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Some developments in general variational inequalities. (English) Zbl 1134.49304
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:
49J40 Variational inequalities
47H10 Fixed-point theorems
47J20 Variational and other types of inequalities involving nonlinear operators (general)
49Q12 Sensitivity analysis for optimization problems on manifolds
90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)
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