Mixed Quasi-variational Inequalities Research Articles

Predictor-corrector algorithms for solving generalized mixed implicit quasi-equilibrium problems

A new class of generalized mixed implicit quasi-equilibrium problems (GMIQEP) with four-functions is introduced and studied. The new class of equilibrium problems includes many known generalized equilibrium problems and generalized mixed implicit quasi-variational inequality problems as many special cases. By employing the auxiliary principle technique, some predictor-corrector iterative algorithms for solving the GMIQEP are suggested and analyzed. The convergence of the suggested algorithm only requires the continuity and the partially relaxed implicit strong monotonicity of the mappings.

ON THE CONVERGENCE ANALYSIS OF THE ITERATIVE METHOD WITH ERRORS FOR GENERAL MIXED QUASIVARIATIONAL INEQUALITIES IN HILBERT SPACES

The purpose of this paper is to investigate the iterative methods for a class of general mixed quasivariational inequalities in a Hilbert space. Utilizing the alternative equivalent formulation between general mixed quasivariational inequalities and implicit fixed-point problems, we suggest and analyze a new modified self-adaptive resolvent method with errors for solving this class of general mixed quasivariational inequalities in conjunction with a technique updating the solution. Moreover, we give the convergence analysis of this method in a Hilbert space. Since this class of general mixed quasivariational inequalities includes a number of known classes of variational inequalities as special cases, our results are more general than some earlier and recent ones in the literature.

ON THE CONVERGENCE ANALYSIS OF THE ITERATIVE METHOD WITH ERRORS FOR GENERAL MIXED QUASIVARIATIONAL INEQUALITIES IN HILBERT SPACES

The purpose of this paper is to investigate the iterative methods for a class of general mixed quasivariational inequalities in a Hilbert space. Utilizing the alternative equivalent formulation between general mixed quasivariational inequalities and implicit fixed-point problems, we suggest and analyze a new modified self-adaptive resolvent method with errors for solving this class of general mixed quasivariational inequalities in conjunction with a technique updating the solution. Moreover, we give the convergence analysis of this method in a Hilbert space. Since this class of general mixed quasivariational inequalities includes a number of known classes of variational inequalities as special cases, our results are more general than some earlier and recent ones in the literature.

Self-adaptive methods for mixed quasi-variational inequalities

It is well known that the variational inequalities involving the nonlinear term φ are equivalent to the fixed-point problems and the resolvent equations. In this paper, we use these alternative equivalent formulations to suggest and analyze some new self-adaptive iterative methods for solving mixed quasi-variational inequalities. Our results can be viewed as significant extensions of the previously known results for mixed quasi-variational inequalities. An example is given to illustrate the efficiency of the proposed method.

An iterative method for generalized set-valued nonlinear mixed quasi-variational inequalities

This paper presents an iterative method for solving the generalized nonlinear set-valued mixed quasi-variational inequality, a problem class that was introduced by Huang et al. (Comp. Math. Appl. 40 (2–3) (2000) 205–215). The method incorporates step size controls that enable application to problems where certain set-valued mappings do not always map to nonempty closed bounded sets.

On General Mixed Quasivariational Inequalities

In this paper, we suggest and analyze several iterative methods for solving general mixed quasivariational inequalities by using the technique of updating the solution and the auxiliary principle. It is shown that the convergence of these methods requires either the pseudomonotonicity or the partially relaxed strong monotonicity of the operator. Proofs of convergence is very simple. Our new methods differ from the existing methods for solving various classes of variational inequalities and related optimization problems. Various special cases are also discussed.

Auxiliary principle technique for multivalued mixed quasi-variational inequalities

In this paper, we suggest and analyze a class of predictor-corrector methods for solving mixed quasi variational inequalities by using the auxiliary principle technique. We prove that the convergence of these predictor-corrector type methods requires only the partial relaxed strongly monotonicity, which is a weaker condition than co-coercivity. Since the mixed quasi variational inequalities include (quasi) variational inequalities as special cases, our results continue to hold for these problems. Our results represent an improvement and refinement of the previously known results. Mathematics subject classification (2000): 49J40.

Predictor–corrector methods for general mixed quasi variational inequalities

In this paper, we use the auxiliary principle technique to suggest a class of predictor–corrector methods for solving general mixed quasi 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 various known and new results for solving various classes of variational inequalities and related problems.

Existence and algorithm of solutions for general multivalued mixed implicit quasi-variational inequalities

A new class of general multivalued mixed implicit quasi-variational inequalities in a real Hilbert space was introduced, which includes the known class of generalized mixed implicit quasi-variational inequalities as a special case, introduced and studied by Ding Xieping. The auxiliary variational principle technique was applied to solve this class of general multivalued mixed implicit quasi-variational inequalities. Firstly, a new auxiliary variational inequality with a proper convex, lower semicontinuous, binary functional was defined and a suitable functional was chosen so that its unique minimum point is equivalent to the solution of such an auxiliary variational inequality. Secondly, this auxiliary variational inequality was utilized to construct a new iterative algorithm for computing approximate solutions to general multivalued mixed implicit quasi-variational inequalities. Here, the equivalence guarantees that the algorithm can generate a sequence of approximate solutions. Finally, the existence of solutions and convergence of approximate solutions for general multivalued mixed implicit quasi-variational inequalities are proved. Moreover, the new convergerce criteria for the algorithm were provided. Therefore, the results give an affirmative answer to the open question raised by M. A. Noor, and extend and improve the earlier and recent results for various variational inequalities and complementarity problems including the corresponding results for mixed variational inequalities, mixed quasi-variational inequalities and quasi-complementarity problems involving the single-valued and set-valued mappings in the recent literature.

Iterative Methods for General Mixed Quasivariational Inequalities

It is well known that mixed quasivariational inequalities are equivalent to implicit fixed-point problems. We use this alternative equivalent formulation to suggest and analyze a new self-adaptive resolvent method for solving mixed quasivariational inequalities in conjunction with a technique updating the solution. We show that the convergence of this method requires pseudomonotonicity, which is a weaker condition than monotonicity. Since mixed quasivariational inequalities include various classes of variational inequalities as special cases, our results continue to hold for these problems.

Resolvent Algorithms for Mixed Quasivariational Inequalities

It is well known that mixed quasivariational inequalities are equivalent to implicit resolvent equations. We use this alternative equivalent formulation to suggest and analyze a new modified resolvent method for solving mixed quasivariational inequalities and related problems. We show that the convergence of this modified method requires only pseudomonotonicity, which is a weaker condition than monotonicity. Since mixed quasivariational inequalities include various classes of variational inequalities as special cases, our result continues to hold for these problems.

Mixed quasi variational inequalities

In this paper, we use the resolvent operator and auxiliary principle techniques to suggest and analyze several iterative algorithms for solving mixed quasi variational inequalities and related problems. We study the convergence criteria of these algorithms under mild conditions. We also study the global stability and existence of a unique solution of these quasi variational inequalities by using the dynamical systems approach. Our results represent refinement and improvement of the previously known results for variational inequalities.

A new class of generalized nonlinear mixed quasi-variational inequalities in Banach spaces

In this paper, we introduce and study a new class of generalized nonlinear mixed quasivariational inequalities. Using the KKM technique, we prove the existence and uniqueness of solutions for this class of generalized nonlinear mixed quasi-variational inequalities in reflexive Banach spaces. Our results include the main results of Verma [1], [2] as special cases. Mathematics subject classification (2000): 47H06, 49J30, 49J40.

Modified resolvent algorithm for general mixed quasi-variational inequalities

It is well known [1] that the general mixed quasi-variational inequalities are equivalent to the implicit resolvent equations. In this paper, we use this alternative formulation to suggest and analyze a modified resolvent algorithm for solving general mixed quasi-variational inequalities. It is shown that the convergence of the new modified method only requires the pseudomonotonicity, which is a weaker condition than monotonicity. Since the general mixed quasi-variational inequalities include the general (mixed) variational inequalities and related optimization problems as special cases, results obtained in this paper continue to hold for these problems. Our results can be viewed as significant extensions of previously known results including those of Noor [1–3] and Solodov and Tseng [4] for various classes of variational inequalities.

Proximal Methods for Mixed Quasivariational Inequalities

Proximal Methods for Mixed Quasivariational Inequalities

Generalized nonlinear mixed quasi-variational inequalities

In this paper, we introduce and study a new class of quasi-variational inequalities, which is called the generalized nonlinear set-valued mixed quasi-variational inequality. Using the resolvent operator technique for maximal monotone mapping, we construct some new iterative algorithms for solving this class of generalized nonlinear set-valued mixed quasi-variational inequalities. We prove the existence of solution for this kind of generalized nonlinear set-valued mixed quasivariational inequalities without compactness and the convergence of iterative sequences generated by the algorithms. We also discuss the convergence and stability of perturbed iterative algorithm for solving a class of generalized nonlinear mixed quasi-variational inequalities.

Existence and algorithm of solutions for generalized mixed implicit quasi-variational inequalities

In this paper, by applying the auxiliary variational principle technique, some existence theorems of solutions for a class of generalized mixed implicit quasi-variational inequalities are proved in Hilbert spaces. A novel and innovative iterative algorithm to compute approximate solutions is suggested and analyzed. The convergence criteria are also given. As special cases of these results, the open problem put forward by Noor is answered positively and some results of existence and algorithm of solutions for generalized implicit quasi-complementarity problems are also obtained. Our results are new and generalize a number of known results of mixed variational inequalities, mixed quasi-variational inequalities and quasi-complementary problems involving single-valued and set-valued mappings in recent literatures.

Error bounds for general mixed quasivariational inequalities

It is well known that mixed quasivariational inequalities are equivalent to the implicit fixed‐point problems. We use this alternative equivalent formulation to suggest and consider some merit functions for general mixed quasivariational inequalities. We use these merit functions to obtain error bounds for the solution under some mild conditions. Some special cases are also discussed.

Existence and iterative algorithms of solutions for generalized mixed quasi-variational inequalities

In this paper,. by using the auxiliary technique of variational inequalities, the existence and iterative algorithms of solutions for a class of generalized mixed quasi-variational inequalities are studied. Our results answer the open problems mentioned by Noor, improve and generalize some recent known results.

Set-valued mixed quasi-variational inequalities and implicit resolvent equations

In this paper, we introduce and study a new class of variational inequalities, which is called the set-valued mixed quasi-variational inequality. The resolvent operator technique is used to establish the equivalence among generalized set-valued mixed quasi-variational inequalities, fixed-point problems and the set-valued implicit resolvent equations. This equivalence is used to study the existence of a solution of set-valued variational inequalities and to suggest a number of iterative algorithms for solving variational inequalities and related optimization problems. The results proved in this paper represent a significant refinement and improvement of the previously known results in this area.