Multivalued general equilibrium problems

In this paper, we use the auxiliary principle technique to suggest some new classes of iterative algorithms for solving multivalued equilibrium problems. The convergence of the proposed methods either requires partially relaxed strongly monotonicity or pseudomonotonicity. As special cases, we obtain a number of known and new results for solving various classes of equilibrium and variational inequality problems. Since multivalued equilibrium problems include equilibrium, variational inequality and complementarity problems as specials cases, our results continue to hold for these problems.     

On nonconvex equilibrium problems

In this paper, we introduce a new class of equilibrium problems, known as mixed quasi nonconvex equilibrium problems. We suggest some iterative schemes for solving nonconvex equilibrium problems by using the auxiliary principle technique. The convergence of the proposed methods either requires partially relaxed strongly monotonicity or pseudomonotonicity. As special cases, we obtain a number of known and new results for solving various classes of equilibrium and variational inequality problems.     

A class of new iterative methods for general mixed variational inequalities

In this paper, we use the auxiliary principle technique to suggest a class of predictor-corrector methods for solving general mixed variational inequalities. The convergence of the proposed methods only requires the partially relaxed strongly monotonicity of the operator, which is weaker than co-coercivity. As special cases, we obtain a number of known and new results for solving various classes of variational inequalities and related problems.     

Multivalued Regularized Equilibrium Problems

In this paper, we introduce a new class of equilibrium problems known as the multivalued regularized equilibrium problems. We use the auxiliary principle technique to suggest some iterative methods for solving multivalued regularized equilibrium problems. The convergence of the proposed methods is studied under some mild conditions. As special cases, we obtain a number of known and new results for solving various classes of regularized equilibrium problems and related optimization problems.     

On equilibrium-like problems

In this article we introduce a new class of equilibrium problems known as mixed quasi invex equilibrium (equilibrium-like) problems with trifunction. This class of equilibrium problems includes invex equilibrium problems, variational inequalities and variational-like inequalities as special cases. We use the auxiliary principle technique to suggest and analyze some iterative schemes for solving invex equilibrium problems and study the convergence criteria of these methods under mild conditions. Our results represent significant and important refinements of the previously known results.     

Invex equilibrium problems

In this paper, we introduce a new class of equilibrium problems, known as invex equilibrium problems in the setting of invexity. This class of equilibrium problems includes equilibrium problems, variational inequalities and variational-like inequalities as special cases. We use the auxiliary principle technique to suggest and analyze some iterative schemes for solving invex equilibrium problems and study the convergence criteria of these methods under some mild conditions. We also consider the concept of well-posedness for invex equilibrium problems. Our results represent significant and important refinements of the previously known results.     

Projection-proximal methods for general variational inequalities

In this paper, we consider and analyze some new projection-proximal methods for solving general variational inequalities. The modified methods converge for pseudomonotone operators which is a weaker condition than monotonicity. The proposed methods include several new and known methods as special cases. Our results can be considered as a novel and important extension of the previously known results. Since the general variational inequalities include the quasi-variational inequalities and implicit complementarity problems as special cases, results proved in this paper continue to hold for these problems.     

Splitting Algorithms for General Pseudomonotone Mixed Variational Inequalities

In this paper, we suggest and analyze a number of resolvent-splitting algorithms for solving general mixed variational inequalities by using the updating technique of the solution. The convergence of these new methods requires either monotonicity or pseudomonotonicity of the operator. Proof of convergence is very simple. Our new methods differ from the existing splitting methods for solving variational inequalities and complementarity problems. The new results are versatile and are easy to implement.     

New extragradient-type methods for general variational inequalities

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.     

Improved Noniterative Algorithm for Solving the Traffic Equilibrium Problem

We provide an equivalent formulation of a previously proposed noniterative algorithm (see A. Maugeri, Appl. Math. Optim. 16, 169–185, 1987) for the traffic equilibrium problem. Moreover, under the strict monotonicity assumption, we provide an improved algorithm which enlarges the range of applicability of the previous algorithm and decreases considerably its computational effort. Our algorithm is based on a general algorithm for variational inequalities (see O. Mancino, G. Stampacchia, J. Optim. Theory Appl. 9, 3–23, 1972), which we further develop and adapt to the traffic equilibrium problem. Both our proofs and the algorithm exploit directly the equilibrium conditions which characterize our problem.     

Auxiliary Problem Principle and Proximal Point Methods

An extension of the auxiliary problem principle to variational inequalities with non-symmetric multi-valued operators in Hilbert spaces is studied. This extension concerns the case that the operator is split into the sum of a single-valued operator , possessing a kind of pseudo Dunn property, and a maximal monotone operator . The current auxiliary problem is k constructed by fixing at the previous iterate, whereas (or its single-valued approximation ^k) k is considered at a variable point. Using auxiliary operators of the form ^k+ , with _k>0, the standard for the auxiliary problem principle assumption of the strong convexity of the function h can be weakened exploiting mutual properties of and h. Convergence of the general scheme is analyzed and some applications are sketched briefly.     

An iterative scheme for variational inequalities

In this paper we introduce and study a general iterative scheme for the numerical solution of finite dimensional variational inequalities. This iterative scheme not only contains, as special cases the projection, linear approximation and relaxation methods but also induces new algorithms. Then, we show that under appropriate assumptions the proposed iterative scheme converges by establishing contraction estimates involving a sequence of norms in Eⁿ induced by symmetric positive definite matrices G_m. Thus, in contrast to the above mentioned methods, this technique allows the possibility of adjusting the norm at each step of the algorithm. This flexibility will generally yield convergence under weaker assumptions.     

A general descent framework for the monotone variational inequality problem

We present a framework for descent algorithms that solve the monotone variational inequality problem VIP _v which consists in finding a solutionv ^* _v satisfyings(v ^*)^T(v–v ^*)0, for allv _v. This unified framework includes, as special cases, some well known iterative methods and equivalent optimization formulations. A descent method is developed for an equivalent general optimization formulation and a proof of its convergence is given. Based on this unified logarithmic framework, we show that a variant of the descent method where each subproblem is only solved approximately is globally convergent under certain conditions.This research was supported in part by individual operating grants from NSERC.     

Iterative methods for variational and complementarity problems

In this paper, we study both the local and global convergence of various iterative methods for solving the variational inequality and the nonlinear complementarity problems. Included among such methods are the Newton and several successive overrelaxation algorithms. For the most part, the study is concerned with the family of linear approximation methods. These are iterative methods in which a sequence of vectors is generated by solving certain linearized subproblems. Convergence to a solution of the given variational or complementarity problem is established by using three different yet related approaches. The paper also studies a special class of variational inequality problems arising from such applications as computing traffic and economic spatial equilibria. Finally, several convergence results are obtained for some nonlinear approximation methods.This research was based on work supported by the National Science Foundation under grant ECS-7926320.     

On the convergence of generalized Schwarz algorithms for solving obstacle problems with elliptic operators

In this paper, multiplicative and additive generalized Schwarz algorithms for solving obstacle problems with elliptic operators are developed and analyzed. Compared with the classical Schwarz algorithms, in which the subproblems are coupled by the Dirichlet boundary conditions, the generalized Schwarz algorithms use Robin conditions with parameters as the transmission conditions on the interface boundaries. As a result, the convergence rate can be speeded up by choosing Robin parameters properly. Convergence of the algorithms is established. This work was supported by 973 national project of China (2004CB719402) and by national nature science foundation of China (10671060).     

General variational inequalities and nonexpansive mappings

In this paper, we suggest and analyze some three-step iterative schemes for finding the common elements of the set of the solutions of the Noor variational inequalities involving two nonlinear operators and the set of the fixed points of nonexpansive mappings. We also consider the convergence analysis of the suggested iterative schemes under some mild conditions. Since the Noor variational inequalities include variational inequalities and complementarity problems as special cases, results obtained in this paper continue to hold for these problems. Results obtained in this paper may be viewed as an refinement and improvement of the previously known results.     

Asymmetric variational inequality problems over product sets: Applications and iterative methods

In this paper, we (i) describe how several equilibrium problems can be uniformly modelled by a finite-dimensional asymmetric variational inequality defined over a Cartesian product of sets, and (ii) investigate the local and global convergence of various iterative methods for solving such a variational inequality problem. Because of the special Cartesian product structure, these iterative methods decompose the original variational inequality problem into a sequence of simpler variational inequality subproblems in lower dimensions. The resulting decomposition schemes often have a natural interpretation as some adjustment processes. This research was based on work supported by the National Science Foundation under grant ECS 811–4571.     

Extragradient algorithms extended to equilibrium problems¶

We make use of the auxiliary problem principle to develop iterative algorithms for solving equilibrium problems. The first one is an extension of the extragradient algorithm to equilibrium problems. In this algorithm the equilibrium bifunction is not required to satisfy any monotonicity property, but it must satisfy a certain Lipschitz-type condition. To avoid this requirement we propose linesearch procedures commonly used in variational inequalities to obtain projection-type algorithms for solving equilibrium problems. Applications to mixed variational inequalities are discussed. A special class of equilibrium problems is investigated and some preliminary computational results are reported.     

Vector-valued perturbed equilibrium problems

The goal of this paper is to prove some general vector-valued perturbed equilibrium principles and some existence results of vector equilibrium points for bifunctions satisfying a new natural notion of lower semi-continuity. We obtain these results by going through a new concept of approximative equilibrium point.     

QPCOMP: A quadratic programming based solver for mixed complementarity problems

QPCOMP is an extremely robust algorithm for solving mixed nonlinear complementarity problems that has fast local convergence behavior. Based in part on the NE/SQP method of Pang and Gabriel [14], this algorithm represents a significant advance in robustness at no cost in efficiency. In particular, the algorithm is shown to solve any solvable Lipschitz continuous, continuously differentiable, pseudo-monotone mixed nonlinear complementarity problem. QPCOMP also extends the NE/SQP method for the nonlinear complementarity problem to the more general mixed nonlinear complementarity problem. Computational results are provided, which demonstrate the effectiveness of the algorithm. This material is based on research supported by National Science Foundation Grant CCR-9157632, Department of Energy Grant DE-FG03-94ER61915, and the Air Force Office of Scientific Research Grant F49620-94-1-0036.