Equilibrium Bifunction Research Articles

Convergence of inertial prox-penalization and inertial forward–backward algorithms for solving monotone bilevel equilibrium problems

The main focus of this paper is on bilevel optimization on Hilbert spaces involving two monotone equilibrium bifunctions. We present a new achievement consisting on the introduction of inertial methods for solving these types of problems. Indeed, two several inertial type methods are suggested: a proximal algorithm and a forward–backward one. Under suitable conditions and without any restrictive assumption on the trajectories, the weak and strong convergence of the sequence generated by the both iterative methods are established. Two particular cases illustrating the proposed methods are thereafter discussed with respect to hierarchical minimization problems and equilibrium problems under a saddle point constraint. Furthermore, numerical examples are given to demonstrate the implementability of our algorithms. The algorithms and their convergence results improve and develop previous results in the field.

New criteria for existence of solutions for equilibrium problems

Equilibrium problems provide a mathematical framework which includes optimization, variational inequalities, fixed point and saddle point problems, and noncooperative games as particular cases. In this paper sufficient conditions for the existence of solutions of an equilibrium problem are given by weakening the assumption of quasiconvexity of the involved equilibrium bifunction. The existence of solutions is established both in presence of compactness of the feasible set as well with a coercivity assumption. The results are obtained in an infinite dimensional setting, and they are based on the so called finite solvability property which is weaker than the recently introduced finite intersection property and in turn, weaker than most common cyclic and proper quasimonotonicity. Some examples are presented to illustrate the various cases in which other existence results for equilibrium problems do not apply. Finally, applications to the solution of quasiequilibrium problems, quasioptimization problems and generalized quasivariational inequalities are discussed.

Fast inertial extragradient algorithms for solving non-Lipschitzian equilibrium problems without monotonicity condition in real Hilbert spaces

Combining inertial-type methods with projection extragradient methods, we propose three inertial extragradient algorithms for solving nonmonotone and non-Lipschitzian equilibrium problems in Hilbert spaces. Under the assumption that the solution set of the associated Minty equilibrium problem is nonempty, we establish the weak and strong convergence of the proposed algorithms, respectively. The convergence is guaranteed without any monotonicity and Lipschitz-type continuity of the equilibrium bifunction. Some numerical experiments illustrate that the presented algorithms are faster than the existing projection extragradient ones.

Hybrid proximal point algorithm for solving split equilibrium problems and its applications

This paper deals with split equilibrium problems in Banach spaces. The presented algorithm is based on the hybrid algorithm and the proximal point algorithm and has been used for finding the solution of split equilibrium problems. Under some standard assumptions on equilibrium bifunctions, it is proven that the generated sequences by the presented scheme are strongly convergent. Finally, the efficiency of the proposed method is demonstrated through some examples. Also, comparative results verify that the proposed method is more effective than the other existing methods in the literature. Furthermore, an application of the presented algorithm in Hilbert spaces and an application of our method to solve the $LASSO$ problem in the field of compressed sensing are given.

A dynamical approach for the quantitative stability of parametric bilevel equilibrium problems and applications

In this paper, we primary establish Hölder and Lipschitz continuity of solutions to abstract dynamical mixed equilibrium problems, DMEP for short, which we apply to obtain quantitative stability for parametric differential variational inclusions. Then, by involving key conditions on Fitzpatrick transform of equilibrium bifunctions introduced and studied in Chbani et al. [From convergence of dynamical equilibrium systems to bilevel hierarchical Ky Fan minimax inequalities and applications. J Minimax Theory Appl. 2019;4:231–270], we derive further quantitative stability for a parametric bilevel equilibrium problem which is regarded in our approach as a limit problem of the dynamic model DMEP. The obtained abstract result on bilevel equilibria is thereby applied to parametric mathematical programs with equilibrium constraints. A specific applied model illustrating the meaning of the involved parameters is also discussed with respect to optimal control problems whose state equation is defined by a variational inequality. We also report a numerical illustration to highlight the convergence and estimation rates we obtained and support our theoretical results.

A new extragradient algorithm with adaptive step-size for solving split equilibrium problems

He (J. Inequal. Appl. 2012:Article ID 162 2012) introduced the proximal point CQ algorithm (PPCQ) for solving the split equilibrium problem (SEP). However, the PPCQ converges weakly to a solution of the SEP and is restricted to monotone bifunctions. In addition, the step-size used in the PPCQ is a fixed constant μ in the interval (0, frac{1}{ | A |^{2} } ). This often leads to excessive numerical computation in each iteration, which may affect the applicability of the PPCQ. In order to overcome these intrinsic drawbacks, we propose a robust step-size { mu _{n} }_{n=1}^{infty } which does not require computation of | A | and apply the adaptive step-size rule on { mu _{n} }_{n=1}^{infty } in such a way that it adjusts itself in accordance with the movement of associated components of the algorithm in each iteration. Then, we introduce a self-adaptive extragradient-CQ algorithm (SECQ) for solving the SEP and prove that our proposed SECQ converges strongly to a solution of the SEP with more general pseudomonotone equilibrium bifunctions. Finally, we present a preliminary numerical test to demonstrate that our SECQ outperforms the PPCQ.

Existence results and two step proximal point algorithm for equilibrium problems on Hadamard manifolds

"In this paper, we study the existence of solutions of equilibrium problems in the setting of Hadamard manifolds under the pseudomonotonicity and geodesic upper sign continuity of the equilibrium bifunction and under different kinds of coercivity conditions. We also study the existence of solutions of the equilibrium problems under properly quasimonotonicity of the equilibrium bifunction. We propose a two-step proximal point algorithm for solving equilibrium problems in the setting of Hadamard manifolds. The convergence of the proposed algorithm is studied under the strong pseudomonotonicity and Lipschitz-type condition. The results of this paper either extend or generalize several known results in the literature."

Modified projected subgradient method for solving pseudomonotone equilibrium and fixed point problems in Banach spaces

Motivated by the work of D.V. Hieu and J.-J. Strodiot [Strong convergence theorems for equilibrium problems and fixed point problems in Banach spaces, J. Fixed Point Theory Appl., (2018), 20:131], we introduce a new projected subgradient method for solving pseudomonotone equilibrium and fixed point problem in Banach spaces. The main iterative steps in the proposed method use a projection method and do not require any Lipschitz-like condition on the equilibrium bifunction. A strong convergence result is proved under mild conditions and we applied our algorithm to solving pseudomonotone variational inequalities in Banach spaces. Also, we provide some numerical examples to illustrate the performance of the proposed method and compare it with other methods in the literature.

An explicit extragradient algorithm for equilibrium problems on Hadamard manifolds

In this paper, we investigate a new extragradient algorithm for solving pseudomonotone equilibrium problems on Hadamard manifolds. The algorithm uses a variable stepsize which is updated at each iteration and based on some previous iterates. The convergence analysis of the proposed algorithm is discussed under mild assumptions. In the case where the equilibrium bifunction is strongly pseudomonotone, the $R$-linear rate of convergence of the new algorithm is formulated. A fundamental experiment is provided to illustrate the numerical behavior of the algorithm. The results presented in this paper generalize some corresponding known results.

Strong convergence theorems for finite families of pseudomonotone equilibrium and fixed point problems in Banach spaces

In this article, we introduce a new linesearch technique with Halpern iteration for finding a common solution of finite families of pseudomonotone equilibrium problems and fixed point of finite family of quasi- $$\phi $$ -nonexpansive mappings in Banach spaces. Under standard assumptions imposed on the equilibrium bifunctions and the quasi- $$\phi $$ -nonexpansive mappings, we proved that the sequence generated by our algorithm converges strongly to the unique solution of the equilibrium and fixed point problems. Numerical example is presented to illustrate the efficiency and accuracy of the proposed algorithm. Our results improve and extend many existing results in the literature in this direction.

Iterative algorithms for split equilibrium problems of monotone operators and fixed point problems of pseudo-contractions

In this paper, we investigate the split equilibrium problems and fixed point problems in Hilbert spaces. We provide a unified framework for solving such problem in which the involved equilibrium bifunctions f and g are pseudomonotone and monotone, respectively, and the operators S and T are pseudocontractive. We suggest an iterative algorithm for solving the split problem and demonstrate its weak convergence.

A self-adaptive inertial subgradient extragradient method for pseudomonotone equilibrium and common fixed point problems

In this paper, we introduce a self-adaptive inertial subgradient extragradient method for solving pseudomonotone equilibrium problem and common fixed point problem in real Hilbert spaces. The algorithm consists of an inertial extrapolation process for speeding the rate of its convergence, a monotone nonincreasing stepsize rule, and a viscosity approximation method which guaranteed its strong convergence. More so, a strong convergence theorem is proved for the sequence generated by the algorithm under some mild conditions and without prior knowledge of the Lipschitz-like constants of the equilibrium bifunction. We further provide some numerical examples to illustrate the performance and accuracy of our method.

A New Linesearch Algorithm for Split Equilibrium Problems

In this paper, we propose a new algorithm for solving a split equilibrium problem involving nonmonotone and monotone equilibrium bifunctions in real Hilbert spaces by using a shrinking projection method with a general Armijo line search rule on the e-subdifferential. We obtain a strong convergence theorem for the new algorithm.

Explicit iterative algorithms for solving equilibrium problems

The paper introduces two new extragradient algorithms for solving a pseudomonotone equilibrium problem with a Lipschitz-type condition recently presented by Mastroeni in auxiliary problem principle. The algorithm uses variable stepsizes which are updated at each iteration and based on some previous iterates. The advantage of the algorithms is that they are done without the prior knowledge of Lipschitz-type constants and also without any linesearch procedure. The convergence of the algorithms is established under mild assumptions. In the case where the equilibrium bifunction is strongly pseudomonotone, the R-linear rate of convergence of the new algorithms is formulated. Several of fundamental experiments are provided to illustrate the numerical behavior of the algorithms and also to compare with others.

Strong convergence theorems for equilibrium problems and fixed point problems in Banach spaces

The paper proposes two new iterative methods for solving pseudomonotone equilibrium problems involving fixed point problems for quasi- $$\phi $$ -nonexpansive mappings in Banach spaces. The proposed algorithms combine the extended extragradient method or the linesearch method with the Halpern iteration. The strong convergence theorems are established under standard assumptions imposed on equilibrium bifunctions and mappings. The results in this paper have generalized and enriched existing algorithms for equilibrium problems in Banach spaces.

On maximal monotonicity of bifunctions on Hadamard manifolds

We study some conditions for a monotone bifunction to be maximally monotone by using a corresponding vector field associated to the bifunction and vice versa. This approach allows us to establish existence of solutions to equilibrium problems in Hadamard manifolds obtained by perturbing the equilibrium bifunction.

Hybrid projected subgradient-proximal algorithms for solving split equilibrium problems and split common fixed point problems of nonexpansive mappings in Hilbert spaces

In this paper, we propose two strongly convergent algorithms which combines diagonal subgradient method, projection method and proximal method to solve split equilibrium problems and split common fixed point problems of nonexpansive mappings in a real Hilbert space: fixed point set constrained split equilibrium problems (FPSCSEPs) in real Hilbert spaces. The computations of first algorthim requires prior knowledge of operator norm. To estimate the norm of an operator is not always easy, and if it is not easy to estimate the norm of an operator, we purpose another iterative algorithm with a way of selecting the step-sizes such that the implementation of the algorithm does not need any prior information as regards the operator norm. The strong convergence properties of the algorithms are established under mild assumptions on equilibrium bifunctions. We also report some applications and numerical results to compare and illustrate the convergence of the proposed algorithms.

Retraction algorithms for solving variational inequalities, pseudomonotone equilibrium problems, and fixed-point problems in Banach spaces

In this paper, using sunny generalized nonexpansive retractions which are different from the metric projection and generalized metric projection in Banach spaces, we present new extragradient and line search algorithms for finding the solution of a J-variational inequality whose constraint set is the common elements of the set of fixed points of a family of generalized nonexpansive mappings and the set of solutions of a pseudomonotone J-equilibrium problem for a J -α-inverse-strongly monotone operator in a Banach space. To prove strong convergence of generated iterates in the extragradient method, we introduce a ϕ ∗-Lipschitz-type condition and assume that the equilibrium bifunction satisfies this condition. This condition is unnecessary when the line search method is used instead of the extragradient method. Using FMINCON optimization toolbox in MATLAB, we give some numerical examples and compare them with several existence results in literature to illustrate the usability of our results.

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.

Stability of Equilibria via Regularity of the Diagonal Subdifferential Operator

In this paper we investigate the Aubin property of the solution map of a parametric equilibrium problem, by providing a connection with a suitable behaviour of the diagonal subdifferential operator associated to the equilibrium bifunction. In particular, we shed some light on the relationship between metric regularity and subregularity of the diagonal subdifferential, on one side, and some properties of the bifunction, on the other side.