Optimality Conditions For Set-Valued Optimization Problems

The theory of vector optimization with set-valued maps finds wide applicationsin differential inclusions, approximation theory, variations, optimization control, andso on, the optimality conditions for set-valued optimization problems in the sense ofvarious solutions are its important components and are the important base ofdeveloping modern algorithms. On the other hand, the concept of convexity playsimportant roles in the optimization theory, hence each of generalizations of convexityreceives researcher's attentions. The thesis is to gain properties of nearlycone-subconvexlike set-valued functions, and under the assumption of nearlycone-subconvexlikeness, to obtain the optimality conditions with derivatives orwithout derivatives for set-valued optimization problems in the sense of Bensonproper efficient element, superly efficient element of normed linear space, stronglyefficient element, strictly efficient element and superly efficient element of locallyconvex space, respectively. For details, these results are givenin the following. 1. Properties for convex cones are discussed, which are used to obtain severalequivalent characterizations for nearly cone-subconvexlike functions (maps). Byapplying an equivalent characterization of the nearly cone-subconvexlike function, animportant property of the nearly cone-subconvexlike function is presented, which andalternative theorem are used to obtain a Lagrange necessary condition for set-valuedoptimization problem to attain its Benson proper efficient element. 2. The super efficiency of normed linear space for set-valued optimizationproblem is investigated. Under the assumption of nearly cone-subconvexlikeness, ascaralization theorem of set-valued optimization problem to attain its superly efficientelement. By applying an equivalent characterization of the nearly cone-subconvexlikefunction, another important property of the nearly cone-subconvexlike function ispresented, which and alternative theorem are used to obtain a Lagrange necessarycondition for set-valued optimization problem to attain its superly efficient element.With the properties of the set of superly efficient points, a Lagrange sufficientcondition is obtained for set-valued optimization problem to attain its superlyefficient element. A kind of unconstrained characterization equivalent to set-valuedoptimization problem is presented in the sense of superly efficient elements. 3. The strong efficiency of set-valued optimization problem in locally convexspaces is investigated. Under the assumption of nearly cone-subconvexlikeness, withproperties of the set of strongly efficient points and alternative theorem, aKuhn-Tucker necessary condition is obtained for set-valued optimization problem toobtain its strongly efficient element. By applying properties of base functional and aNamioka decomposition theorem of a functional in a biordered linear space, aKuhn-Tucker sufficient condition is obtained for set-valued optimization problem toobtain its strongly efficient element. By applying an equivalent characterization of thenearly cone-subconvexlike function, another important property of the nearlycone-subconvexlike function is presented, which and a Namioka decompositiontheorem of a functional in a biordered linear space are used to obtain a Lagrangenecessary condition for set-valued optimization problem to attain its stronglyefficient element. With the properties of the set of strongly efficient points, aLagrange sufficient conditionis obtained for set-valued optimization problem to attainits strongly efficient element. Several kinds of Kuhn-Tucker and Lagrangeunconstrained characterizations equivalent to set-valued optimization problem arepresented in the sense of strongly efficient elements. 4. The strict efficiency of set-valued optimization problem in locally convexspaces is investigated. Under the assumption of nearly cone-subconvexlikeness, witha separation theorem for convex se...