A General Inertial Projection-Type Algorithm for Solving Equilibrium Problem in Hilbert Spaces with Applications in Fixed-Point Problems

A plethora of applications from mathematical programming, such as minimax, and mathematical programming, penalization, fixed point to mention a few can be framed as equilibrium problems. Most of the techniques for solving such problems involve iterative methods that is why, in this paper, we introduced a new extragradient-like method to solve equilibrium problems in real Hilbert spaces with a Lipschitz-type condition on a bifunction. The advantage of a method is a variable stepsize formula that is updated on each iteration based on the previous iterations. The method also operates without the previous information of the Lipschitz-type constants. The weak convergence of the method is established by taking mild conditions on a bifunction. For application, fixed-point theorems that involve strict pseudocontraction and results for pseudomonotone variational inequalities are studied. We have reported various numerical results to show the numerical behaviour of the proposed method and correlate it with existing ones.This research work was financially supported by Spanish Government for Grant RTI2018-094336-B-I00 (MCIU/AEI/FEDER, UE) and to the Basque Government for Grant IT1207-19

Projection-proximal methods for general variational inequalities

AbstractIn this paper, we consider and analyze some new projection-proximal methods for solving general variational inequalities. The modified methods converge for pseudomonotone operators which is a weaker condition than monotonicity. The proposed methods include several new and known methods as special cases. Our results can be considered as a novel and important extension of the previously known results. Since the general variational inequalities include the quasi-variational inequalities and implicit complementarity problems as special cases, results proved in this paper continue to hold for these problems

Inertial subgradient extragradient with projection method for solving variational inequality and fixed point problems

In this paper, we introduce a new modified inertial Mann-type method that combines the subgradient extragradient method with the projection contraction method for solving quasimonotone variational inequality problems and fixed point problems in real Hilbert spaces. We establish strong convergence of the proposed method under some mild conditions without knowledge of the operator norm. Finally, we give numerical experiments to illustrate the efficiency of the method over the existing one in the literature

A new inertial condition on the subgradient extragradient method for solving pseudomonotone equilibrium problem

In this paper we study the pseudomonotone equilibrium problem. We consider a new inertial condition for the subgradient extragradient method with self-adaptive step size for approximating a solution of the equilibrium problem in a real Hilbert space. Our proposed method contains inertial factor with new conditions that only depend on the iteration coefficient. We obtain a weak convergence result of the proposed method under weaker conditions on the inertial factor than many existing conditions in the literature. Finally, we present some numerical experiments for our proposed method in comparison with existing methods in the literature. Our result improves, extends and generalizes several existing results in the literature

Modified mildly inertial subgradient extragradient method for solving pseudomonotone equilibrium problems and nonexpansive fixed point problems

This paper presents and examines a newly improved linear technique for solving the equilibrium problem of a pseudomonotone operator and the fixed point problem of a nonexpansive mapping within a real Hilbert space framework. The technique relies two modified mildly inertial methods and the subgradient extragradient approach. In addition, it can be viewed as an advancement over the previously known inertial subgradient extragradient approach. Based on common assumptions, the algorithm's weak convergence has been established. Finally, in order to confirm the efficiency and benefit of the proposed algorithm, we present a few numerical experiments

A novel iterative approach for resolving generalized variational inequalities

For figuring out general variational inequalities, we propose a novel and innovative iterative method. First, we demonstrate that the fixed point formulation and general vaiational inequality are equivalent. The fixed point formulation is used to formulate the explicit and implicit schemes. The general variational inequalities are the basis for the new algorithms. The newly developed algorithm is demonstrated numerically. For figuring out general variational inequalities, these new methods are innovative. Additionally, the convergence analysis is provided under certain favorable conditions

Iterative Methods for the Elasticity Imaging Inverse Problem

Cancers of the soft tissue reign among the deadliest diseases throughout the world and effective treatments for such cancers rely on early and accurate detection of tumors within the interior of the body. One such diagnostic tool, known as elasticity imaging or elastography, uses measurements of tissue displacement to reconstruct the variable elasticity between healthy and unhealthy tissue inside the body. This gives rise to a challenging parameter identification inverse problem, that of identifying the Lamé parameter μ in a system of partial differential equations in linear elasticity. Due to the near incompressibility of human tissue, however, common techniques for solving the direct and inverse problems are rendered ineffective due to a phenomenon known as the “locking effect”. Alternative methods, such as mixed finite element methods, must be applied to overcome this complication. Using these methods, this work reposes the problem as a generalized saddle point problem along with a presentation of several optimization formulations, including the modified output least squares (MOLS), energy output least squares (EOLS), and equation error (EE) frameworks, for solving the elasticity imaging inverse problem. Subsequently, numerous iterative optimization methods, including gradient, extragradient, and proximal point methods, are explored and applied to solve the related optimization problem. Implementations of all of the iterative techniques under consideration are applied to all of the developed optimization frameworks using a representative numerical example in elasticity imaging. A thorough analysis and comparison of the methods is subsequently presented