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We propose a prototypical Split Inverse Problem (SIP) and a new variational problem, called the Split Variational Inequality Problem (SVIP), which is a SIP. It entails finding a solution of one inverse problem (e.g., a Variational Inequality Problem (VIP)), the image of which under a given bounded linear transformation is a solution of another inverse problem such as a VIP. We construct iterative algorithms that solve such problems, under reasonable conditions, in Hilbert space and then discuss special cases, some of which are new even in Euclidean space.

We consider a parametric quasi-variational inequality (QVI) without any convexity assumption. Using the concept of optimal value function , we transform the problem into that of solving a nonsmooth system of inequalities. Based on this reformulation, new coderivative estimates as well as robust stability conditions for the optimal solution map of this QVI are developed. Also, for an optimization problem with QVI constraint, necessary optimality conditions are constructed and subsequently, a tailored semismooth Newton-type method is designed, implemented, and tested on a wide range of optimization examples from the literature. In addition to the fact that our approach does not require convexity, its coderivative and stability analysis do not involve second order derivatives, and subsequently, the proposed Newton scheme does not need third order derivatives, as it is the case for some previous works in the literature.

The paper aims to introduce inverse tensor variational inequalities and analyze their application to an economic control equilibrium model. More precisely, some existence and uniqueness results are established and the well-posedness analysis is investigated. Moreover, the Tikhonov regularization method is extended to tensor inverse problems to study them when they are ill-posed. Lastly, the policymaker’s point of view for the oligopolistic market equilibrium problem is introduced. The equivalence between the equilibrium conditions and a suitable inverse tensor variational inequality is established.

In this paper, we study a class of generalized monotone variational inequality (GMVI) problems whose operators are not necessarily monotone (e.g., pseudo-monotone). We present non-Euclidean extragradient (N-EG) methods for computing approximate strong solutions of these problems, and demonstrate how their iteration complexities depend on the global Lipschitz or Hölder continuity properties for their operators and the smoothness properties for the distance generating function used in the N-EG algorithms. We also introduce a variant of this algorithm by incorporating a simple line-search procedure to deal with problems with more general continuous operators. Numerical studies are conducted to illustrate the significant advantages of the developed algorithms over the existing ones for solving large-scale GMVI problems.

The inverse mixed variational inequality problem comes from classical variational inequality, and it has many applications. In this paper, we propose new algorithms to study the inverse mixed variational inequality problems in Hilbert spaces, and these algorithms are based on the generalized projection operator. Next, we establish convergence theorems under inverse strong monotonicity conditions. In addition, we also provide inertial-type algorithms for the inverse mixed variational inequality problems with conditions that differ from the above convergence theorems.

In this work, we reformulate the problem of existence of maximal elements for preference relations as a variational inequality problem in the sense of Stampacchia. Similarly, we establish the uniqueness of maximal elements using a variational inequality problem in the sense of Minty. In both of these approaches, we use the normal cone operator to find existence and uniqueness results, under mild assumptions. In addition, we provide an algorithm for finding such maximal elements, which is inspired by the steepest descent method for minimization. Under certain conditions, we prove that the sequence generated by this algorithm converges to a maximal element.

This paper presents an iterative algorithm for approximating a solution to a split equality variational problem involving a finite family of pseudomonotone mappings in Hilbert spaces. We demonstrate the strong convergence of the sequence produced by the algorithm to a solution of the problem in Hilbert spaces, under the assumption that the mappings are uniformly continuous. Additionally, we apply our main findings to solve split variational inequality and split equality zero point problems for a finite family of pseudomonotone mappings in Hilbert spaces, expanding on existing literature.

In this paper, first, we introduce a new iterative algorithm involving demicontractive mappings in Hilbert spaces and, second, we prove some strong convergence theorems of the proposed method with the Armijo-line search to show the existence of a solution of the split common fixed point problem. Finally, we give some numerical examples to illustrate our main results.

The Generalized Nash equilibrium problem is an important model that has its roots in the economic sciences but is being fruitfully used in many different fields. In this survey paper we aim at discussing its main properties and solution algorithms, pointing out what could be useful topics for future research in the field.

The aim of this paper is to provide new characterizations of the curvature dimension condition in the context of metric measure spaces (X,\\mathsf d,\\mathfrak m). On the geometric side, the authors' new approach takes into account suitable weighted action functionals which provide the natural modulus of K-convexity when one investigates the convexity properties of N-dimensional entropies. On the side of diffusion semigroups and evolution variational inequalities, the authors' new approach uses the nonlinear diffusion semigroup induced by the N-dimensional entropy, in place of the heat flow. Under suitable assumptions (most notably the quadraticity of Cheeger's energy relative to the metric measure structure) both approaches are shown to be equivalent to the strong \\mathrm {CD}^{*}(K,N) condition of Bacher-Sturm.