Variable Metric Derivative And Its Application In Set-valued Optimization Theory

Set-valued optimization problem ,researches on which touch upon such subjects a linear and nonlinear analysis,topoloeical vector lattice, partial ordering theory in abstract spaces and can be concluded as extreme value problem of set-valued function, is the joint requirement of the development of parameter optimization ,control theory ,best approximation, nonsmooth analysis , fixed point theory , varation problem and mathematical economics ,is the focal point problem of multiobjective optimization field , and therefore has important learning value and certain degree of difficulty. The optimality condition of set-valued optimization is the central content , the absolutely necessary theory foundation of modern optimization method,and,moreover, one of the most difficult problems of set-valued optimization. But the optimality conditions which have the same construction form as the vector single-value optimization has not found to be obtained.The main reason we think is that there is not a good analytical tool to be found. The purposes of this paper are to extend the Clarke?s tangent cone ,Adjacent tangent cone and the Contingent tangent cone introduced by Clarke,R.T.Rockafellar and G.Bouligand ,on the basis of the derivative thought given by Fermat, define three kinds of variable metric derivatives for a set-valued function with the help of the epigraph or subgraph of the set-valued function, from which establish the optimality conditions for the efficient element of nonconvex set-valued optimization under weak efficiency ,Benson proper efficiency and the super efficiency .The weak duality theorem , the direct duality theorems and the inverse duality theorems for wolfe type duality,Mondeir type duality and a kind of modified Lagrangian duality are presented respectively. For detail ,we conclude them as follows: 1. The concept of changing measure (1,a)larke tangent cone, (1,a)djacent cone and (1,a)ontingent tangent cone ,which extend the classic Clarke cone , the Adjacent cone and the Contingent cone , are defined. As an example the existence of the extended cone are illustrated. 2. Three kinds of changing measure derivatuive ,i.e. the a? order Clarke derivative ,the arder Adjacent derivative and the arder contingent derivative of a set-valued map are established with the (1,a)larke tangent cone, (l,a)djacent tangent cone , (l,a) Contingent tangent cone and epigraph (the subgraph) of the set-valued map. 3. The concept of pseudoconvexity for set-valued map is introduced and is applied to the discussion of optimality condition and duality for a set-valued map. 4. A kind of generalized alternative theorem which can be applied to establish the optimality condition of set-valued optimization is developed. 5. The nonderivative type necessary and sufficient optimality conditions which refine the classic results under weak efficiency, Benson proper efficiency and super efficiency are established for nonconvex set-valued optimization problems. 6. The necessary and sufficient optimality conditions with the changing measure derivatives are established under weak efficiency , Benson proper efficiency and super efficiency for a set-valued map. The Fritz John and Kuhnucker conditions established in this paper have the same forms as the case of vector single valued optimization and therefore perfect the...