The Research Of Algorithms On General Variational Inequalities

General variational inequalities are an important and useful generalization of variational inequalities. It has been shown that general variational inequalities provide us with a unified, simple, and natural framework to study a wide class of problems including unilateral, moving, obstacle, free, equilibrium, and economics arising in various areas of pure and applied sciences. It is also one of focal point problems paid close attention by scholars in the field of applied mathematics.This paper is devoted to studying systematically the algorithms of general variational inequality problems, including general strongly monotone variational inequality, general monotone variational inequality, general pseudomonotone variational inequality, and multivalued general mixed quasi-variational inequality, which are the unity and extension of a large number of known variational inequalities and mixed variational inequalities.A modified projection method for solving general variational inequalities is presented. A practical and robust stepsize choice strategy, termed self-adaptive procedure, is developed. The resulting algorithm is globally convergent for strongly monotone operator. Numerical results and comparison with some existing projection-type methods are also given to illustrate the efficiency of the proposed method.A modified prediction-correction method is proposed for general monotone variational inequalities by using the better prediction and correction stepsizes. Preliminary numerical experiments indicate that the improvements are significant.Some new projection methods are presented for solving general variational inequalities based on the operator splitting technique, including three-steps and k-steps iterative algorithms. The modified methods converge for g-pseudomonotone and g-Lipschitz continuous operators, which is much weaker than g-monotonicity. It is worthwhile to point out that the new iterative methods are different from the existing projection methods, and our proof of convergence is very simple compared with other methods.Some new predictor-corrector methods and three-steps iterative algorithms for solving multivalued general mixed quasi-variational inequalities is presented by using the auxiliary principle and resolvent operator technique, respectively. If the bifunction involving the multivalued general mixed quasi-variational inequalities is skew-symmetric, then the new predictor methods is shown that the convergence of the new method requires the partically relaxed strong monotonicity property of the operator, which is a weaker condition than cocoecivity. Moreover, we correct some mistakes of Nbor. The three-steps iterative algorithms converge strongly under suitable conditions.A new smoothing Newton algorithm for sloving box constrained variational inequality is proposed by using Chen-Harker-Kanzow-Smale smoothing function. This method has the advantage that it has only to deal with a smooth function at any iteration and it never requires a procedure to decrease an approximate parameter. Under suitable conditions, it's globally convergent.