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Zbl 1134.49304
Noor, Muhammad Aslam
Some developments in general variational inequalities. (English)
[J] Appl. Math. Comput. 152, No. 1, 199-277 (2004). ISSN 0096-3003

Summary: General variational inequalities provide us with a unified, natural, novel and simple framework to study a wide class of equilibrium problems arising in pure and applied sciences. In this paper, we present a number of new and known numerical techniques for solving general variational inequalities using various techniques including projection, Wiener-Hopf equations, updating the solution, auxiliary principle, inertial proximal, penalty function, dynamical system and well-posedness. We also consider the local and global uniqueness of the solution and sensitivity analysis of the general variational inequalities as well as the finite convergence of the projection-type algorithms. Our proofs of convergence are very simple as compared with other methods. Our results present a significant improvement of previously known methods for solving variational inequalities and related optimization problems. Since the general variational inequalities include (quasi) variational inequalities and (quasi) implicit complementarity problems as special cases, results presented here continue to hold for these problems. Several open problems have been suggested for further research in these areas.

MSC 2000:: *49J40 Variational methods including variational inequalities
47H10 Fixed point theorems for nonlinear operators on topol.linear spaces
47J20 Inequalities involving nonlinear operators
49Q12 Sensitivity analysis
90C33 Complementarity problems

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Zentralblatt MATH Berlin [Germany]