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We are focused on solving a general class of bilevel variational inequalities involving quasimonotone operators in real Hilbert spaces. A strong convergent iterative method for solving the problem is presented and analysed. Our work generalizes several existing results in the literature and holds two major mathematical advantages. 1) Any generated sequence by the algorithm preserves the Fejér monotonicity property; and 2) There is no need to execute a line-search or know a-prior the strongly monotone coefficient or Lipschitz constant. Numerical experiments with comparisons to existing/related methods illustrate the performances of the proposed method and in particular, application to optimal control problems suggests the practical potential of our scheme. [ABSTRACT FROM AUTHOR]

In the last two decades, several properties of operators that are weaker than monotonicity have received attention by researchers from many areas including mathematical economics, with the goal to develop new tools applicable in convex analysis and related topics. This paper puts in perspective notions that are extensions of monotoniticity but not beyond quasimonotonicity like pseudomonotonicity, semistrict quasimonotonicity, strict quasimonotonicity and proper quasimonotonicity, and discusses systematically when the sum of two operators satisfying one of those properties, inherits the same property. The case of properly quasimonotone operators deserves a special attention since this notion, being stronger than quasimonotonicity, suffices to obtain many results, including the solvability of variational inequality problems. Several examples showing the optimality in some sense of our results, are presented. [ABSTRACT FROM AUTHOR]

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.

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. [ABSTRACT FROM AUTHOR]

The primary goal of this research is to investigate the approximate numerical solution of variational inequalities using quasimonotone operators in infinite-dimensional real Hilbert spaces. In this study, the sequence obtained by the proposed iterative technique for solving quasimonotone variational inequalities converges strongly toward a solution due to the viscosity-type iterative scheme. Furthermore, a new technique is proposed that uses an inertial mechanism to obtain strong convergence iteratively without the requirement for a hybrid version. The fundamental benefit of the suggested iterative strategy is that it substitutes a monotone and non-monotone step size rule based on mapping (operator) information for its Lipschitz constant or another line search method. This article also provides a numerical example to demonstrate how each method works.

Abstract In this article, we introduce an inertial-type algorithm that combines the extragradient subgradient method, the projection contraction method, and the viscosity method. The proposed method is used for solving quasimonotone variational inequality problems in infinite dimensional real Hilbert spaces such that it does not depend on the Lipschitz constant of the cost operator. Further, we prove the strong convergence results of the new algorithm. Our strong convergence results are achieved without imposing strict conditions on the control parameters and inertial factor of our algorithm. We utilize our algorithm to solve some problems in applied sciences and engineering such as image restoration and optimal control. Some numerical experiments are carried out to support our theoretical results. Our numerical illustrations show that our new method is more efficient than many existing methods.

Projection methods have been one of the popular methods for solving variational inequality problems in the literature. Several projection methods have been studied for variational inequality problems and most of these methods require at least two evaluations of the cost operator and (or) two projections per iteration. One of the major interests of researchers in variational inequalities is to study projection methods with less computational complexity in terms of reduced number of evaluations of the cost operator and reduced number of projections onto feasible set per iteration especially when the cost operator is non-monotone. In this paper, we propose a simple projection method which requires only one evaluation of the cost operator and one projection onto the feasible set per iteration. We obtain weak convergence results when the cost operator is quasimonotone and Lipschitz continuous with self-adaptive line search strategy. We also give weak convergence analysis when the cost operator is non-monotone and Lipschitz continuous. Some numerical implementations using standard tests are given to illustrate our method and compare with other recently related methods. [ABSTRACT FROM AUTHOR]

In this article, we introduce an inertial-type algorithm that combines the extragradient subgradient method, the projection contraction method, and the viscosity method. The proposed method is used for solving quasimonotone variational inequality problems in infinite dimensional real Hilbert spaces such that it does not depend on the Lipschitz constant of the cost operator. Further, we prove the strong convergence results of the new algorithm. Our strong convergence results are achieved without imposing strict conditions on the control parameters and inertial factor of our algorithm. We utilize our algorithm to solve some problems in applied sciences and engineering such as image restoration and optimal control. Some numerical experiments are carried out to support our theoretical results. Our numerical illustrations show that our new method is more efficient than many existing methods. [ABSTRACT FROM AUTHOR]

In this paper, we consider an iterative approximation problem of split variational inequalities in Hilbert spaces. In order to solve this split problem, we construct an iterative algorithm which combines a forward-backward-forward method and a self-adaptive rule to update the step-sizes. We prove that the constructed algorithm converges strongly to a solution of the split variational inequalities under some mild assumptions. [ABSTRACT FROM AUTHOR]

In this paper, we introduce an inertial relaxed Tseng extragradient method involving only a single projection for solving bilevel variational inequality problems with Lipschitz continuous and quasimonotone mapping in Hilbert spaces. Under some mild standard assumptions, we obtain a strong convergence result for solving bilevel quasimonotone variational inequality problems. The main advantages of the proposed iterative method are that it requires only one projection onto the feasible set and the use self adaptive step-size rule based on operator knowledge rather than a Lipschitz constant or some line search method. Moreover, some interesting preliminary numerical experiments and comparisons were presented. [ABSTRACT FROM AUTHOR]

The paper proposes multiple new extragradient methods for solving a variational inequality problem involving quasimonotone operators in infinite-dimensional real Hilbert spaces. These methods contain variable stepsize rules that are revised at each iteration and are dependent on prior iterations. These algorithms have the benefit of not requiring prior knowledge of the Lipschitz constant or any line-search approach. Simple conditions are used to demonstrate the algorithm's convergence. A collection of simple experiments is presented to show the numerical behavior of the algorithms. [ABSTRACT FROM AUTHOR]

For most projection methods, the operator of a variational inequality problem is assumed to be monotone (or pseudomonotone) and Lipschitz continuous. In this paper, we present a projection-like method to solve quasimonotone variational inequality problems without Lipschitz continuity. Under some mild assumptions, we prove that the sequence generated by the proposed algorithm converges to a solution. Numerical experiments are provided to show the effectiveness of the method. [ABSTRACT FROM AUTHOR]

In this paper we study some properties of the adjusted normal cone operator of quasiconvex functions. In particular, we introduce a new notion of maximal quasimotonicity for set-valued maps different from similar ones recently appeared in the literature, and we show that it is enjoyed by this operator. Moreover, we prove the s+w* cone upper semicontinuity of the normal cone operator in the domain of f in case the set of global minima has non empty interior. [ABSTRACT FROM AUTHOR]

It is well-known that the sum of two quasiconvex functions is not quasiconvex in general, and the same occurs with the minimum. Although apparently these two statements (for the sum or minimum) have nothing in common, they are related, as we show in this paper. To develop our study, the notion of quasiconvex family is introduced, and we establish various characterizations of such a concept: one of them being the quasiconvexity of the pointwise infimum of arbitrary translations of quasiconvex functions in the family; another is the convexity of the union of any two of their sublevel sets; a third one is the quasiconvexity of the sum of the quasiconvex functions, composed with arbitrary nondecreasing functions. As a by-product, any of the aforementioned characterizations, besides providing quasiconvexity of the sum, also implies the semistrict quasiconvexity of the sum if every function in the family has the same property. Three concrete applications in quasiconvex optimization are presented: First, we establish the convexity of the (Benson) proper efficient solution set to a quasiconvex vector optimization problem; second, we derive conditions that allow us to reduce a constrained optimization problem to one with a single inequality constraint, and finally, we show a class of quasiconvex minimization problems having zero duality gap. [ABSTRACT FROM AUTHOR]

In this paper we study the uniqueness of solutions to ordinary differential equations which fail to satisfy both accretivity condition and the uniqueness condition of Nagumo, Osgood and Kamke. The evolution systems considered here are governed by a continuous operators $A$ defined on $mathbb{R}^N$ such that $A$ is a derivor; i.e., $-A$ is quasi-monotone with respect to $(mathbb{R}^{+})^N$.

As an extension of the Leray-Schauder degree, we introduce a topological degree theory for a class of demicontinuous operators of generalized $(S_{+})$ type in real reflexive Banach spaces, based on the recent Berkovits degree. Using the degree theory, we show that the Borsuk theorem holds true for this class. Moreover, we study the Dirichlet boundary value problem involving the p-Laplacian by way of an abstract Hammerstein equation. [ABSTRACT FROM AUTHOR]

A quasivariational inequality is a variational inequality in which the constraint set depends on the variable. Based on fixed point techniques, we prove various existence results under weak assumptions on the set-valued operator defining the quasivariational inequality, namely quasimonotonicity and lower or upper sign-continuity. Applications to quasi-optimization and traffic network are also considered. [ABSTRACT FROM AUTHOR]

Whenever the data of a Stampacchia variational inequality, that is, the set-valued operator and/or the constraint map, are subject to perturbations, then the solution set becomes a solution map, and the study of the stability of this solution map concerns its regularity. An important literature exists on this topic, and classical assumptions, for monotone or quasimonotone set-valued operators, are some upper or lower semicontinuity. In this paper, we limit ourselves to perturbations on the constraint map, and it is proved that regularity results for the solution maps can be obtained under some very weak regularity hypothesis on the set-valued operator, namely the lower or upper sign-continuity. [ABSTRACT FROM AUTHOR]

The concept of maximal quasimonotonicity of set-valued map is introduced and studied. Regularity properties of this class of operators is investigated, in particular the ccvc property, an adaptation to the quasimonotone case of the classical notion of cusco map. In an Asplund space, we provide sufficient conditions for a ccvc quasimonotone operator to be single-directional on a $$G_\delta $$ -dense subset. [ABSTRACT FROM AUTHOR]

In this paper, we propose two interior proximal algorithms inspired by the logarithmic-quadratic proximal method. The first method we propose is for general linearly constrained quasiconvex minimization problems. For this method, we prove global convergence when the regularization parameters go to zero. The latter assumption can be dropped when the function is assumed to be pseudoconvex. We also obtain convergence results for quasimonotone variational inequalities, which are more general than monotone ones. [ABSTRACT FROM AUTHOR]