A Class Of Generalized (h,Φ)-η Subgradient And Generalized (h,Φ)-η Pre-Invex Functions With Their Applications To Optimization Problems

This paper consisting of seven chapters,discusses a class of generalized (h, φ)-η; subgradient and (h, φ)-η pre-invex functions and their applications to optimization problems.The first chapter consisting of three sections is about introduction.In the first section, six generalizations of convexity concepts and their relative properties are reviewed;In the second section,three generalizations of derivative and gradient concepts and their relative properties are reviewed;In the third section,we state plainly some main results of this paper.The second chapter consisting of seven sections,discusses a class of generalized (h, φ)-η subgradient and its applications to optimization problems.In the first section ,Ben-Tal generalized algebraic operations are introduced,and properties of Ben-Tal generalized algebraic operations are discussed,which are our main tools of algebraic operations;In the second section,with the properties of Ben-Tal generalized algebraic operations,the notion of (h, φ)— directional derivative is introduced,and relative basic properties are discussed;In the third section ,with the notion of (h, φ)— directional derivative,a class of (h, φ)- subgradient is defined,and relative basic properties are discussed;In the fourth section,applications of (h, φ)— subgradient to optimization problems are discussed,three constraint qualifications of (h, φ) — K — T conditions and the optimization conditions of (h, φ)—programming problem are gotten;In the fifth section,a class of generalized (h, φ)-η directional derivative which is generalization of (h, φ)— directional derivative is defined,and relative basic properties are discussed;In the sixth section ,with the notion of generalized (h, φ)-η directional derivative,a class of (h, φ)-η subgradient and regular weakly (h, φ) — Lipschitz function are defined,and relative basic properties are discussed;In the seventh section,we first present two inequations on (h, φ)-η pre-invex function and (h, φ)-η prequasi-linear function,then the optimality conditions of non-smooth (h, φ)— semi-infinite programming problem and a constraint qualification of (h, φ)- K — T conditions are gotten.