Study On Optimality And Duality For Several Classes Of Nonlinear Optimization

Nonlinear optimization problem is one of the main research felds of optimizationtheory and applications, which is widely used in the areas of fnancial investment, ecolog-ical protection, project evaluation, etc. The study of this topic involves many disciplines,such as: convex analysis, nonlinear analysis, and so on. Therefore, the research on thistopic has important theoretical signifcance and practical value.Optimality conditions are an important foundation for developing optimization algo-rithms, which and duality are important components in optimization theory. The studyon optimality and duality for nonlinear optimization is always one of the hot researchissues of optimization.The study of this paper focuses on the optimality and duality for several classesof nonlinear optimization problems under diferent generalized convexity. The resultspresented in this paper improve, extend and unify many authors’ recent research results.The major works of this paper are as follows:1. Study on a class of E-convex multiobjective optimization problem with inequalityand equality constraints. Firstly, a sufcient optimality condition for this problem is givenunder the assumption of the E-convex conditions, Then, a class of Wolfe type dual modelof this problem is established, and theorems of weak duality, strong duality and converseduality are obtained. Furthermore, the results presented in this paper hold in ordinarycircumstances.2. Firstly, a concept of local star-shaped E-invex set is introduced, and some of itsbasic properties are discussed. Based on this concept, a new class of concepts of semilocalE-preinvex and other related functions is introduced, which is by examples illustratedthat the class of functions not only exists, but also they are real generalizations of someknown generalized convex functions, and some important characterizations are also inves-tigated. Then, under the assumption of semilocal E-preinvexity, three kinds of nonlinearoptimization problems are considered:(1) A necessary and sufcient optimality condi-tion for a class of nonlinear optimization problem with no constraints is established.(2)Some optimality conditions for a class of nonlinear optimization problem with inequalityconstraints are derived and Mond-Weir type duality theorems for this problem are estab-lished.(3) Several sufcient optimality conditions for a class of nonlinear multiobjectivefractional programming problems are obtained, and utilizing the method of Bector,etc. adual model for this problem is formulated, and some duality results are gained.3. Study on a class of nonsmooth vector optimization problem with cone constraints.Firstly, several generalized cone-invex functions are introduced by using Clarke’s gener- alized gradients. Then, the relationships between them are discussed and some knowngeneralized convex functions are extended by examples. Finally, several sufcient opti-mality conditions and Mond-Weir type duality results for this problem are establishedunder the assumptions of the generalized cone-invexity.4. Study on a class of nondiferentiable multiobjective programming problem withinequality constraints. Firstly, by using dI-invexity, a class of concept of generalized dI-V-I type univexity is introduced, and some sufcient optimality conditions for this problemare derived under the univexity assumptions. Finally, a Mond-Weir type dual model isformulated and weak duality, converse duality and strict duality theorems are proved.5. Study on a class of minmax fractional programming problem. Firstly, by in-troducing a new concept of second-order generalized (F, α, ρ, θ)-d-V-I type univexity, asecond-order dual model of this problems is formulated. Then, by using functional sub-linearity, weak duality, strong duality and strict converse duality theorems are obtainedunder the assumption of second-order generalized I type univexity.