Duality methods for solving variational inequalities

Journal of Mathematical Analysis and Applications, 1999

In this paper, we suggest and analyze some new iterative methods for solving general monotone mixed variational inequalities, which are being used to study odd-order and nonsymmetric boundary value problems arising in pure and applied sciences. These new methods can be viewed as generalizations and extensions of Ž . the methods of He, Solodov and Tseng, and Noor for solving monotone mixed variational inequalities.

Journal of Mathematical Analysis and Applications, 2003

In this paper, we consider and analyze a new class of extragradient-type methods for solving general variational inequalities. The modified methods converge for pseudomonotone operators which is weaker condition than monotonicity. Our proof of convergence is very simple as compared with other methods. The proposed methods include several new and known methods as special cases. Our results present a significant improvement of previously known methods for solving variational inequalities and related optimization problems.  2002 Elsevier Science (USA). All rights reserved.

Applied Mathematics & Optimization, 1990

A general framework for the treatment of a class of elliptic variational inequalities by an augmented Lagrangian method, when inequalities with infinite-dimensional image space are augmented, is developed. Applications to the obstacle problem, the elastoplastic torsion problem, and the Signorini problem are given.

Journal of Mathematical Analysis and Applications, 2009

In this paper a general analysis of duality for an extended ε-variational inequality problem based on the notions of ε-convexity and ε-conjugacy is performed. Optimal solutions of both the primal and dual problems are also related to the saddle point of an associated Lagrangian. Gap functions for these problems are proposed. An existence theorem for the extended ε-variational inequality is also established by means of the KKM lemma.

Korpelevich's extragradient method has been studied and extended extensively due to its applicability to the whole class of monotone variational inequalities. In the present paper, we propose a variant extragradient-type method for solving monotone variational inequalities. Convergence analysis of the method is presented under reasonable assumptions on the problem data. MSC: 47H05; 47J05; 47J25

Applicable Analysis, 2019

The goal of the note is to introduce a new modified subgradient extragradient algorithm for solving variational inequalities in Hilbert spaces. A result on the strong convergence of the algorithm is proved without the knowledge of Lipschitz constant of the operator. Several numerical experiments for the proposed algorithm are presented.

2006

This work aims to investigate some applications of the conjugate duality for scalar and vector optimization problems to the construction of gap functions for variational inequalities and equilibrium problems. The basic idea of the approach is to reformulate variational inequalities and equilibrium problems into optimization problems depending on a fixed variable, which allows us to apply duality results from optimization problems. Based on some perturbations, first we consider the conjugate duality for scalar optimization. As applications, duality investigations for the convex partially separable optimization problem are discussed. Afterwards, we concentrate our attention on some applications of conjugate duality for convex optimization problems in finite and infinite-dimensional spaces to the construction of a gap function for variational inequalities and equilibrium problems. To verify the properties in the definition of a gap function weak and strong duality are used. The remainder of this thesis deals with the extension of this approach to vector variational inequalities and vector equilibrium problems. By using the perturbation functions in analogy to the scalar case, different dual problems for vector optimization and duality assertions for these problems are derived. This study allows us to propose some set-valued gap functions for the vector variational inequality. Finally, by applying the Fenchel duality on the basis of weak orderings, some variational principles for vector equilibrium problems are investigated.

arXiv (Cornell University), 2022

In this paper, we propose a general extra-gradient scheme for solving monotone variational inequalities (VI), referred to here as Approximation-based Regularized Extra-gradient method (ARE). The first step of ARE solves a VI subproblem with an approximation operator satisfying a p th -order Lipschitz bound with respect to the original mapping, further coupled with the gradient of a (p + 1) th -order regularization. The optimal global convergence is guaranteed by including an additional extra-gradient step, while a p th -order superlinear local convergence is shown to hold if the VI is strongly monotone. The proposed ARE is inclusive and general, in the sense that a variety of solution methods can be formulated within this framework as different manifestations of approximations, and their iteration complexities would follow through in a unified fashion. The ARE framework relates to the first-order methods, while opening up possibilities to developing higher-order methods specifically for structured problems that guarantee the optimal iteration complexity bounds.

Abstract and Applied Analysis, 2014

We consider the problem of seeking a symmetric positive semidefinite matrix in a closed convex set to approximate a given matrix. This problem may arise in several areas of numerical linear algebra or come from finance industry or statistics and thus has many applications. For solving this class of matrix optimization problems, many methods have been proposed in the literature. The proximal alternating direction method is one of those methods which can be easily applied to solve these matrix optimization problems. Generally, the proximal parameters of the proximal alternating direction method are greater than zero. In this paper, we conclude that the restriction on the proximal parameters can be relaxed for solving this kind of matrix optimization problems. Numerical experiments also show that the proximal alternating direction method with the relaxed proximal parameters is convergent and generally has a better performance than the classical proximal alternating direction method.

Lecture Notes in Computational Science and Engineering, 2007

We review our recent results concerning optimal algorithms for numerical solution of both coercive and semi-coercive variational inequalities by combining dual-primal FETI algorithms with recent results for bound and equality constrained quadratic programming problems. The convergence bounds that guarantee the scalability of the algorithms are presented. These results are confirmed by numerical experiments.