Some Gap Functions And Error Bounds For Generalized Vector Equilibrium Problems

In vector optimization,the stability analysis for vector equilibrium problems is a main topic.Error bounds for vector equilibrium problems have played important roles in stability analysis.By using error bounds,one can obtain an upper estimate of the distance between an arbitrary feasible point and the solution set of a vector equilibrium problem or a vector variational inequality,which is important for the algorithm research in vector optimization.Additional,gap functions and regularized gap functions have turned out to be very useful in deriving the error bounds.In this thesis,we mainly study gap functions and error bounds for gneralized vector equilibrium problems,by using scalarization approaches.First of all,by extending generalized mixed variational inequalities to vector cases,a new kind of generalized vector variational inequality is formed.Using linear scalarization methods,gap functions and regularized gap functions for the generalized vector variational inequality are established,and regularized gap functions are verified by using generalized f-projection operators.In addition,the error bounds for generalized vector variational inequalities are obtained in terms of the gap functions and the regularized gap functions within a new kind of monotonicity assumption.Moreover,by using nonlinear scalarization techniques and a minimax strategy,error bound results in terms of gap functions for a generalized mixed vector equilibrium problem are established,where the solutions for vector problems can be general sets under natural assumptions,but are not limited to singletons.The other essentially equivalent approach via a separation principle is analyzed.We use the idea to establish gap functions in finite dimensional spaces.As an application,we also get the gap functions and error bounds for vector equilibrium problems.