In this paper, we introduce three new iterative methods for finding a common point of the set of fixed points of a symmetric generalized hybrid mapping and the set of solutions of an equilibrium problem in a real Hilbert space. Each method can be considered as an combination of Ishikawa's process with the proximal point algorithm, the extragradient algorithm with or without linesearch. Under certain conditions on parameters, the iteration sequences generated by the proposed methods are proved to be weakly convergent to a solution of the problem. These results extend the previous results given in the literature. A numerical example is also provided to illustrate the proposed algorithms.
In this paper we study some novel parallel and sequential hybrid methods for finding a common fixed point of a finite family of asymptotically quasi $ϕ$-nonexpansive mappings. The results presented here modify and extend some previous results obtained by several authors.
In this paper, a novel parallel hybrid iterative method is proposed for finding a common element of the set of solutions of a system of equilibrium problems, the set of solutions of variational inequalities for inverse strongly monotone mappings and the set of fixed points of a finite family of nonexpansive mappings in Hilbert space. Strong convergence theorem is proved for the sequence generated by the scheme. Finally, a parallel iterative algorithm for two finite families of variational inequalities and nonexpansive mappings is established.
In this paper we propose and analyze three parallel hybrid extragradient methods for finding a common element of the set of solutions of equilibrium problems involving pseudomonotone bifunctions and the set of fixed points of nonexpansive mappings in a real Hilbert space. Based on parallel computation we can reduce the overall computational effort under widely used conditions on the bifunctions and the nonexpansive mappings.A simple numerical example is given to illustrate the proposed parallel algorithms.
In this paper, we introduce a Bayesian abstract fuzzy economy model and we prove the Bayesian fuzzy equilibrium existence. As applications, we prove the existence of the solutions for two types of random quasi-variational inequalities with random fuzzy mappings and we also obtain random fixed point theorems.
In this paper, we propose new algorithms for finding a common point of the solution set of a pseudomonotone equilibrium problem and the set of fixed points of a symmetric generalized hybrid mapping in a real Hilbert space. The convergence of the iterates generated by each method is obtained under assumptions that the fixed point mapping is quasi-nonexpansive and demiclosed at $0$, and the bifunction associated with the equilibrium problem is weakly continuous. The bifunction is assumed to be satisfying a Lipschitz-type condition when the basic iteration comes from the extragradient method. It …
We consider the problem of finding a fixed point of a nonexpansive mapping, which is also a solution of a pseudo-monotone equilibrium problem, where the bifunction in the equilibrium problem is the sum of two ones. We propose a splitting algorithm combining the gradient method for equilibrium problem and the Mann iteration scheme for fixed points of nonexpansive mappings. At each iteration of the algorithm, two strongly convex subprograms are required to solve separately, one for each of the component bifunctions. Our main result states that, under paramonotonicity property of the given bifunc…
In this paper, we introduce some new hybrid algorithms for finding a solution of a system of equilibrium problems. In these algorithms, by constructing specially cutting-halfspaces, we avoid using the extra-steps as in the extragradient method and the Armijo linesearch method which are inherently costly when the feasible set has a complex structure. The strong convergence of the algorithms is established.
In this paper, we introduce a new hybrid algorithm for solving equilibrium problems. The algorithm combines the extragradient method and the hybrid (outer approximation) method. In this algorithm, only an optimization program is solved at each iteration without the extra-steps like as in the extragradient method and the Armijo linesearch method. A specially constructed half-space in the hybrid method is the reason for the absence of an optimization program in our algorithm. The strong convergence theorem is established and several numerical experiments are implemented to illustrate the converg…
The paper proposes a novel hybrid method for solving equilibrium problems and fixed point problems. By constructing specially cutting-halfspaces, in this algorithm, only an optimization program is solved at each iteration without the extra-steps as in some previously known methods. The strongly convergence theorem is established and some numerical examples are presented to illustrate its convergence.
In this paper, we propose two novel parallel hybrid methods for finding a common element of the set of solutions of a finite family of generalized equilibrium problems for monotone bifunctions $\left\{f_i\right\}_{i=1}^N$ and $α$ - inverse strongly monotone operators $\left\{A_i\right\}_{i=1}^N$ and the set of common fixed points of a finite family of (asymptotically) $κ$- strictly pseudocontractive mappings $\left\{S_j\right\}_{j=1}^M$ in Hilbert spaces. The strong convergence theorems are established under the standard assumptions imposed on equilibrium bifunctions and operators. A numerical…
In this paper, we consider some equilibrium problems (or saddle point problems), in which the domains of the considered mappings are limited at some regions. These restricted regions are defined by some mappings which are called the constrained mappings in the given problems. For this reason, we introduce the concepts of constrained ordered equilibrium problems, that are useful extensions of ordered equilibrium problems and ordinary equilibrium problems. Then, by using some fixed point theorems on posets, we prove several theorems for existence of solutions to some constrained ordered equilibr…
In this paper, we propose new extragradient algorithms for solving a split equilibrium and nonexpansive mapping SEPNM($C, Q, A, f, g, S, T)$ where $C, Q$ are nonempty closed convex subsets in real Hilbert spaces $\mathcal{H}_1, \mathcal{H}_2 $ respectively, $A : \mathcal{H}_1 \to \mathcal{H}_2$ is a bounded linear operator, $f$ is a pseudomonotone bifunction on $C$ and $g$ is a monotone bifunction on $Q$, $S, T$ are nonexpansive mappings on $C$ and $Q$ respectively. By using extragradient method combining with cutting techniques, we obtain algorithms for the problem. Under certain conditions o…
In this paper we propose two strongly convergent parallel hybrid iterative methods for finding a common element of the set of fixed points of a family of quasi $ϕ$-asymptotically nonexpansive mappings $\{F(S_j)\}_{j=1}^N$, the set of solutions of variational inequalities $\{VI(A_i,C)\}_{i=1}^M$ and the set of solutions of equilibrium problems $\{EP(f_k)\}_{k=1}^K$ in uniformly smooth and 2-uniformly convex Banach spaces. A numerical experiment is given to verify the efficiency of the proposed parallel algorithms.
We study distributed algorithms for seeking a Nash equilibrium in a class of non-cooperative convex games with strongly monotone mappings. Each player has access to her own smooth local cost function and can communicate to her neighbors in some undirected graph. To deal with fast distributed learning of Nash equilibria under such settings, we introduce a so called augmented game mapping and provide conditions under which this mapping is strongly monotone. We consider a distributed gradient play algorithm for determining a Nash equilibrium (GRANE). The algorithm involves every player performing…
Urban air mobility is a concept that promotes aerial modes of transport in urban areas. In these areas, the location and capacity of the vertiports--where the travelers embark and disembark the aircraft--not only affect the flight delays of the aircraft, but can also aggravate the congestion of ground vehicles by creating extra ground travel demands. We introduce a mathematical model for selecting the location and capacity of the vertiports that minimizes the traffic congestion in hybrid air-ground transportation networks. Our model is based on a mathematical program with bilinear equilibrium …
The main purpose of this paper is to find conditions for Holder calmness of the solution mapping, viewed as a function of the boundary data, of a hemivariational inequality governed by the Navier-Stokes operator. To this end, a more abstract model is studied first: a class of parametric equilibrium problems defined by trifunctions. The presence of trifunctions allows the extension of the monotonicity notions and of the duality principle in the theory of equilibrium problems.
In this paper, we propose two iterative methods for finding a common solution of a finite family of equilibrium problems for pseudomonotone bifunctions. The first is a parallel hybrid extragradient-cutting algorithm which is extended from the previously known one for variational inequalities to equilibrium problems. The second is a new cyclic hybrid extragradient-cutting algorithm. In the cyclic algorithm, using the known techniques, we can perform and develop practical numerical experiments.
Disaster management is a complex problem demanding sophisticated modeling approaches. We propose utilizing a hybrid method involving inverse optimization to parameterize the cost functions for a road network's traffic equilibrium problem and employing a modified version of Fan and Liu (2010)'s two-stage stochastic model to make protection decisions using the information gained from inverse optimization. We demonstrate the framework using two types of cost functions for the traffic equilibrium problem and show that accurate parameterizations of cost functions can change spending on protection d…
In this paper, we study the mathematical program with equilibrium constraints (MPEC) formulated as a mathematical program with a parametric generalized equation involving the regular normal cone. Compared with the usual way of formulating MPEC through a KKT condition, this formulation has the advantage that it does not involve extra multipliers as new variables, and it usually requires weaker assumptions on the problem data. Using the so-called first order sufficient condition for metric subregularity, we derive verifiable sufficient conditions for the metric subregularity of the involved set-…
In this paper, we establish the lower semicontinuity of the solution mapping and of the approximate solution mapping for parametric fixed point problems under some suitable conditions. As applications, the lower semicontinuity result applies to the parametric vector quasi-equilibrium problem, and allows to prove the existence of solutions for generalized Stackelberg games.
In this paper, we introduce two parallel extragradient-proximal methods for solving split equilibrium problems. The algorithms combine the extragradient method, the proximal method and the hybrid (outer approximation) method. The weak and strong convergence theorems for iterative sequences generated by the algorithms are established under widely used assumptions for equilibrium bifunctions.
We consider the stochastic generalized Nash equilibrium problem (SGNEP) with joint feasibility constraints and expected-value cost functions. We propose a distributed stochastic projected reflected gradient algorithm and show its almost sure convergence when the pseudogradient mapping is monotone and the solution is unique. The algorithm is based on monotone operator splitting methods tailored for SGNEPs when the expected-value pseudogradient mapping is approximated at each iteration via an increasing number of samples of the random variable. Finally, we show that a preconditioned variant of o…
We consider the stochastic generalized Nash equilibrium problem (SGNEP) with expected-value cost functions. Inspired by Yi and Pavel (Automatica, 2019), we propose a distributed GNE seeking algorithm based on the preconditioned forward-backward operator splitting for SGNEP, where, at each iteration, the expected value of the pseudogradient is approximated via a number of random samples. As main contribution, we show almost sure convergence of our proposed algorithm if the pseudogradient mapping is restricted (monotone and) cocoercive. For non-generalized SNEPs, we show almost sure convergence …
