Study On Optimality And Stability Of Cone Constrained Multi-objective Optimization Problems

Cone constrained multi-objective optimization problems are optimization problems whose objective functions are vector-valued with orders induced by closed convex cones and constraint sets are defined by conic constraints.This type of problems has many backgrounds in financial analysis and engineering designing.Among a large number of literatures,most results are fo-cused on multi-objective optimization problems whose orders are induced by R+p,and problems whose orders are induced by abstract convex pointed cones.Results for orders induced by other specific cones are seldom witnessed.Optimality and stability are two main topics in theory of multi-objective optimization,whereas the study about second-order optimality,especially about the second-order sufficient optimality conditions,is not completed;and results about stability of multi-objective optimization are only restricted to the upper semi-continuity and lower semi-continuity for mappings of efficient solutions.This dissertation will study second-order optimality for cone constrained multi-objective optimization problems including those whose orders are induced byg R+p,second-order cone and symmetric semi-definite matrix cone,and the strong regularity and the isolated calmness of Karush-Kuhn-Tucker(KKT)system for inequality and equality constrained R+p induced multi-objective problems.The main results obtained in this dissertation are summarized as follows:1.Chapter 2 establishes second-order necessary and second-order sufficient optimality con-ditions for cone constrained multi-objective optimization whose order is induced by R+p.First of all,we derive,for an abstract constrained multi-objective optimization problem,two basic nec-essary optimality theorems for weak efficient solutions with one about the first-order necessary optimality and the other abut the first-order necessary optimality,and a second-order sufficien-t optimality theorem for efficient solutions.Secondly,basing on the optimality results for the problem,we demonstrate,for cone constrained multi-objective optimization problems,the first-order and second-order necessary optimality conditions under Robinson constraint qual-ification as well as the second-order sufficient optimality conditions under outer second-order regularity for the conic constraint.Finally,using the optimality conditions for cone constrained multi-objective optimization obtained,we establish optimality conditions,including the first-order necessary,the second-order necessary and the second-order sufficient optimality condi-tions,for polyhedral cone,second-order cone and semi-definite cone constrained multi-objective optimization problems.2.Chapter 3 considers the multi-objective optimization problems whose orders are induced by a product of finite second-order cones(denoted by Q-multi-objective optimization problem-s),and establishes the first-order necessary,the second-order necessary optimality conditions for weak efficient solutions and the second-order sufficient optimality conditions for efficient solu-tions.For Q-multi-objective optimization problems with explicit constraints,we demonstrate the first-order necessary and the second-order necessary optimality conditions under Robinson con-straint qualification,and the second-order sufficient optimality conditions when the constraint sets satisfy outer second-order regularity condition.As applications,we obtain optimality con-ditions for Q-multi-objective optimization problems with polyhedron cone,second-order cone and semidefinite cone constraints.3.Chapter 4 establishes the first-order necessary,the second-order necessary and second-order sufficient optimality conditions for multi-objective optimization problems whose orders are induced by symmetric semidefinite coneS+m(denoted by S+m-multi-objective optimization problems).For abstract constrained T-multi-objective optimization problems,two basic nec-essary optimality theorems for weak efficient solutions and a second-order sufficient optimality conditions for efficient solutions are developed.For S+m-multi-objective optimization problems with cone constraints,we demonstrate the first-order necessary and the second-order necessary optimality conditions under Robinson constraint qualification,and the second-order sufficient optimality conditions when the constraint sets satisfy outer second-order regularity condition.As applications,we obtain optimality conditions for S+m-multi-objective optimization problems with polyhedron cone,second-order cone and semidefinite cone constraints.4.Chapter 5 studies the strong regularity and the isolated calmness of Karush-Kuhn-Tucker system for R+P-multiobjective optimization problems with equality and inequality constraints.Under the strict complementarity the existence of a differentiable KKT solution mapping is proved and the derivative formula for the KKT solution mapping is established;We demonstrate,without the strict complementarity condition,that the linear independence constraint qualifica-tion and the strong second-order sufficient optimality conditions are equivalent to the strong regularity of the KKT system;Moreover,other three equivalent conditions to the strong regular-ity of the KKT system are proved:the normal mapping corresponding to the KKT system is a Lipschitz homeomorphism,the strong stability of the corresponding optimization problem,or the uniform second-order growth condition for the corresponding problem.Finally,we demonstrate the equivalence between the isolated calmness of the KKT solution mapping and the second-order sufficient optimality conditions with strict Robinson constraint qualification.