm Research Articles

Double inertial subgradient extragradient algorithm for solving equilibrium problems and common fixed point problems with application to image restoration

This paper presents a double inertial method for solving equilibrium problems and common fixed point problems in Hilbert spaces. On the basis of the subgradient extragradient method, we modify the self adaptive rule and use an additional parameter to select appropriate step size. Under reasonable assumptions, we establish both weak and linear convergence properties for the proposed algorithm. Finally, numerical experiments are conducted to validate the rationality and effectiveness of the proposed method over the existing ones in the literature.

Extension of the subgradient extragradient algorithm for solving variational inequalities without monotonicity

Extension of the subgradient extragradient algorithm for solving variational inequalities without monotonicity

A new double inertial subgradient extragradient algorithm for solving split pseudomonotone equilibrium problems and fixed point problems

The purpose of this article is to suggest a modified subgradient extragradient method that includes double inertial extrapolations and viscosity approach for finding the common solution of split equilibrium problem and fixed point problem. The strong convergence result of the suggested method is obtained under some standard assumptions on the control parameters. Our method does not require solving two strongly convex optimization problems in the feasible sets per iteration, and the step-sizes do not depend on bifunctional Lipschitz-type constants. Furthermore, unlike several methods in the literature, our method does not depend on the prior knowledge of the operator norm of the bounded linear operator. Instead, the step-sizes are self adaptively updated. We apply our method to solve split variational inequality problem. Lastly, we conduct some numerical test to compare our method with some well known methods in the literature.

Self adaptive alternated inertial algorithm for solving variational inequality and fixed point problems

<abstract><p>We introduce an alternated inertial subgradient extragradient algorithm of non-Lipschitz and pseudo-monotone operators to solve variational inequality and fixed point problems. We also demonstrated that, under certain conditions, the sequence produced by our algorithm exhibits weak convergence. Moreover, some numerical experiments have been proposed to compare our algorithm with previous algorithms in order to demonstrate the effectiveness of our algorithm.</p></abstract>

Modified inertial subgradient extragradient algorithms for generalized equilibria systems with constraints of variational inequalities and fixed points

<abstract><p>In this research, we studied modified inertial composite subgradient extragradient implicit rules for finding solutions of a system of generalized equilibrium problems with a common fixed-point problem and pseudomonotone variational inequality constraints. The suggested methods consisted of an inertial iterative algorithm, a hybrid deepest-descent technique, and a subgradient extragradient method. We proved that the constructed algorithms converge to a solution of the considered problem, which also solved some hierarchical variational inequality.</p></abstract>

A modified viscosity type inertial subgradient extragradient algorithm for nonmonotone equilibrium problems and application to cardiovascular disease detection

A modified viscosity type inertial subgradient extragradient algorithm for nonmonotone equilibrium problems and application to cardiovascular disease detection

Weak and strong convergence of a modified double inertial projection algorithm for solving variational inequality problems

We propose a new double inertial projection algorithm by combining the subgradient extragradient algorithm with the projection contraction algorithm. On one hand, our algorithm requires the mapping to be Lipschitz continuous, but without the Lipschitz constant. On the other hand, this algorithm only requires the mapping is quasimonotone.Under some mild conditions, we obtain a weak convergence result of the algorithm. In addition, we give a strong convergence result of the algorithm when the mapping is strongly pseudomonotone. Some numerical experiments are given to show the effectiveness of the proposed algorithm.

Subgradient Extra-Gradient Algorithm for Pseudomonotone Equilibrium Problems and Fixed-Point Problems of Bregman Relatively Nonexpansive Mappings

In this article, we introduce a new subgradient extra-gradient algorithm to find the common element of a set of fixed points of a Bregman relatively nonexpansive mapping and the solution set of an equilibrium problem involving a Pseudomonotone and Bregman–Lipschitz-type bifunction in reflexive Banach spaces. The advantage of the algorithm is that it is run without prior knowledge of the Bregman–Lipschitz coefficients. Finally, two numerical experiments are reported to illustrate the efficiency of the proposed algorithm.

New Convergence Theorems for Pseudomonotone Variational Inequality on Hadamard Manifolds

In this paper, we propose an efficient viscosity type subgradient extragradient algorithm for solving pseudomonotone variational inequality on Hadamard manifolds which is of symmetrical characteristic. Under suitable conditions, we obtain the convergence of the iteration sequence generated by the proposed algorithm to a solution of a pseudomonotone variational inequality on Hadamard manifolds. We also employ our main result to solve a constrained convex minimization problem and present a numerical experiment to illustrate the asymptotic behavior of the algorithm. Our results develop and improve some recent results.

Inertial subgradient extragradient method for solving pseudomonotone variational inequality problems in Banach spaces

In this paper, an inertial subgradient extragradient algorithm is proposed to solve the pseudomonotone variational inequality problems in Banach space. This iterative scheme employs a new line-search rule. Strong convergence theorems for the proposed algorithms are established under the assumptions that the operators are non-Lipschitz continuous. Furthermore, several numerical experiments are given to show that our method has better convergence performance than the known ones in the literatures.

An accelerated subgradient extragradient algorithm for solving bilevel variational inequality problems involving non-Lipschitz operator

In this paper, an accelerated subgradient extragradient algorithm with a new non-monotonic step size is proposed to solve bilevel variational inequality problems involving non-Lipschitz continuous operator in Hilbert spaces. The proposed algorithm with a new non-monotonic step size has the advantage of requiring only one projection onto the feasible set during each iteration and does not require prior knowledge of the Lipschitz constant of the mapping involved. Under suitable and weaker conditions, the proposed algorithm achieves strong convergence. Some numerical tests are provided to demonstrate the efficiency and advantages of the proposed algorithm against existing related algorithms.

Accelerated Subgradient Extragradient Algorithm for Solving Bilevel System of Equilibrium Problems

In this research paper, we propose a novel approach termed the inertial subgradient extragradient algorithm to solve bilevel system equilibrium problems within the realm of real Hilbert spaces. Our algorithm is capable of circumventing the necessity for prior knowledge about the Lipschitz constant of the involving bifunction and only computes the minimization of strong bifunctions onto the feasible set that is required. Under appropriate conditions, we establish strong convergence theorems for our proposed algorithms. To validate our algorithms, we illustrate a series of numerical examples. Through these examples, we demonstrate the performance of the algorithms we have put forth in this paper.

Modified Inertial-Type Subgradient Extragradient Methods for Variational Inequalities and Fixed Points of Finite Bregman Relatively Nonexpansive and Demicontractive Mappings

In this paper, we design two inertial-type subgradient extragradient algorithms with the linear-search process for resolving the two pseudomonotone variational inequality problems (VIPs) of and the common fixed point problem (CFPP) of finite Bregman relatively nonexpansive operators and Bregman relatively demicontractive operators in Banach spaces of both p-uniform convexity and uniform smoothness, which are more general than Hilbert ones. By the aid of suitable restrictions, it is shown that the sequences fabricated by the suggested schemes converge weakly and strongly to a solution of a pair of VIPs with a CFPP constraint, respectively. Additionally, the illustrative instance is furnished to back up the practicability and implementability of the suggested methods. This paper reveals the competitive advantage of the proposed algorithms over the existing algorithms; that is, the existing hybrid projection method for a single VIP with an FPP constraint is extended to develop the modified inertial-type subgradient extragradient method for a pair of VIPs with an CFPP constraint.

New subgradient extragradient algorithm for solving variational inequalities in Hadamard manifold

This paper introduces a schematic approximation method for a solution to a variational inequality problem in the framework of Hadamard manifold. The method is a combination of the subgradient extragradient technique and Popov extragradient method. Using this method, some convergence algorithms were proved when the cost operators are pseudomonotone and strongly pseudomonotone, respectively. In the construction of this method, the dependence on Lipschitz constants of the operators is dispensed with by the use of a monotone decreasing step size. We give an application of our main result to the constrained convex minimization problem. Finally, we report some numerical examples to illustrate the efficiency and applicability of the method.

Modified Subgradient Extragradient Algorithms with A New Line-Search Rule for Variational Inequalities

Modified Subgradient Extragradient Algorithms with A New Line-Search Rule for Variational Inequalities

A Halpern-type algorithm for a common solution of nonlinear problems in Banach spaces

Abstract In this article, we propose a Halpern-type subgradient extragradient algorithm for solving a common element of the set of solutions of variational inequality problems for continuous monotone mappings and the set of f-fixed points of continuous f-pseudocontractive mappings in reflexive real Banach spaces. In addition, we prove a strong convergence theorem for the sequence generated by the algorithm. As a consequence, we obtain a scheme that converges strongly to a common f-fixed point of continuous f-pseudocontractive mappings and a scheme that converges strongly to a common zero of continuous monotone mappings in Banach spaces. Furthermore, we provide a numerical example to illustrate the implementability of our algorithm.

An inertial subgradient extragradient algorithm for modified variational inclusion problem and mixed equilibrium problem in real Hilbert space

An inertial subgradient extragradient algorithm for modified variational inclusion problem and mixed equilibrium problem in real Hilbert space

Modified subgradient extragradient algorithms for systems of generalized equilibria with constraints

<abstract><p>In this paper, we introduce the modified Mann-like subgradient-like extragradient implicit rules with linear-search process for finding a common solution of a system of generalized equilibrium problems, a pseudomonotone variational inequality problem and a fixed-point problem of an asymptotically nonexpansive mapping in a real Hilbert space. The proposed algorithms are based on the subgradient extragradient rule with linear-search process, Mann implicit iteration approach, and hybrid deepest-descent technique. Under mild restrictions, we demonstrate the strong convergence of the proposed algorithms to a common solution of the investigated problems, which is a unique solution of a certain hierarchical variational inequality defined on their common solution set.</p></abstract>

A complementarity model for a supply chain network equilibrium problem with electronic commerce

In this paper, we develop a competitive supply chain network equilibrium model with multiple products, in which electronic commerce is considered. After analysing equilibrium conditions of the manufactures, retailers, and demand markets, and giving economic interpretations, equilibrium conditions of the supply chain network formulated as an equivalent complementarity model are established. In addition, a modified subgradient extragradient algorithm is proposed to find the solution of the complementarity problem, and the convergence of the proposed algorithm is proved. We also give some numerical examples to demonstrate the efficiency of the given model and algorithm.

A self-adaptive inertial subgradient extragradient algorithm for solving bilevel equilibrium problems

A self-adaptive inertial subgradient extragradient algorithm for solving bilevel equilibrium problems