Research On The Optimality Of Approximate Quasi-weak Efficient Solutions For Vector Equilibrium Problems

The vector equilibrium problem is a generalized mathematical model with broad application prospects,including problems such as vector optimization,variational inequality,and vector complementarity.In practical problems,the solution of the problem cannot be obtained accurately,so it is of great value and significance to discuss the approximate solution of the problem.In optimization theory,optimality conditions are often discussed by scholars as an important content.Therefore,this thesis studies the optimality of approximate quasi-weakly efficient solutions to the vector equilibrium problem.The specific contents include:1.Based on the Michel-Penot sub-differentiation,the optimality conditions and penalties for approximate quasi-weakly efficient solutions of the vector equilibrium problem with equality and inequality constraints(CVEP)are discussed.Firstly,under the assumptions of constrained quality and generalized pseudo-convexity,quasi-convexity and approximate quasi-type-I functions,a strong KKT-type optimality necessary condition and two optimality sufficient conditions for the approximate quasi-weakly efficient solution of CVEP are given,respectively.Secondly,under the assumption of a higher-order stabilization condition,the equivalence between the penalty problem and the stabilization condition of CVEP is proved.2.The optimality conditions and duality theorems of approximate quasi-weakly efficient solutions for the multi-objective vector optimization problem with inequality constraints(CMVOP)are studied based on Clarke sub-differentiation.Firstly,with the help of the complex function algorithm of Clarke sub-differentiation and the concept of approximate pseudo-quasi-type-I function,the optimality conditions for the approximate quasi-weakly efficient solution of CMVOP are obtained.Secondly,an approximate mixed dual model is introduced,and three dual theorems between the dual problem and the CMVOP for approximate quasi-weak efficient solutions are established.3.Qualitative analysis of the solutions of a class of inverse mixed quasi-variational inequality problems(IMQVIP)based on the Lipschitz continuity of mapping.Firstly,the existence and uniqueness theorems of the solution of IMQVIP are proved by using the fixed point theorem.Secondly,under the assumption of strong monotonic map and relaxed strong monotonic map,respectively,the convergence of IMQVIP projection iterative algorithm is verified and the error bound of IMQVIP is given.