| Multi-objective Optimization problem is a branch of research-ing more than one objective function in its domain. In many practi-cal problems, such as military, science, economics, management and industrial design, and other fields. We judge a scheme is good or not, it's very difficult to just use only one yardstick, so recently many scholars have committed to multi-objective programming research. Multi-objective programming research means has been increasingly rich and colorful. In this paper we used the tools abstract differen-tial to studied the concave function and weakly convex function, we get some results different from the past. We consider the following multi-objective programming problem:1.In the first chapter we described the origin and the devel-opment of the multi-objective programming.we also introduced the application of the abstract convex optimization.2.In the second chapter we introduced the symbols and the prepare knowledge we need in this paper3. The third chapter told the Sufficient and necessary condi-tions we get in the conditions of (ε, L)-subdifferential and (ε, L)-normal set;Theorem3.1:Let L is a subspace of a real value functions de-fined on Rn.Let x∈U,suppose that -fi(x)is HL-convex on U,then xis a weakly efficient solution of problem (VP)if and only if for all i the following can not hold:we get the sufficient and necessary conditions of the concave function multi-objective problem in the conditions of (ε, L)-subdifferential and (ε, L)-normal set. The result is also right in the conditions of theε-subdifferential and e-normal set.Theorem3.2:Let L is a subspace of real valued functions defined on Rn. x∈U,If-fi(x) is a HL-convex function in U, the following results is the Sufficient and necessary condition of that x is Global weakly efficient solution if and only if there areτi≥0:We also get the following results: (1)If x is the global weakly efficient solution of (VP),then (2) then xis the Sufficient and necessary condition of the global weakly efficient solution of (VP)is that (1) holds.Corollary3.3:Let L is a subspace of real valued functions de-fined on Rn. x∈U, if-fi(x) is HL-convex,if xis the global weakly efficient solution of (VP)then (?)τi≥0 make the following holds:Corollary3.4:Let L is a subspace of real valued functions de-fined on holds,then xis the global weakly efficient solution of (VP). Theorem3.3:Let L is a subspace of real valued functions defined in Rn.x∈U,If f∈L,then4.We introduced the some properties of the L-subdifferential andε-normal set in weakly convex functions. We concentrate on functions of the form (4.1) Ai∈Sn, pi:/Rn→R∪{+∞}is a convex function, then fi is a wealy convex function.if Aiin positive semidefinite, then fi is convex.Q=Diag(q)is the set of symmetric n×n matricesTheorem4.1:If fi is a function defined in (4.1), proper convex function,let x∈domp, domp=∩dompi,then Theorem4.2:fi=1/2+<αi,x>,(?)τipi(x)is a proper convex function,let x∈domp,domp=∩dompi,thenTheorem4.3:If lq,b,x∈L,lq,b,x(x)=1/2+,q,b∈RnTheorem4.1 and Theorem4.2 have been changed as follows:(4.2)(4.3)Theorem4.4: L={l:l(x)=1/2+<β,x>,Q=Diag(q),q,β∈Rn},lq,b,x∈L,lq,b,x(x)=1/2(Qx,x)++pi(x),If(?)q,b∈Rn,such that then xis the global weakly efficient solution of (VP1).corollary5.1.1:Let fi(x)=1/2+<αi,x).[CQ]if exists q∈Rn,such that then xis the global weakly efficient solution of (VP1).5.§5.2 We Mainly discussed Lagrange Saddle points and the solution of the weakly convex function multi-objective program-ming's if equivalence or not.We considered the following problem:Ai, Bj∈Sn, U(?)Rn,pi(x), qi(x)is proper convex funciton。Theorem5.2.1:If (x,λ)is a saddle point of L(x,λ), then xis the global weakly efficient solution of (VP2) Theorem5.2.2:Let U0:={x∈U|gi(x),i=1,2….m},x∈U0,if there such that then(x,λ)is a saddle point of L(x,λ). A=(A1,A2,…,Ak),B=(B1,B2,…,Bm),Ai∈Sn,i=1,2,…k,Bj∈Sn,j=1,2,…mlq,b,x(x)=1/2(Qx,x)+ |
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