A novel iterative approach for resolving generalized variational inequalities

In this paper we introduce and consider a new class of variational inequalities with four operators. This class is called the extended general mixed quasi variational inequality. We show that the extended general mixed quasi variational inequality is equivalent to the mixed point problem. We use this alternative equivalent formulation to discuss the existence of a solution of extended general mixed quasi variational inequality and also develop several iterative methods for solving extended general mixed quasi variational inequality and its variant forms. We consider the convergence analysis of the proposed iterative methods under appropriate conditions. We also introduce a new class of resolvent equation, which is called the extended general implicit resolvent equation and establish an equivalent relation between the extended general implicit resolvent equation and the extended general mixed quasi variational inequality. Some special cases are also discussed.

Differentiable non-convex functions and general variational inequalities

We suggest and analyze a new self‐adaptive method for solving general mixed variational inequalities, which can be viewed as an improvement of the method of (Noor 2003). Global convergence of the new method is proved under the same assumptions as Noor′s method. Some preliminary computational results are given to illustrate the efficiency of the proposed method. Since the general mixed variational inequalities include general variational inequalities, quasivariational inequalities, and nonlinear (implicit) complementarity problems as special cases, results proved in this paper continue to hold for these problems.

Iterative methods for solving extended general mixed variational inequalities

Extended general variational inequalities

In this paper, we introduce and consider a new system of general nonconvex variational inequalities involving four different operators. We establish the equivalence between the system of general nonconvex variational inequalities and the fixed points problem using the projection technique. This alternative equivalent formulation is used to suggest and analyze some new explicit iterative methods for this system of nonconvex variational inequalities. We also study the convergence analysis of the new iterative method under certain mild conditions. Several special cases are also considered. Our results can be viewed as a refinement and improvement of the previously known results for variational inequalities.

In this paper, we introduce a new class of general mixed vector F-implicit complementarity problems and general mixed vector F-implicit variational inequality problems, and study the equivalence between of them under certain assumptions in Banach spaces. We also derive some new existence theorems of solutions for the general mixed vector F-implicit complementarity problems and the general mixed vector F-implicit variational inequality problems by using the FKKM theorem under some suitable assumptions without monotonicity. Moreover, we establish sufficient conditions for the upper semicontinuity and lower semicontinuity of the solution mapping of the general mixed vector F-implicit variational inequality problems.

Existence theorem and algorithm for a general implicit variational inequality in Banach space

Based on the classical proximal point algorithm (PPA), some PPA-based numerical algorithms for general variational inequalities (GVIs) have been developed recently. Inspired by these algorithms, in this article we propose some proximal algorithms for solving linearly constrained GVIs (LCGVIs). The resulted subproblems are regularized proximally, and they are allowed to be solved either exactly or approximately.

In this paper, we introduce a new class of variational inequalities, which is called the general quasi-variational inequality. We establish the equivalence among the general quasi variational inequality and implicit fixed point problems and the Wiener–Hopf equations. We use this equivalent formulation to discuss the existence of a solution of the general quasi-variational inequality. This equivalent formulation is used to suggest and analyze some iterative algorithms for solving the general quasi-variational inequality. We also discuss the convergence analysis of these iterative methods. Several special cases are also discussed.

In this paper, using the projection operator, we introduce two new dynamical systems for extended general quasi variational inequalities. These dynamical systems are called extended general implicit projected dynamical system and extended general implicit Wiener-Hopf dynamical system. We prove that these new dynamical systems converge globally exponentially to a unique solution of the extended general quasi variational inequalities under some suitable conditions. Some special cases are also discussed. The ideas and technique of this paper may stimulate further research.

In this paper, we introduce the concept of well-posedness for general variational inequalities and obtain some results under pseudomonotonicity. It is well known that monotonicity implies pseudomonotonicity, but the converse is not true. In this respect, our results represent an improvement and refinement of the previous known results. Since the general variational inequalities include (quasi) variational inequalities and complementarity problems as special cases, results obtained in this paper continue to hold for these problems.

We apply the variational characterization of the metric projection to prove some results about the solvability of general variational inequalities and the existence of maximum and minimum solutions to some general variational inequalities in the Hilbert lattices.

This paper introduces a novel Picard-type iterative algorithm for solving general variational inequalities in real Hilbert spaces. The proposed algorithm enhances both the theoretical framework and practical applicability of iterative algorithms by relaxing restrictive conditions on parametric sequences, thereby expanding their scope of use. We establish convergence results, including a convergence equivalence with a previous algorithm, highlighting the theoretical relationship while demonstrating the increased flexibility and efficiency of the new approach. The paper also addresses gaps in the existing literature by offering new theoretical insights into the transformations associated with variational inequalities and the continuity of their solutions, thus paving the way for future research. The theoretical advancements are complemented by practical applications, such as the adaptation of the algorithm to convex optimization problems and its use in real-world contexts like machine learning. Numerical experiments confirm the proposed algorithm’s versatility and efficiency, showing superior performance and faster convergence compared to an existing method.

This paper first introduces a new problem that is the general split variational inequality problem and invents a mathematical tool for solving our new problem which is Lemma 2.5. Then, we establish and prove a strong convergence theorem aimed at finding an element of the set of the solution of the general split variational inequality problem. Furthermore, we apply our main theorem to demonstrate a strong convergence theorem for finding solutions to the split variational inequality problem, the split feasibility problem, and the minimization problem. Finally, we provide numerical examples to advocate our main result