The Research Of The Algorithms For Solving 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 in the whole.1. In the First part of this paper (Chapter 2,3) , 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 ideal point 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 e-constraint algorithm with satisfactoriness and the Kuhn - Tucker conditions (in the case of convex program).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 :Where y_i is the solution of the following equation:minWhere x, F₀ and Ω₀ is the decision variable,the objective function and the constraint set of the upper-level programming respectively;y_i,F₁ⁱ and Ω_i . (i = 1,2, ...,p) is the decision variable the objective function and the constraint set of the lower-level programming respectively;Each F₁ⁱ (i = 1,2, ...,p) is convex vector-valued function. Each Ω_i(i = 1,2, ...,p) is a convex set.The decision principle of this model as follows:(l)There is the positive direct Stackelberg leader-follower game between the upper-level decision maker and the lower-level decision makers. The upper-level decision maker(UDM) is the leader and each lower-level decision maker is the follower.(2)There is noncooperate Nash complete equilibrium game among the lower-level decision makers.With the ε-constraint algorithm with satisfactoriness and the Kuhn-Tucker conditions, The problem (l)-(4) can be convert to :imn(/oi(a;, yu j/2, ? ? ?, yP), ? ? ?, foN0{x, V\, V2, ? ? ?, Up)) (5)vVi/n(a;1yi)y2,---,yp)- E 4l)Vyi^l)(x,yi,y2,-,yp) = 0, t = l,2,...,p. (7)tx^/if (x,yi,y2,---,yp))=0, i = l,2,...,p,j = 1,2, ...,U (8)?i-°>0, i = l,2,...,p,j = l,2,...,ti (9)(z,yi,y2,---,yP) eft- i = i,2,---,p (10)Where Q- = {(x,yi,y2, â–  â–  â–  ,yP)\hf(x,yi,y2r â–  â–  ,yp) > 0,i = 1,2, ...,p,j = 1,2, ...,tj} is the constraint set composed by the formula (2.9) and the formula (2.10).[seeing the original paper].y₊ C ,. ,. ,.(!) ,.(!) ,.(1) ,.(P) ,.(P) ,,(P)^ - c Dm jL O;Leta, — v ₁ 5Z Sij{i — 1)2, ...,iVo), called the mean moving subtraction of' ° j=i the ith subject. Obviously all Qj > 0;Let \ = -^-,i = 1,2,..., JVo, each3=1Qi is lined in turn from large to little. Let an > ?i2 > ?â– â–  > ?iNoi an(iNothen have A^ > \& > ... > A^orelated and also E ^ik = 1-fei(l)If for alii € {1,2,..., iV0}, A* > 0,let the coefficient of (fon(z) - /fa) be Ajat0, the coefficient of (/bi2(z) —Z^) ^e -^i(M)-i)>???? ana* the coefficientof (foiNoi*) - foiNo) be Ail-No(2)If there exits Bit € {1,2,..., iV0}, Afc = 0, ï¿¡ <5fcj = 0- Because 6kj > 0, * 4j = 0,j = l,2,...,yV0. This tell us fkj = fkk = fok(zj*) = fok(zk*). Let the coefficient of (fok(z) — fok) be Afc = 1 in the problem (Pa)- The other coefficient of (foi(z) — /^) is given like (1), and then for all i €It cab be shown from the experence, the weak efficient solution given like this is more satisfactory to us .In chapter 3,1 use the maximal module ideal point algorithm with power to solve the upper-level model and the lower-level model respectively.2.The Second part of this paper (chapter 4) 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 con-diton under the reasonable constraint qualification. Some relative theorems and their proofs are given and the interact algorithm using the l\ penalty function is given to solve this problem.The model we are discussing is:min f{x,y) = (fi(x,y)J2(x,y),...,fp{x,y)) (12a)s.t. G(x,y)<0 (126) (12)y is the solution of the variational ineqality (F(x,-),C(x)). (12c)12.: mark : For all given x e X, e C(x) = {y € Rm : g(x,y) < 0},andiv 6 C(x),(v-y,F(x,y)) >0Where the mapping / : Rn x Rm —> i?p,the vecter-valued mapping G â–  Rn x Rm —> R? the vecter-valued mapping g : Rn x Rm —> Rl are continuous differential. Suppose / and G are continuous and twice differential maps for all (x,y G)i?n+m. For every x 6 X,z = 1,2,...,/,&(ï¿¡, ?) is convex in the second argument.With the KKT conditions ,the problem (12) is converted to :min^A f(x, y) s.t. G{x,y) <0 F(x,y)+ ZXiV ygi(x, y) = 0i=lAi>0, 1 = 1,2,...,! (13)9i(x,y)<0, i-1,2,...,/ Ai5t(x,y) = 0, ,i = l,2,...,ZThrough defining the sequentially bounded constraint qualification (SBCQ), there are the following results:Theorem 3 Let F and each gi be continuous.Suppose that each gi(^.-) is convex for all x € X,and SJygi{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 defined above,and then the problem (12)is equivalent to the problem (13). Let the linear summation problem with the power of the problem (13) isAnd then there isTheorem 4 About every given power coefficient a = (ai, a2,..., ap)T 6 A++ (or A+), the most superior solution of the problem (14) is the efficient solution (or the weak efficient solution) of the problem (13). Where, A+ = {a = (a1,a2,...,ap)T : Va* > 0, ï¿¡ at = 1},A++ = {a =p(ai,a2,..., otp)T : Va^ > 0, E aj = 1}t=i Suppose the penalty term:l v{x, y,\) = -J2 min(-Gi(x, y), 0)+^ I min(-ft(i, y), Aj)|+|F(x, y)+^Z xi^y9i{x, v)\Let the Zi penalty function be^ y,\) (15)Where /x > 0 is a penalty scalar,which is a very big positive number. So the problem (14) can be converted tominp/i(x,y,A) (16)i,j/,ADenote z = (x, y, A).Theorem 5 (the convergent theorem) Let the feasible region of theproblem is not empty, and there is a e > 0, subjecting to the setjs compact .Let {//&} is a strict increasing positive sequence of numbers limiting infty . For every k,there is a optimal solution z^ to the problem (16). 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 (13).