| In this thesis, the duality assertions and Farkas-type results for generalizedequilibrium problems, fractional programming problems and composed convexoptimization problems are investigated. And the higher-order optimality conditions andhigher-order Fritz John type optimality conditions for a set-valued optimization problemare also investigated. Moreover, the generalized differentiability of perturbationmappings for parametric vector optimization problems are also investigated. This thesisis divided into seven chapters. It is organized as follows:In Chapter1, at first, the development and researches on the topic of duality forgeneralized equilibrium problems and some optimization problems are recalled. Then,the development and researches on the topic of optimality and sensitivity for vectoroptimization problems are reviewed. Finally, we give the motivations and list the mainresearch works.In Chapter2, a generalized equilibrium problem with DC functions is considered.By using the method of Fenchel conjugate function, a dual scheme for the generalizedequilibrium problem is introduced. And the weak and strong duality assertions areobtained. Then, by using the obtained duality assertions, some Farkas-type resultswhich characterize the optimal value of the generalized equilibrium problem are given.Finally, the proposed approach is applied to a convex optimization problem and ageneralized variational inequality problem.In Chapter3, a constrained fractional programming problem is considered. Byusing an idea due to Dinkelbach[1], we first associate the fractional programmingproblem with a constrained optimization problem. Adopting different tactics, two typesof dual problems of the constrained optimization problem are constructed. After that, byusing the properties of the epigraph of the conjugate functions, the duality assertions forthe constrained optimization problem are investigated. Finally, by using the dualityassertions, some Farkas-type results for the fractional programming problem areobtained.In Chapter4, a composed convex optimization problem is considered. By using theproperties of the epigraph of the conjugated functions, some new constraintqualifications are introduced. Then, by using these new constraint qualifications, somenecessary and sufficient conditions which characterize the stable strong and total dualities for the composed convex optimization problem are obtained. Moreover, weshow that our general results encompass as special cases some recently obtained resultsin the literature.In Chapter5, Studniarski derivatives of set-valued maps are first defined and theirproperties are discussed. Then, by virtue of these derivatives and strict local minimality,higher-order optimality conditions and higher-order Fritz John type optimalityconditions are obtained for a set-valued optimization problem whose constraintcondition is determined by a set-valued map. Moreover, the relationships betweenStudniarski derivative of a set-valued map and its profile map are investigated. Somesensitivity results for a parametrized vector optimization are also obtained.In Chapter6, a family of parameterized set-valued optimization problems, whoseconstraint set depends on a parameter, are considered. Some calculus rules are obtainedfor calculating the second-order contingent derivatives of the composition and sum oftwo set-valued mappings. Then, by using these calculus rules, some results concerningsensitivity analysis are established, and an explicit expression for the second-ordercontingent derivative of the (weak) perturbation mapping in the set-valued optimizationproblems is obtained. Moreover, some calculus rules of the generalized second-ordercontingent epiderivative for frontier and efficient solution maps in parametric vectoroptimization problems are established.In Chapter7, the results of this thesis are briefly summarized. Some problems whichare remained and thought over in future are put forward. |
