The Algorithms For Solving Bilevel Multiobjective Programming Problems

In this paper,the algorithms are researched to solve the Mathematical Programs with Equilibrium Constraints (MPEC problem )with a variational inequality and bilevel multiobjective program. The main content of this paper consists of two part on the whole.l.The first part of this paper (chapter 2) is to offer a new problem, whose object is multi-objective and whose constraint is nonlinear inequality and variational inequality. This problem belongs to MPEC problem. There also offer Karush-Kuhn-Tucker (KKT) conditons of this program and they can be considered sufficient and necessary conditon under the reasonable constraint qualification. Some relative theorems and their proofs are given and the interact algorithm using the L1 penalty function is given to solve this problem.The model we are discussing is:y is the solution of the variational ineqality (F(x, -),C(x)). (1c) (lc)mark : For all given x X, C(x) = {y Rm : g(x, y) < 0},and Vi; e C(x),Where the mapping f : R" x Rm - Rp,the vecter-valued mapping G : Rn x Rm - Rq the vecter-valued mapping g : Rn Rm - Rl are continuousdifferential. Suppose / and G are continuous and twice differential maps for all (x,y )Rn+m. For every x X,i = 1,2,...,l,gi(x,) is convex in the second argument.With the KKT conditions ,the problem (1) is converted to :a.t. G(x,y)<0Through denning the sequentially bounded constraint qualification (SBCQ), there are the following results:Theorem 1 Let F and each gi be continuous. Suppose that each gi(x, -) is convex for all x X,and ygi(x,y) exits and is continuous at all points (x,y) in an open set containing D .Assume that the SBCQ holds on D for the set- valued map M denned above, and then the problem (l)is equivalent to the problem (2).Let the linear summation problem with the power of the problem (2) iss.t. G(x,y)<0 (3)F(x,y)+ And then there isTheorem 2 About every given power coefficient = (1 , 2, ..., p)T A++ (or A+), the most superior solution of the problem (3) is the efficient solution (or the weak efficient solution) of the problem (2). Where, A+ = {a =Suppose the penalty term:Let the l1 penalty function be(4)Where > 0 is a penalty scalar,which is a very big positive number. So the problem (3) can be converted toDenote z = (x,y, ).Theorem 3 (the convergent theorem) Let the feasible region of the problem isnot empty, and there is a > 0, subjecting to the setis compact .Let { k} is a strict increasing positive sequence of numbers limiting infty . For every k,there is a optimal solution z(k) to the problem (5). Then there is a conergent subsequence {z} of {z}, and the limit of any conergent subsequence like this is the optimal solution of the problem (2).2. In the second part of this paper (Chapter 3,4) , I use the algorithms -which had been given before -to solve the bilevel singular-objective decision making and singular-level multiobjective programming problems. The maxmal module idealpoint algorithms is given to solve bilevel multiobjective programming problem: At first, bilevel multiobjective programming problem is converted to the singular-level constrait programming problem by the means of -constraint algorithm with satisfactoriness and the Kuhn - Tucker conditions (in the case of convex pro-gram).When the constraint is a compact set , then the maximal module ideal point algorithm is offerd to solve the weak efficient solution of the problem. And then, the analyst and the decision makers interact and use the gradual tolerant constraint algorithm to check the satisfactoriness of the solution. The model we are discussing is :min (6) Where yi is the solution of the following equation:(8) Where x, F0and 0 is the decision variable,the objective function and the constraint set of the upper-level programming respectively; yi,F1(i0 and i,(i = 1,2, ...,p) is the decision variable.the objective function and the constraint set of the lower-level programming respectively; Each F1(i) (i = 1,2, ...,p) is convex vector-valued function. Each i(z = 1,...