scholarly journals Generalized multivalued equilibrium-like problems: auxiliary principle technique and predictor-corrector methods

2016 ◽  
Vol 2016 (1) ◽  
Author(s):  
Vahid Dadashi ◽  
Abdul Latif
Keyword(s):  
Auxiliary Principle ◽  
Auxiliary Principle Technique ◽  
Predictor Corrector

scholarly journals Mixed quasi-equilibrium-like problems

2006 ◽  
Vol 2006 ◽  
pp. 1-8 ◽  
Author(s):  
Muhammad Aslam Noor
Keyword(s):  
Optimization Problems ◽  
New Method ◽  
Quasi Equilibrium ◽  
Convergence Criteria ◽  
Auxiliary Principle ◽  
Auxiliary Principle Technique ◽  
Mild Conditions ◽  
Special Cases ◽  
Bregman Function ◽  
Predictor Corrector

We use the auxiliary principle technique in conjunction with the Bregman function to suggest and analyze a three-step predictor-corrector method for solving mixed quasi-equilibrium-like problems. We also study the convergence criteria of this new method under some mild conditions. As special cases, we obtain various new and known methods for solving variational-like inequalities and related optimization problems.

scholarly journals Three-step iterations for mixed quasi variational-like inequalities

2005 ◽  
Vol 2005 (14) ◽  
pp. 2299-2306
Author(s):  
Muhammad Aslam Noor
Keyword(s):  
Variational Inequalities ◽  
Optimization Problems ◽  
New Method ◽  
Convergence Criteria ◽  
Auxiliary Principle ◽  
Auxiliary Principle Technique ◽  
Mild Conditions ◽  
Special Cases ◽  
Bregman Function ◽  
Predictor Corrector

In this paper, we use the auxiliary principle technique in conjunction with the Bregman function to suggest and analyze a three-step predictor-corrector method for solving mixed quasi variational-like inequalities. We also study the convergence criteria of this new method under some mild conditions. As special cases, we obtain various new and known methods for solving variational inequalities and related optimization problems.

scholarly journals Mixed quasi invex equilibrium problems

2004 ◽  
Vol 2004 (57) ◽  
pp. 3057-3067 ◽  
Author(s):  
Muhammad Aslam Noor
Keyword(s):  
Variational Inequalities ◽  
Equilibrium Problems ◽  
Strong Monotonicity ◽  
Auxiliary Principle ◽  
Auxiliary Principle Technique ◽  
Iterative Schemes ◽  
New Class ◽  
Special Cases ◽  
Weaker Conditions

We introduce a new class of equilibrium problems, known asmixed quasi invex equilibrium(orequilibrium-like) problems. This class of invex equilibrium problems includes equilibrium problems, variational inequalities, and variational-like inequalities as special cases. Several iterative schemes for solving invex equilibrium problems are suggested and analyzed using the auxiliary principle technique. It is shown that the convergence of these iterative schemes requires either pseudomonotonicity or partially relaxed strong monotonicity, which are weaker conditions than the previous ones. As special cases, we also obtained the correct forms of the algorithms for solving variational-like inequalities, which have been considered in the setting of convexity. In fact, our results represent significant and important refinements of the previously known results.

scholarly journals On a class of multivalued variational inequalities

1998 ◽  
Vol 11 (1) ◽  
pp. 79-93 ◽  
Author(s):  
Muhammad Aslam Noor
Keyword(s):  
Variational Inequalities ◽  
Optimization Problems ◽  
Iterative Algorithms ◽  
Auxiliary Principle ◽  
Existence Of A Solution ◽  
Auxiliary Principle Technique ◽  
New Class ◽  
Special Cases ◽  
Projection Techniques ◽  
Multivalued Variational Inequalities

In this paper, we introduce and study a new class of variational inequalities, which are called multivalued variational inequalities. These variational inequalities include as special cases, the previously known classes of variational inequalities. Using projection techniques, we show that multivalued variational inequalities are equivalent to fixed point problems and Wiener-Hopf equations. These alternate formulations are used to suggest a number of iterative algorithms for solving multivalued variational inequalities. We also consider the auxiliary principle technique to study the existence of a solution of multivalued variational inequalities and suggest a novel iterative algorithm. In addition, we have shown that the auxiliary principle technique can be used to find the equivalent differentiable optimization problems for multivalued variational inequalities. Convergence analysis is also discussed.

scholarly journals Some New Classes of Extended General Mixed Quasi-Variational Inequalities

2012 ◽  
Vol 2012 ◽  
pp. 1-9 ◽  
Author(s):  
Muhammad Aslam Noor ◽  
Khalida Inayat Noor
Keyword(s):  
Variational Inequality ◽  
Variational Inequalities ◽  
Auxiliary Principle ◽  
Existence Of A Solution ◽  
Auxiliary Principle Technique ◽  
New Class ◽  
Special Cases ◽  
Quasi Variational Inequality

We consider and study a new class of variational inequality, which is called the extended general mixed quas-variational inequality. We use the auxiliary principle technique to study the existence of a solution of the extended general mixed quasi-variational inequality. Several special cases are also discussed. Results proved in this paper may stimulate further research in this area.

scholarly journals Existence and Iterative Algorithm of Solutions for a System of Generalized Nonlinear Mixed Variational-Like Inequalities

2012 ◽  
Vol 2012 ◽  
pp. 1-14
Author(s):  
Yan-Mei Du
Keyword(s):  
Banach Spaces ◽  
Iterative Algorithm ◽  
Existence Result ◽  
Existence Of Solutions ◽  
Auxiliary Principle ◽  
Auxiliary Principle Technique

We introduce and study a system of generalized nonlinear mixed variational-like inequality problems (SGNMVLIPs) in Banach spaces. The auxiliary principle technique is applied to study the existence and iterative algorithm of solutions for the SGNMVLIP. First, the existence of solutions of the auxiliary problems for the SGNMVLIP is shown. Second, an iterative algorithm for solving the SGNMVLIP is constructed by using this existence result. Finally, not only the existence of solutions of the SGNMVLIP is shown but also the convergence of iterative sequences generated by the algorithm is also proven. The technique and results presented in this paper generalize and unify the corresponding techniques and results given in the literature.

scholarly journals Extended Perturbed Mixed Variational-Like Inequalities for Fuzzy Mappings

2021 ◽  
Vol 2021 ◽  
pp. 1-16
Author(s):  
Muhammad Bilal Khan ◽  
Muhammad Aslam Noor ◽  
Khalida Inayat Noor ◽  
Ahmad Termimi Ab Ghani ◽  
Lazim Abdullah
Keyword(s):  
Iterative Algorithms ◽  
Existence Results ◽  
Convergence Criterion ◽  
Fuzzy Mapping ◽  
Numerical Techniques ◽  
Auxiliary Principle ◽  
Auxiliary Principle Technique

In this article, our aim is to consider a class of fuzzy mixed variational-like inequalities (FMVLIs) for fuzzy mapping known as extended perturbed fuzzy mixed variational-like inequalities (EPFMVLIs). As exceptional cases, some new and classically defined “FMVLIs” are also attained. We have also studied the auxiliary principle technique of auxiliary “EPFMVLI” for “EPFMVLI.” By using this technique and some new analytic results, some existence results and efficient numerical techniques of “EPFMVLI” are established. Some advanced and innovative iterative algorithms are also obtained, and the convergence criterion of iterative sequences generated by algorithms is also proven. In the end, some new and previously known existence results and algorithms are also studied. Results secured in this paper can be regarded as purification and development of previously familiar results.

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