Bilevel Variational Inequality Research Articles

An Alternated Inertial Projection and Contraction Algorithm for Solving Quasimonotone Bilevel Variational Inequalities with Application to Optimal Control Problems

An Alternated Inertial Projection and Contraction Algorithm for Solving Quasimonotone Bilevel Variational Inequalities with Application to Optimal Control Problems

Correction: A modified inertial Tseng technique of Bilevel variational inequality problem with application to image processing

Correction: A modified inertial Tseng technique of Bilevel variational inequality problem with application to image processing

A relaxed projection method for solving bilevel variational inequality problems

A relaxed projection method for solving bilevel variational inequality problems

Two novel two-step inertial algorithms for a class of bilevel variational inequalities

This paper introduces two novel two-step inertial and self-adaptive step sizes algorithms for solving a strongly monotone variational inequality over the solution set of a split convex feasibility problem with multiple output sets in real Hilbert spaces. We establish strong convergence properties of the proposed method under mild conditions and employ it to solve monotone variational inequalities over the solution set of the split feasibility problem. The method is further applied to the image classification problem via the support vector machine learning. Numerical results are given to demonstrate the accelerating behaviors of our method over other related methods in the literature.

A modified inertial Tseng technique of Bilevel variational inequality problem with application to image processing

A modified inertial Tseng technique of Bilevel variational inequality problem with application to image processing

Two new inertial relaxed CQ algorithms for a class of bilevel variational inequalities with the split feasibility problem constraints

Two new inertial relaxed CQ algorithms for a class of bilevel variational inequalities with the split feasibility problem constraints

On Bilevel Monotone Inclusion and Variational Inequality Problems

In this article, the problem of solving a strongly monotone variational inequality problem over the solution set of a monotone inclusion problem in the setting of real Hilbert spaces is considered. To solve this problem, two methods, which are improvements and modifications of the Tseng splitting method, and projection and contraction methods, are presented. These methods are equipped with inertial terms to improve their speed of convergence. The strong convergence results of the suggested methods are proved under some standard assumptions on the control parameters. Also, strong convergence results are achieved without prior knowledge of the operator norm. Finally, the main results of this research are applied to solve bilevel variational inequality problems, convex minimization problems, and image recovery problems. Some numerical experiments to show the efficiency of our methods are conducted.

An inertial-type algorithm for a class of bilevel variational inequalities with the split variational inequality problem constraints

In this paper, we investigate the problem of solving strongly monotone variational inequalities over the solution set of the split variational inequality problem with multiple output sets in real Hilbert spaces and establish a new iterative algorithm for it. Our algorithms are accelerated by the inertial technique and eliminate the dependence on the norm of the transformation operators and the strongly monotone and Lipschitz continuous constants of the involved operator by employing a self-adaptive step size criterion. The strong convergence result is given under some mild conditions widely used in the convergence analysis. Two corollaries for the solutions of the split variational inequality problem and the split feasibility problem with multiple output sets are also obtained using our main result. Finally, some numerical experiments have been conducted to illustrate the effectiveness of the proposed algorithms and compare them with the related ones.

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.

Relaxed Single Projection Methods for Solving Bilevel Variational Inequality Problems in Hilbert Spaces

Relaxed Single Projection Methods for Solving Bilevel Variational Inequality Problems in Hilbert Spaces

A new inertial relaxed Tseng extrgradient method for solving quasi-monotone bilevel variational inequality problems in Hilbert spaces

A new inertial relaxed Tseng extrgradient method for solving quasi-monotone bilevel variational inequality problems in Hilbert spaces

Hybrid Alternated Inertial Projection and Contraction Algorithm for Solving Bilevel Variational Inequality Problems

This paper considers an alternated inertial-type extrapolation algorithm for solving bilevel pseudomonotone variational inequality problem in the framework of real Hilbert spaces with split variational inequality and fixed-point constraints of demimetric mapping. The algorithm which involves alternated inertial uses self-adjustment stepsize condition that depends solely on the information from previous iterative step. The bilevel problem considered in the work consists of upper-level problem with an underlying operator which is pseudomonotone, while the lower problem is associated with strongly monotone mapping and the fixed-point constraint of demimetric mapping. Our algorithm is anchored on modified projection and contraction techniques, fused with alternated inertial and relaxation. Under some suitable conditions on the algorithm control parameters, we obtain a strong convergence result of the proposed method without the prior knowledge of the operator norm or the coefficients of the underlying operators within the scope of real Hilbert spaces. Finally, some numerical examples are presented to illustrate the gain of our method in comparison to some related algorithms in the literature.

Convergence of the projection and contraction methods for solving bilevel variational inequality problems

In this work, we analyze some convergent properties of a projection and contraction algorithm for solving a variational inequality problem, where the feasible domain is the solution set of an affine variational inequality problem. We prove that, for solving the problem where the second cost mapping is affine and not necessary for monotone properties, any iterative sequence generated by the algorithm converges to a unique solution provided that the first cost mapping is strongly monotone and Lipschitz continuous. Computational errors of the algorithm are showed. Finally, some preliminary numerical experiences and comparisons are also reported.

A self adaptive method for solving a class of bilevel variational inequalities with split variational inequality and composed fixed point problem constraints in Hilbert spaces

<p style='text-indent:20px;'>In this work, we propose a new inertial method for solving strongly monotone variational inequality problems over the solution set of a split variational inequality and composed fixed point problem in real Hilbert spaces. Our method uses stepsizes that are generated at each iteration by some simple computations, which allows it to be easily implemented without the prior knowledge of the operator norm as well as the Lipschitz constant of the operator. In addition, we prove that the proposed method converges strongly to a minimum-norm solution of the problem without using the conventional two cases approach. Furthermore, we present some numerical experiments to show the efficiency and applicability of our method in comparison with other methods in the literature. The results obtained in this paper extend, generalize and improve results in this direction.</p>

STRONG CONVERGENCE OF MULTI-PARAMETER PROJECTION METHODS FOR VARIATIONAL INEQUALITY PROBLEMS

In this paper, we introduce a multi-parameter projection method for solving a variational inequality problem, and establish its strong convergence in a Hilbert space under appropriate conditions. The method involves two projectionsteps with different variable stepsizes where one of them is computed explicitly on a specifically structural half-space. The proof of strong convergence of the method is based on the regularization solutions depending on parameters of the original problem. It turns out that the solution obtained by the method is the solution of a bilevel variational inequality problem whose constraint is the solution set of our considered problem. In order to support the obtained theoretical results, we perform some experiments on transportation equilibrium and optimal control problems, and also involve comparisons. Numerical results show the computational effectiveness and the fast convergence of the new method over some existing ones.

Two projection-based methods for bilevel pseudomonotone variational inequalities involving non-Lipschitz operators

Two projection-based methods for bilevel pseudomonotone variational inequalities involving non-Lipschitz operators

An inertial scheme for solving bi-level variational inequalities and the fixed point problem with pseudomonotone and $\varrho$-demimetric mappings

An inertial scheme for solving bi-level variational inequalities and the fixed point problem with pseudomonotone and $\varrho$-demimetric mappings

Revisiting inertial subgradient extragradient algorithms for solving bilevel variational inequality problems

Revisiting inertial subgradient extragradient algorithms for solving bilevel variational inequality problems

Two adaptive modified subgradient extragradient methods for bilevel pseudomonotone variational inequalities with applications

We consider the bilevel variational inequality problem with a pseudomonotone operator in real Hilbert spaces and investigate two modified subgradient extragradient methods with inertial terms. Our first scheme requires the operator to be Lipschitz continuous (the Lipschitz constant does not need to be known) while the second one only requires it to be uniformly continuous. The proposed methods employ two adaptive stepsizes making them work without the prior knowledge of the Lipschitz constant of the mapping. The strong convergence properties of the iterative sequences generated by the proposed algorithms are obtained under mild conditions. Some numerical tests and applications are given to demonstrate the advantages and efficiency of the stated schemes over previously known ones.

Variational inequality over the set of common solutions of a system of bilevel variational inequality problem with applications

Variational inequality over the set of common solutions of a system of bilevel variational inequality problem with applications