Some Studies On Kuhn-Tucker Optimality Conditions Of Multi-objective Optimization

In this thesis,we mainly investigate the Kuhn-Tucker optimality conditions.Under some regularity assumptions,we obtain the second-order Kuhn-Tucker optimality conditions as well as the second-order strong Kuhn-Tucker optimality conditions for continuously Fréchet differentiable multi-objective problems.In addition,we establish an approximate strong Kuhn-Tucker optimality necessary condition of a local efficient solution,and discuss the strong Kuhn-Tucker optimality conditions for cone constrained vector optimization problems.The thesis is divided into seven chapters as follows:In Chapter 1,the introduction part,we introduce the background of the multi-objective optimization,the related advances on the Kuhn-Tucker optimality and the main content of the present thesis.In Chapter 2,we mainly introduce some basic concepts used in the thesis,including the concepts of solutions of multi-objective optimization problems,the concepts of second-order tangent approximations and the concepts of second-order directional derivatives.In addition,we discuss the first-order Kuhn-Tucker optimality necessary conditions for multi-objective problems,and also prodive several useful properties of radial second-order directional derivatives.In Chapter 3,we introduce a new kind of sequential approximate Kuhn-Tucker condition for a multi-objective problem.First,we prove that every local efficient solution satisfies the approximate strong Kuhn-Tucker optimality condition.But such an optimality condition may not hold at a weak efficient solution.Secondly,we use a so-called cone-continuity regularity condition to guarantee that the limit of an approximate strong Kuhn-Tucker sequence converges to a strong Kuhn-Tucker point.Finally,under the appropriate assumptions,we show that the approximate strong Kuhn-Tucker condition is also a sufficient condition of properly efficient points for convex multiobjective optimization problems.In Chapter 4,using dual cones and their properties,we state a generalization of Tucker's theorem of the alternative to conic systems.This allows us to discuss the strong Kuhn-Tucker type optimality conditions,suitable for cone constrained vector optimization problems.We first establish the strong Kuhn-Tucker theorems of the alternativetype at points not necessarily efficient.Subsequently,provided a regularity condition holds,then we achieve the strong Kuhn-Tucker necessary optimality conditions at Benson properly efficient solutions.In addition,as an application,Benson-properly efficient solutions are characterized through linear scalarization.In Chapter 5,by virtue of the radial second-order directional derivative and the projective second-order tangent cone,we construct a second-order constraint qualification of Abadie type.Using such a constraint qualification,we achieve the secondorder Kuhn-Tucker optimality conditions for weak efficient solutions.Secondly,we also obtain the second-order sufficient optimality conditions of a kind of strict local efficient solution.By means of the Motzkin alternative theorem,both the necessary conditions and the sufficient conditions are shown in equivalent pairs of primal and dual formulations.In Chapter 6,using the radial second-order directional derivative and the secondorder projective second-order tangent cone,we construct a second-order regularity condition of Abadie type.Such a regularity condition allows us to prove that the second-order strong Kuhn-Tucker optimality necessary condition holds at a Borwein properly efficient solution.By means of the Tucker alternative theorem,the necessary conditions are shown in equivalent pairs of primal and dual formulations.In addition,we provide a counterexample to illustrate that the strong Kuhn-Tucker optimality necessary condition may not hold if the Abadie type regularity condition is weakened to the Guignard type one.Finally,we also achieve a second-order strong Kuhn-Tucker optimality sufficient condition for local Geoffrion properly efficient solutions.In Chapter 7,we summarize the work in this paper and put forward the content of the future investigation.