A NEW SELF-ADAPTIVE ITERATIVE METHOD FOR GENERAL MIXED QUASI VARIATIONAL INEQUALITIES

The general mixed quasi variational inequality containing a nonlinear term φ is a useful and an important generalization of variational inequalities. The projection method can not be applied to solve this problem due to the presence of nonlinear term. It is well known that the variational inequalities involving the nonlinear term φ are equivalent to the fixed point problems and resolvent equations. In this article, the authors use these alternative equivalent formulations to suggest and analyze a new self-adaptive iterative method for solving general mixed quasi variational inequalities. Global convergence of the new method is proved. An example is given to illustrate the efficiency of the proposed method.

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 Xie-ping . 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 .

ALGORITHM OF SOLUTIONS FOR MIXEDNONLINEAR VARIATIONAL-LIKE INEQUALITIES IN REFLEXIVE BANACH SPACE

In this paper, the author studies a class of mixed nonlinear variational-like inequalities in reflexive Banach space. By applying a minimax inequality obtained by the author, some existence uniqueness theorems of solutions for the mixed nonlinear variational-like inequalities are proved. Next, by applying the auxiliary problem technique, rite author suggests an innovative iterative algorithm to compute the approximate solutions of the mixed nonlinear variational-like inequalities. Finally, the convergence criteria is also discussed.

General Convergence Analysis for Three-step Projection Methods and Applications to Variational Problems

First a general model for a three-step projection method is introduced, and second it has been applied to the approximation solvability of a system of nonlinear variational inequality problems in a Hilbert space setting. Let H be a real Hilbert space and K be a nonempty closed convex subset of H. For arbitrarily chosen initial points x0, y0, z0 ∈ K, compute sequences xn, yn, zn such thatT ： K→ H is a nonlinear mapping onto K. At last three-step models are applied to some variational inequality problems.

New cooperative projection neural network for nonlinearly constrained variational inequality

This paper proposes a new cooperative projection neural network （CPNN）, which combines automatically three individual neural network models with a common projection term. As a special case, the proposed CPNN can include three recent recurrent neural networks for solving monotone variational inequality problems with limit or linear constraints, respectively. Under the monotonicity condition of the corresponding Lagrangian mapping, the proposed CPNN is theoretically guaranteed to solve monotone variational inequality problems and a class of nonmonotone variational inequality problems with linear and nonlinear constraints. Unlike the extended projection neural network, the proposed CPNN has no limitation on the initial point for global convergence. Compared with other related cooperative neural networks and numerical optimization algorithms, the proposed CPNN has a low computational complexity and requires weak convergence conditions. An application in real-time grasping force optimization and examples demonstrate good performance of the proposed CPNN.

The Polar φ-Brunn-Minkowski Inequalities

A general framework of polar Brunn-Minkowski theory that unifies the Orlicz Brunn-Minkowski inequality, the Lp Brunn-Minkowski inequality for p ∈(0,1) and the logBrunn-Minkowski inequality all for polar bodies is provided. It is shown that this general polar φ Brunn-Minkowski inequality is equivalent to a general polar φ Minkowski mixed volume inequality.