Research On Theory And Algorithms For Several Hierarchical Optimization Problems

In this dissertation,we study the theory and algorithms for bilevel programs,trilevel programs,and bilevel variational inequalities.The research content of this dissertation includes six parts as follows:In the first part,we study a class of bilevel programs with a convex lower level problem violating Slater's CQ.Since the lower level problem with a relaxing feasible region satisfies Slater's CQ,an approximate solution of the original bilevel programs is obtained by solving a perturbation bilevel programs.The lower convergences of the constraint set mapping and the solution set mapping of the lower level problem of the perturbation bilevel programs have been discussed.We show that,the solutions of a sequence of the perturbation bilevel programs converge to a solution of the original bilevel programs under some appropriate conditions.And this convergence result has been applied to a simple trilevel programs.In the second part,we study a class of bilevel programs with a nonconvex lower level problem containing the inequality constraints.By using the penalty function method,we transform the lower level problems into a program with a box constraint.And we propose an algorithm for this problem based on the integral entropy function.This algorithm is an improvements on the algorithm proposed by Lin et al.(Mathematical Programming,144(1):277-305,2014).Numerical experiments show that the improved algorithm is feasible.In the third part,we study a class of multiobjective bilevel programs which both levels are multiobjective optimization problem.We first transform this problem into a semivectorial bilevel programs by applying scalarization approach to the lower level multiobjective optimization problem.Then two necessary optimality conditions are obtained via replacing the lower level scalar problem by it 's KKT conditions and optimal-valued function respectively.In the forth part,we study the optimality conditions for a class of trilevel programs which all levels are nonlinear programs.We firstly transform this problem into a bilevel programs by applying KKT approach to the lower-level problem.Then we obtain a necessary optimality condition via the differential calculus of Mordukhovich.Finally,a theorem for existence of optimal solution is derived via Weierstrass Theorem.In the fifth part,we study the optimality conditions for a class of pessimistic trilevel programs which middle-level is a pessimistic problem.Applying KKT approach to the lower level problem,we firstly translate this problem into a pessimistic bilevel programs.Then,we obtain a necessary optimality condition via the differential calculus of Mordukhovich.Finally,we obtain an existence theorem of optimal solution by direct method.In the sixth part,we study a class of bilevel variational inequalities with hierarchical nesting structure.We firstly get the existence of solution for this problem by using Himmelberg fixed point theorem.Then the unique of solution for upper-level variational inequality is given under some mild conditions.Using gap functions of upper-level and lower-level variational inequalities,we transform bilevel variational inequalities into one-level variational inequality.Moreover,we propose two iterative algorithms to find the solutions of the bilevel variational inequalities.Finally,the convergence of the proposed algorithm is derived under some mild conditions.