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 Finite and Strong Convergence of a Proximal Method for Equilibrium Problems

We introduce the notions of conditioning and well-posedness for equilibrium problems. Using these concepts, we obtain finite and strong convergence results for the proximal method that improve, develop, and unify several theorems in optimization and nonlinear analysis.     

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.     

A note on finite termination of iterative algorithms in mathematical programming

Two well-known results on the unified treatment of finite termination of a class of algorithms for solving convex programming problems and for solving variational inequality problems are reconsidered. In particular, some of the underlying assumptions employed in the existing literature are shown to be redundant.     

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.     

Interior proximal extragradient method for equilibrium problems

The Bregman function-based Proximal Point Algorithm (BPPA) is an efficient tool for solving equilibrium problems and fixed-point problems. Extending rather classical proximal regularization methods, the main additional feature consists in an application of zone coercive regularizations. The latter allows to treat the generated subproblems as unconstrained ones, albeit with a certain precaution in numerical experiments. However, compared to the (classical) Proximal Point Algorithm for equilibrium problems, convergence results require additional assumptions which may be seen as the price to pay for unconstrained subproblems. Unfortunately, they are quite demanding – for instance, as they imply a sort of unique solvability of the given problem. The main purpose of this paper is to develop a modification of the BPPA, involving an additional extragradient step with adaptive (and explicitly given) stepsize. We prove that this extragradient step allows to leave out any of the additional assumptions mentioned above. Hence, though still of interior proximal type, the suggested method is applicable to an essentially larger class of equilibrium problems, especially including non-uniquely solvable ones.     

A proximal method with separable Bregman distances for quasiconvex minimization over the nonnegative orthant

We present an interior proximal method with Bregman distance, for solving the minimization problem with quasiconvex objective function under nonnegative constraints. The Bregman function is considered separable and zone coercive, and the zone is the interior of the positive orthant. Under the assumption that the solution set is nonempty and the objective function is continuously differentiable, we establish the well definedness of the sequence generated by our algorithm and obtain two important convergence results, and show in the main one that the sequence converges to a solution point of the problem when the regularization parameters go to zero.     

Strong Convergence of an Inexact Proximal Point Algorithm for Equilibrium Problems in Banach Spaces

We develop an inexact proximal point algorithm for solving equilibrium problems in Banach spaces which consists of two principal steps and admits an interesting geometric interpretation. At a certain iterate, first we solve an inexact regularized equilibrium problem with a flexible error criterion to obtain an axillary point. Using this axillary point and the inexact solution of the previous iterate, we construct two appropriate hyperplanes which separate the current iterate from the solution set of the given problem. Then the next iterate is defined as the Bregman projection of the initial point onto the intersection of two halfspaces obtained from the two constructed hyperplanes containing the solution set of the original problem. Assuming standard hypotheses, we present a convergence analysis for our algorithm, establishing that the generated sequence strongly and globally converges to a solution of the problem which is the closest one to the starting point of the algorithm.     

An interior point method with Bregman functions for the variational inequality problem with paramonotone operators

We present an algorithm for the variational inequality problem on convex sets with nonempty interior. The use of Bregman functions whose zone is the convex set allows for the generation of a sequence contained in the interior, without taking explicitly into account the constraints which define the convex set. We establish full convergence to a solution with minimal conditions upon the monotone operatorF, weaker than strong monotonicity or Lipschitz continuity, for instance, and including cases where the solution needs not be unique. We apply our algorithm to several relevant classes of convex sets, including orthants, boxes, polyhedra and balls, for which Bregman functions are presented which give rise to explicit iteration formulae, up to the determination of two scalar stepsizes, which can be found through finite search procedures. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.Corresponding author.     

Proximal-Point Algorithm Using a Linear Proximal Term

Proximal-point algorithms (PPAs) are classical solvers for convex optimization problems and monotone variational inequalities (VIs). The proximal term in existing PPAs usually is the gradient of a certain function. This paper presents a class of PPA-based methods for monotone VIs. For a given current point, a proximal point is obtained via solving a PPA-like subproblem whose proximal term is linear but may not be the gradient of any functions. The new iterate is updated via an additional slight calculation. Global convergence of the method is proved under the same mild assumptions as the original PPA. Finally, profiting from the less restrictions on the linear proximal terms, we propose some parallel splitting augmented Lagrangian methods for structured variational inequalities with separable operators. B.S. He was supported by NSFC Grant 10571083 and Jiangsu NSF Grant BK2008255.