Some Optimality And Duality For Constrained Optimization Problems

In this thesis, various second-order derivatives for set-valued maps are discussed.And the efficiency, weak efficiency, sharp efficiency and weak sharp efficiency withcorresponding second-order constraint qualifications, optimality conditions andgeneralized Fermat rules for constrained set-valued optimization problems areinvestigated. Simultaneously, the calmness conditions, error bound properties andMordukhovich stationary point conditions for multiobjective optimization problemswith equilibrium constraints are considered. Moreover, the image space analysisapproach and unified duality theories for nonlinear programming problems are alsoinvestigated. The thesis is divided into six chapters and organized as follows:In Chapter1, the development and present researches on the related topics ofoptimization problems are firstly recalled. Then the efficiency, second-order optimalityconditions and generalized Fermat rules for vector and set-valued optimizationproblems, and the Lagrange-type dualities and image space analysis for nonlinearprogramming problems are reviewed. Lastly, the motivations and the main researchwork of this thesis are also given.In Chapter2, some notations, definitions, and basic assumptions and properties arerecalled. These definitions mainly refer to the various stability conditions in vectoroptimization problems, the first and second order contingent derivatives andcoderivatives for set-valued maps and the separation functions for image space analysis.In Chapter3, the second-order optimality conditions for constrained set-valuedoptimization problems are considered. At first, we propose the concepts of second-orderlower derivative, asymptotic second-order derivative, second-order semidifferentiabilityand asymptotic second-order semidifferentiability for set-valued maps, and thenestablish some second-order optimality conditions, with no gap between the necessaryand sufficient conditions, for the sharp efficiency of set-valued optimization problemswith inclusion constraints. Subsequently, by virtue of the composed idea, on one hand,we present the second-order composed contingent derivative for set-valued maps,establish a second-order Kurcyusz-Robinson-Zowe constraint qualification forset-valued optimization problems with generalized inequality constraints and obtain acorresponding second-order Karush-Kuhn-Tucker optimality condition. On the otherhand, by using the epigraph, we introduce the second-order composed contingent epiderivative for set-valued maps, study some of its properties and establish somesecond-order optimality conditions for set-valued optimization problems with abstractconstraints.In Chapter4, the generalized Fermat rules for constrained optimization problemsare considered. On one hand, by means of the canonical perturbation scheme forconstraint systems, we propose a calmness condition for a general multiobjectiveoptimization problem with equilibrium constraints and obtain some existences for twoclasses of multiobjective exact penalty functions. Moreover, we also establish aMordukhovich stationary condition for weak efficiency of the problem by virtue of theMordukhovich generalized differentiation and normal cones. On the other hand, byusing distance functions, we present the notions of sharp efficiency and weak sharpefficiency for set-valued optimization problems with abstract constraints. And then, byvirtue of proposing the concept of uniformly strong normality for sets, we establishsome strong Fermat rules for sharp efficiency and quasi-strong Fermat rules for weaksharp efficiency without any convexity assumptions by using various generalizeddifferentiation and normal cones. Simultaneously, we further obtain some equivalentcharacterizations for the convex problems in the primal and dual spaces.In Chapter5, the unified duality theory for nonlinear programming problemswithout any convex data is considered. By virtue of the image space analysis and theregular weak separation functions, we firstly establish a unified duality scheme. Then,under some appropriate assumptions, we obtain some equivalent characterizations notonly for the generalized Lagrange multiplier and saddle point, but also for the zeroduality gap property between the primal and dual problems in the form of the regularweak separation for some suitable sets in the image space. Moreover, we also obtain anequivalent relationship between the zero duality gap property and the lowersemicontinuity of the perturbation function at the origin by means of the correspondingcanonical perturbation scheme. Finally, we give a unified interpretation for three specialduality schemes, including the Lagrange-type duality, the Wolfe duality and theMond-Weir duality.In Chapter6, the main results of this thesis are briefly summarized. Some problemswhich are remained and thought over in future are put forward.