Constraint Qualification In A General Class Of Nonconvex Nondifferentiable Mathematical Programming Problem And Its Corresponding Theorems

For a class of nondifferentiable and nonconvex multiobjective mathematical 0programming problem (VP), that is, the problem of minimizing a local Lipschitzian vector function subject to a set of differentiable nonlinear inequalities and equalities in a convex set, generalized Kuhn-Tucker type necessary and sufficient optimality condition, generalized saddle-point type necessary and sufficient optimality condition, generalized Lagrange duality and generalized Mond-Weir duality are concerned. The dissertation consists of four chapters.In chapter 1, the recent advances in the study of nondifferentiable and nonconvex multiobjective mathematical programming problem are stated. Then the work we do on this problem is introduced.In chapter 2, the author introduces some definitions, notions and several preliminary results, which will be needed later in the sequel. In section 1, a few basic auxiliary results on nonsmooth analysis are recalled. In section 2, combining the definition of (?)-convexity, η-invexity and d-univexity, the author proposes the definition of generalized ((?),ρ,θ) — d-univexity, and then presents some basic non-smooth alternative theorems for systems involving inequality constraints and systems involving mixed constraints (inequality constraints and equality constraints). In section 3 some basic knowledge on multiobjective mathematical programming problem is introduced. In section 4, the author presents other preliminaries which will be needed in the sequel.In chapter 3, generalized Kuhn-Tucker type optimality conditions are given for weak efficiency, efficiency and proper efficiency of problem (VP) under the conditions of a generalized Kuhn-Tucker constraint qualification and a generalized Arrow-Hurwicz-Uzawa constraint qualification. Based on the nonsmooth alternative theorems, generalized saddle-point type optimality conditions are proposed for weak efficiency, efficiency and proper efficiency of problem (VP).