Modified basic projection methods for a class of equilibrium problems

Projection methods are a popular class of methods for solving equilibrium problems. In this paper, we propose approximate one projection methods for solving a class of equilibrium problems, where the cost bifunctions are paramonotone, the feasible sets are defined by a continuous convex function inequality and not necessarily differentiable in the Euclidean space \(\mathcal R^{s}\). At each main iteration step in our algorithms, the usual projections onto the feasible set are replaced by computing inexact subgradients and one projection onto the intersection of two halfspaces containing the solution set of the equilibrium problems. Then, by choosing suitable parameters, we prove convergence of the whole generated sequence to a solution of the problems, under only the assumptions of continuity and paramonotonicity of the bifunctions. Finally, we present some computational examples to illustrate the assumptions of the proposed algorithms.     

Extragradient and linesearch methods for solving split feasibility problems in Hilbert spaces

Utilizing the Tikhonov regularization method and extragradient and linesearch methods, some new extragradient and linesearch algorithms have been introduced in the framework of Hilbert spaces. In the presented algorithms, the convexity of optimization subproblems is assumed, which is weaker than the strong convexity assumption that is usually supposed in the literature, and also, the auxiliary equilibrium problem is not used. Some strong convergence theorems for the sequences generated by these algorithms have been proven. It has been shown that the limit point of the generated sequences is a common element of the solution set of an equilibrium problem and the solution set of a split feasibility problem in Hilbert spaces. To illustrate the usability of our results, some numerical examples are given. Optimization subproblems in these examples have been solved by FMINCON toolbox in MATLAB.     

A Two-Phase Algorithm for a Variational Inequality Formulation of Equilibrium Problems

We introduce an explicit algorithm for solving nonsmooth equilibrium problems in finite-dimensional spaces. A particular iteration proceeds in two phases. In the first phase, an orthogonal projection onto the feasible set is replaced by projections onto suitable hyperplanes. In the second phase, a projected subgradient type iteration is replaced by a specific projection onto a halfspace. We prove, under suitable assumptions, convergence of the whole generated sequence to a solution of the problem. The proposed algorithm has a low computational cost per iteration and, some numerical results are reported.     

An explicit algorithm for monotone variational inequalities

We introduce a fully explicit method for solving monotone variational inequalities in Hilbert spaces, where orthogonal projections onto the feasible set are replaced by projections onto suitable hyperplanes. We prove weak convergence of the whole generated sequence to a solution of the problem, under only the assumptions of continuity and monotonicity of the operator and existence of solutions.     

Iterative algorithms for solving fixed point problems and variational inequalities with uniformly continuous monotone operators

Using the double projection and Halpern methods, we prove two strong convergence results for finding a solution of a variational inequality problem involving uniformly continuous monotone operator which is also a fixed point of a quasi-nonexpansive mapping in a real Hilbert space. In our proposed methods, only two projections onto the feasible set in each iteration are performed, rather than one projection for each tentative step during the Armijo-type search, which represents a considerable saving especially when the projection is computationally expensive. We also give some numerical results which show that our proposed algorithms are efficient and implementable from the numerical point of view.     

New extragradient methods for solving variational inequality problems and fixed point problems

In this paper, we introduce two new iterative algorithms for finding a common element of the set of fixed points of a quasi-nonexpansive mapping and the set of solutions of the variational inequality problem with a monotone and Lipschitz continuous mapping in real Hilbert spaces, by combining a modified Tseng’s extragradient scheme with the Mann approximation method. We prove weak and strong convergence theorems for the sequences generated by these iterative algorithms. The main advantages of our algorithms are that the construction of solution approximations and the proof of convergence of the algorithms are performed without the prior knowledge of the Lipschitz constant of cost operators. Finally, we provide numerical experiments to show the efficiency and advantage of the proposed algorithms.     

A new shrinking gradient-like projection method for equilibrium problems

The paper proposes a new shrinking gradient-like projection method for solving equilibrium problems. The algorithm combines the generalized gradient-like projection method with the monotone hybrid method. Only one optimization program is solved onto the feasible set at each iteration in our algorithm without any extra-step dealing with the feasible set. The absence of an optimization problem in the algorithm is explained by constructing slightly different cutting-halfspace in the monotone hybrid method. Theorem of strong convergence is established under standard assumptions imposed on equilibrium bifunctions. An application of the proposed algorithm to multivalued variational inequality problems (MVIP) is presented. Finally, another algorithm is introduced for MVIPs in which we only use a value of main operator at the current approximation to construct the next approximation. Some preliminary numerical experiments are implemented to illustrate the convergence and computational performance of our algorithms over others.     

Hybrid Proximal-Type and Hybrid Shrinking Projection Algorithms for Equilibrium Problems,Maximal Monotone Operators,and Relatively Nonexpansive Mappings

In this article, we introduce two hybrid proximal-type algorithms and two hybrid shrinking projection algorithms by using the hybrid proximal-type method and the hybrid shrinking projection method, respectively, for finding a common element of the set of solutions of an equilibrium problem, the set of fixed points of a relatively nonexpansive mapping, and the set of solutions to the equation 0 ∈ Tx for a maximal monotone operator T defined on a uniformly smooth and uniformly convex Banach space. The strong convergence of the sequences generated by the proposed algorithms is established. Our results improve and generalize several known results in the literature.     

Shooting method for solving equilibrium programming problems

A new iterative method is proposed for solving equilibrium programming problems. The sequence of points it generates is proved to converge weakly to the solution set of the equilibrium problem under study. If the initial point has at least one projection onto the solution set of the equilibrium problem, the sequence generated by the method is shown to converge strongly to the set of these projections. The partial gradient of the initial data is assumed to be invertible and strictly monotone, which differs from the classical skew-symmetry condition.     

Existence and algorithm of solutions for mixed equilibrium problems and bilevel mixed equilibrium problems in Banach spaces

Some classes of mixed equilibrium problems and bilevel mixed equilibrium problems are introduced and studied in reflexive Banach spaces. First, by using a minimax inequality, some new existence results of solutions and the behavior of solution set for the mixed equilibrium problems and the bilevel mixed equilibrium problems are proved under suitable assumptions without the coercive conditions. Next, by using auxiliary principle technique, some new iterative algorithms for solving the mixed equilibrium problems and the bilevel mixed equilibrium problems are suggested and analyzed. The strong convergence of the iterative sequences generated by the proposed algorithms is proved under suitable assumptions without the coercive conditions. These results are new and generalize some recent results in this field.     

Numerical solutions to large-scale differential Lyapunov matrix equations

In this paper, we introduce an extension of multiple set split variational inequality problem (Censor et al. Numer. Algor. 59, 301–323 2012) to multiple set split equilibrium problem (MSSEP) and propose two new parallel extragradient algorithms for solving MSSEP when the equilibrium bifunctions are Lipschitz-type continuous and pseudo-monotone with respect to their solution sets. By using extragradient method combining with cutting techniques, we obtain algorithms for these problems without using any product space. Under certain conditions on parameters, the iteration sequences generated by the proposed algorithms are proved to be weakly and strongly convergent to a solution of MSSEP. An application to multiple set split variational inequality problems and a numerical example and preliminary computational results are also provided.     

A Projected Subgradient Algorithm for Bilevel Equilibrium Problems and Applications

In this paper, we propose a new algorithm for solving a bilevel equilibrium problem in a real Hilbert space. In contrast to most other projection-type algorithms, which require to solve subproblems at each iteration, the subgradient method proposed in this paper requires only to calculate, at each iteration, two subgradients of convex functions and one projection onto a convex set. Hence, our algorithm has a low computational cost. We prove a strong convergence theorem for the proposed algorithm and apply it for solving the equilibrium problem over the fixed point set of a nonexpansive mapping. Some numerical experiments and comparisons are given to illustrate our results. Also, an application to Nash–Cournot equilibrium models of a semioligopolistic market is presented.     

Auxiliary Principle and Algorithm for Mixed Equilibrium Problems and Bilevel Mixed Equilibrium Problems in Banach Spaces

A new class of bilevel mixed equilibrium problems is introduced and studied in real Banach spaces. By using the auxiliary principle technique, new iterative algorithms for solving the mixed equilibrium problems and bilevel mixed equilibrium problems are suggested and analyzed. Strong convergence of the iterative sequences generated by the algorithms is proved under suitable conditions. The behavior of the solution set of the bilevel mixed equilibrium problem is also discussed.     

Iterative Algorithms for Mixed Equilibrium Problems,Strict Pseudocontractions and Monotone Mappings

In this paper, we introduce some iterative algorithms for finding a common element of the set of solutions of a mixed equilibrium problem, the set of fixed points of a strict pseudocontraction and the set of solutions of a variational inequality for a monotone, Lipschitz continuous mapping. We obtain both weak and strong convergence theorems for the sequences generated by these processes in Hilbert spaces.     

On extragradient-viscosity methods for solving equilibrium and fixed point problems in a Hilbert space

Abstract

In this paper, new numerical algorithms are introduced for finding the solution of a variational inequality problem whose constraint set is the common elements of the set of fixed points of a demicontractive mapping and the set of solutions of an equilibrium problem for a monotone mapping in a real Hilbert space. The strong convergence of the iterates generated by these algorithms is obtained by combining a viscosity approximation method with an extragradient method. First, this is done when the basic iteration comes directly from the extragradient method, under a Lipschitz-type condition on the equilibrium function. Then, it is shown that this rather strong condition can be omitted when an Armijo-backtracking linesearch is incorporated into the extragradient iteration. The particular case of variational inequality problems is also examined.     

A new class of bilevel generalized mixed equilibrium problems in Banach spaces

A new class of bilevel generalized mixed equilibrium problems involving set-valued mappings is introduced and studied in a real Banach space. By using the auxiliary principle technique, new iterative algorithms for solving the generalized mixed equilibrium problems and bilevel generalized mixed equilibrium problems involving set-valued mappings are suggested and analyzed. Existence of solutions and strong convergence of the iterative sequences generated by the algorithms are proved under quite mild conditions. The behavior of the solution set of the generalized mixed equilibrium problems and bilevel generalized mixed equilibrium problems is also discussed. These results are new and generalize some recent results in this field.     

Hybrid pseudoviscosity approximation schemes for equilibrium problems and fixed point problems of infinitely many nonexpansive mappings

In this paper, we introduce hybrid pseudoviscosity approximation schemes with strongly positive bounded linear operators for finding a common element of the set of solutions of an equilibrium problem and the set of common fixed points of infinitely many nonexpansive mappings in the setting of Hilbert spaces. We prove the strong convergence of the sequences generated by our scheme to a solution of an equilibrium problem which is also a common fixed point of infinitely many nonexpansive mappings. Our results can be treated as extension and improvement of the corresponding results appeared in the literature in the recent past.     

Charged ball method for solving some computational geometry problems

The concept of replacement of the initial stationary optimization problem with some nonstationary mechanical system tending with time to the position of equilibrium, which coincides with the solution of the initial problem, makes it possible to construct effective numerical algorithms. First, differential equations of the movement should be derived. Then we pass to the difference scheme and define the iteration algorithm. There is a wide class of optimization methods constructed in such a way. One of the most known representatives of this class is the heavy ball method. As a rule, such type of algorithms includes parameters that highly affect the convergence rate. In this paper, the charged ball method, belonging to this class, is proposed and investigated. It is a new effective optimization method that allows solving some computational geometry problems. A problem of orthogonal projection of a point onto a convex closed set with a smooth boundary and the problem of finding the minimum distance between two such sets are considered in detail. The convergence theorems are proved, and the estimates for the convergence rate are found. Numerical examples illustrating the work of the proposed algorithms are given.     

Duality-based algorithms for total-variation-regularized image restoration

Image restoration models based on total variation (TV) have become popular since their introduction by Rudin, Osher, and Fatemi (ROF) in 1992. The dual formulation of this model has a quadratic objective with separable constraints, making projections onto the feasible set easy to compute. This paper proposes application of gradient projection (GP) algorithms to the dual formulation. We test variants of GP with different step length selection and line search strategies, including techniques based on the Barzilai-Borwein method. Global convergence can in some cases be proved by appealing to existing theory. We also propose a sequential quadratic programming (SQP) approach that takes account of the curvature of the boundary of the dual feasible set. Computational experiments show that the proposed approaches perform well in a wide range of applications and that some are significantly faster than previously proposed methods, particularly when only modest accuracy in the solution is required.     

A proximal point algorithm for generalized fractional programs

In this paper, the problem of solving generalized fractional programs will be addressed. This problem has been extensively studied and several algorithms have been proposed. In this work, we propose an algorithm that combines the proximal point method with a continuous min–max formulation of discrete generalized fractional programs. The proposed method can handle non-differentiable convex problems with possibly unbounded feasible constraints set, and solves at each iteration a convex program with unique dual solution. It generates two sequences that approximate the optimal value of the considered problem from below and from above at each step. For a class of functions, including the linear case, the convergence rate is at least linear.