A continuation method for monotone variational inequalities

This paper presents a continuation method for monotone variational inequality problems based on a new smooth equation formulation. The existence, uniqueness and limiting behavior of the path generated by the method are analyzed.This work was supported by the National Science Foundation Presidential Young Investigator Award ECE-8552773 and by a grant from the Burlington Northern Railroad.     

A combined relaxation method for variational inequalities with nonlinear constraints

A simple iterative method for solving variational inequalities with a set-valued, nonmonotone mapping and a convex feasible set is proposed. This set can be defined by nonlinear functions. The method is based on combining and extending ideas contained in various relaxation methods of nonsmooth optimization. Also a modification of the averaging method for the problem under consideration is proposed. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.This research was supported in part by RFFI grant No. 95-01-00061.     

A variational inequality formulation for unconfined seepage problems in porous media

In the existing variational inequality formulations for the unconfined seepage problem in porous media, the seepage point, namely the exit point of the free surface, is a singular point and how to locate the seepage point exactly has been an open issue. By generalizing Darcy’s law applied solely to the saturated zone in an earth dam to the entire dam including the no-flow zone, a new variational inequality formulation is presented. The new formulation imposes a boundary condition of Signorini’s type on the potential seepage boundary and the seepage point turns out to be such a point that makes both inequalities in Signorini’s complementary condition become equalities. Singularity of the seepage point is accordingly eliminated. A strategy is developed for overcoming the mesh-dependency in the finite element implementation.     

Local analysis of Newton-type methods for variational inequalities and nonlinear programming

This paper presents some new results in the theory of Newton-type methods for variational inequalities, and their application to nonlinear programming. A condition of semistability is shown to ensure the quadratic convergence of Newton's method and the superlinear convergence of some quasi-Newton algorithms, provided the sequence defined by the algorithm exists and converges. A partial extension of these results to nonsmooth functions is given. The second part of the paper considers some particular variational inequalities with unknowns (x, ), generalizing optimality systems. Here only the question of superlinear convergence of {x ^k } is considered. Some necessary or sufficient conditions are given. Applied to some quasi-Newton algorithms they allow us to obtain the superlinear convergence of {x ^k }. Application of the previous results to nonlinear programming allows us to strengthen the known results, the main point being a characterization of the superlinear convergence of {x ^k } assuming a weak second-order condition without strict complementarity.     

A Gauss-Seidel iteration method for nonlinear variational inequality problems over rectangles

This paper presents a simple yet practically useful Gauss-Seidel iterative method for solving a class of nonlinear variational inequality problems over rectangles and of nonlinear complementarity problems. This scheme is a nonlinear generalization of a robust iterative method for linear complementarity problems developed by Mangasarian. Global convergence is presented for problems with Z-functions. It is noted that the suggested method can be viewed as a specific case of a class of linear approximation methods studied by Pang and others.     

A B-differentiable equation-based,globally and locally quadratically convergent algorithm for nonlinear programs,complementarity and variational inequality problems

This paper presents a globally convergent, locally quadratically convergent algorithm for solving general nonlinear programs, nonlinear complementarity and variational inequality problems. The algorithm is based on a unified formulation of these three mathematical programming problems as a certain system of B-differentiable equations, and is a modification of the damped Newton method described in Pang (1990) for solving such systems of nonsmooth equations. The algorithm resembles several existing methods for solving these classes of mathematical programs, but has some special features of its own; in particular, it possesses the combined advantage of fast quadratic rate of convergence of a basic Newton method and the desirable global convergence induced by one-dimensional Armijo line searches. In the context of a nonlinear program, the algorithm is of the sequential quadratic programming type with two distinct characteristics: (i) it makes no use of a penalty function; and (ii) it circumvents the Maratos effect. In the context of the variational inequality/complementarity problem, the algorithm provides a Newton-type descent method that is guaranteed globally convergent without requiring the F-differentiability assumption of the defining B-differentiable equations.This work was based on research supported by the National Science Foundation under Grant No. ECS-8717968.     

A variational inequality approach for the determination of oligopolistic market equilibrium

This paper presents an alternative approach to that by Murphy, Sherali and Soyster [13] for computing market equilibria with mathematical programming methods. This approach is based upon a variational inequality representation of the problem and the use of a diagonalization/relaxation algorithm.     

A new method for a class of nonlinear variational inequalities with fuzzy mappings

In this paper, we construct a new iterative algorithm of solution for a new class of nonlinear variational inequalities with fuzzy mappings and give some convergence analysis of iterative sequences generated by algorithm.     

A robust secant method for optimization problems with inequality constraints

This paper presents a secant method, based on R. B. Wilson's formula for the solution of optimization problems with inequality constraints. Global convergence properties are ensured by grafting the secant method onto a phase I - phase II feasible directions method, using a rate of convergence test for crossover control.This research was sponsored by the National Science Foundation, Grant No. ENG-73-08214 and Grant No. (RANN)-ENV-76-04264, and by the Joint Services Electronics Program. Contract No. F44620-76-C-0100.     

A new hybrid iterative method for mixed equilibrium problems and variational inequality problem for relaxed cocoercive mappings with application to optimization problems

In this paper, we introduce and study a new hybrid iterative method for finding a common element of the set of solutions of a mixed equilibrium problem, the set of fixed points of an infinite family of nonexpansive mappings and the set of solutions of variational inequalities for a ξ-Lipschitz continuous and relaxed (m,v)-cocoercive mappings in Hilbert spaces. Then, we prove a strong convergence theorem of the iterative sequence generated by the proposed iterative algorithm which solves some optimization problems under some suitable conditions. Our results extend and improve the recent results of Yao et al. [Y. Yao, M.A. Noor, S. Zainab and Y.C. Liou, Mixed equilibrium problems and optimization problems, J. Math. Anal. Appl (2009). doi:10.1016/j.jmaa.2008.12.005] and Gao and Guo [X. Gao and Y. Guo, Strong convergence theorem of a modified iterative algorithm for Mixed equilibrium problems in Hilbert spaces, J. Inequal. Appl. (2008). doi:10.1155/2008/454181] and many others.     

A hybrid extragradient method for approximating the common solutions of a variational inequality, a system of variational inequalities, a mixed equilibrium problem and a fixed point problem

In this paper, we give a hybrid extragradient iterative method for finding the approximate element of the common set of solutions of a generalized equilibrium problem, a system of variational inequality problems, a variational inequality problem and a fixed point problem for a strictly pseudocontractive mapping in a real Hilbert space. Further we establish a strong convergence theorem based on this method. The results presented in this paper improves and generalizes the results given in Yao et al. [36] and Ceng et al. [7], and some known corresponding results in the literature.     

Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory,algorithms and applications

Over the past decade, the field of finite-dimensional variational inequality and complementarity problems has seen a rapid development in its theory of existence, uniqueness and sensitivity of solution(s), in the theory of algorithms, and in the application of these techniques to transportation planning, regional science, socio-economic analysis, energy modeling, and game theory. This paper provides a state-of-the-art review of these developments as well as a summary of some open research topics in this growing field.The research of this author was supported by the National Science Foundation Presidential Young Investigator Award ECE-8552773 and by the AT&T Program in Telecommunications Technology at the University of Pennsylvania.The research of this author was supported by the National Science Foundation under grant ECS-8644098.     

A unified framework for linear control problems with state-variable inequality constraints

This paper briefly reviews the literature on necessary optimality conditions for optimal control problems with state-variable inequality constraints. Then, it attempts to unify the treatment of linear optimal control problems with state-variable inequality constraints in the framework of continuous linear programming. The duality theory in this framework makes it possible to relate the adjoint variables arising in different formulations of a problem; these relationships are illustrated by the use of a simple example. This framework also allows more general problems and admits a simplex-like algorithm to solve these problems.This research was partially supported by Grant No. A4619 from the National Research Council of Canada to the first author. The first author also acknowledges the support provided by the Brookhaven National Laboratory, where he conducted his research.     

Application of regularization technique to variational adjoint method: A case for nonlinear convection-diffusion problem

A discrete assimilation system for a one-dimensional variable coefficient convection-diffusion equation is constructed. The variational adjoint method combined with the regularization technique is employed to retrieve the initial condition and diffusion coefficient with the aid of a set of simulated observations. Several numerical experiments are performed: (a) retrieving both the initial condition and diffusion coefficient jointly (Experiment JR), (b) retrieving either of them separately (Experiment SR), (c) retrieving only the diffusion coefficient with the iteration count increased to 800 (Experiment NoR-SR), and (d) retrieving only the diffusion coefficient with the consideration of a regularization term based on the Experiment NoR-SR (Experiment AdR-SR). The results indicate that within the limit of 100 iterations, the retrieval quality of the Experiment SR is better than those from the Experiment JR. Compared with the initial condition, the diffusion coefficient is a little difficult to retrieve, whereas we still achieve the desired result by increasing the iterations or integrating the regularization term into the cost functional for the improvement with respect to the diffusion coefficient. Further comparisons between the Experiment NoR-SR and AdR-SR show that the regularization term can really help not only improve the precision of retrieval to a large extent, but also speed up the convergence of solution, even if some perturbations are imposed on those observations.     

Convergence theorems of shrinking projection methods for equilibrium problem, variational inequality problem and a finite family of relatively quasi-nonexpansive mappings

We introduce a W-mapping for a finite family of relatively quasi-nonexpansive mappings and construct an iterative scheme for finding a common element of the solution set of equilibrium problem, the solution set of the variational inequality problem for an inverse-strongly-monotone operator and set of common fixed points of a finite family of relatively quasi-nonexpansive mappings. Strong convergence theorems are presented in a 2-uniformly convex and uniformly smooth Banach space. Our results generalize and extend relative results.     

A simplicial decomposition algorithm for solving the variational inequality formulation of the general traffic assignment problem for large scale networks

Summary The class of simplicial decomposition methods has been shown to constitute efficient tools for the solution of the variational inequality formulation of the general traffic assignment problem. This paper presents a particular implementation of such an algorithm, with emphasis on its ability to solve large scale problems efficiently. The convergence of the algorithm is monitored by the primal gap function, which arises naturally in simplicial decomposition schemes. The gap function also serves as an instrument for maintaining a reasonable subproblem size, through its use in column dropping criteria. The small dimension and special structure of the subproblems also allows for the use of very efficient algorithms; several algorithms in the class of linearization methods are presented. When restricting the number of retained extremal flows in a simplicial decomposition scheme, the number of major iterations tends to increase. For large networks the shortest path calculations, leading to new extremal flow generation, require a large amount of the total computation time. A special study is therefore made in order to choose the most efficient extremal flow generation technique. Computational results on symmetric problems are presented for networks of some large cities, and on asymmetric problems for some of the networks used in the literature. Computational results for bimodal models of some large cities leading to asymmetric problems are also discussed.     

A viscosity of extragradient approximation method for finding equilibrium problems, variational inequalities and fixed point problems for nonexpansive mappings

In this paper, we investigate the problem for finding the set of solutions for equilibrium problems, the set of solutions of the variational inequalities for k-Lipschitz continuous mappings and fixed point problems for nonexpansive mappings in a Hilbert space. We introduce a new viscosity extragradient approximation method which is based on the so-called viscosity approximation method and extragradient method. We show that the sequence converges strongly to a common element of the above three sets under some parameters controlling conditions. Finally, we utilize our results to study some convergence problems for finding the zeros of maximal monotone operators. Our results are generalization and extension of the results of Kumam [P. Kumam, Strong convergence theorems by an extragradient method for solving variational inequalities and equilibrium problems in a Hilbert space, Turk. J. Math. 33 (2009) 85–98], Wangkeeree [R. Wangkeeree, An extragradient approximation method for equilibrium problems and fixed point problems of a countable family of nonexpansive mappings, Fixed Point Theory and Applications, 2008, Article ID 134148, 17 pages, doi:10.1155/2008/134148], Yao et al. [Y. Yao, Y.C. Liou, R. Chen, A general iterative method for an finite family of nonexpansive mappings, Nonlinear Analysis 69 (5–6) (2008) 1644–1654], Qin et al. [X. Qin, M. Shang, Y. Su, A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces, Nonlinear Analysis (69) (2008) 3897–3909], and many others.     

Vector network equilibrium problems with capacity constraints of arcs and nonlinear scalarization methods

In this paper, a vector network equilibrium problem with capacity constraints of arcs is studied. By virtue of Fan-Browder’ fixed point theorem, an existence result of a (weak) vector equilibrium flow is derived. By using nonlinear scalarization methods, some sufficient and necessary conditions for a weak vector equilibrium flow are obtained.