Generalized convexity in optimization

Abstract

In this thesis, some new generalized convex functions and generalized monotone functions are introduced, and their properties and various relations are established. These new generalized convexities and generalized monotonicities are then applied to the study of optimality conditions and duality theory in optimization. The thesis consists of four parts. In part 1, real-valued generalized convexities such as the concepts of semistrictly preinvex functions and generalized preinvex functions are introduced and their properties are given. The preinvexity of a real-valued function is characterized by an intermediate-point preinvexity condition. Some properties of semistrictly preinvex functions and semipreinvex functions are discussed. In particular, the relationship between a semistrictly preinvex function and a preinvex function is investigated. It is shown that a function is semistrictly preinvex if and only if it satisfies a strict invexity inequality for any two points with distinct function values. We also prove that the ratio of two semipreinvex functions is semipreinvex. It is worth noting that these characterizations reveal various interesting relationships among prequasiinvex, semistrictly prequasiinvex, and strictly prequasiinvex functions. These relationships are useful in the study of optimization problems. Part 2 studies set-valued generalized convexities. Generalized subconvexlike and nearly subconvexlike functions are introduced. In particular, these two classes of generalized convexities are invariant under multiplication and division provided that the multiplier or the divisor is a positive function. We note here that no other generalized convex function in the family of convexlike functions and its generalizations possesses this prominent property. A potential application of the near subconvexlikeness or generalized subconvexlikeness is in fractional programming. Furthermore, theorems of the alternative are established. Applications are given to obtain Lagrangian multiplier theorems for set-valued optimization problems and scalarization theorems for a weakly efficient solution, Benson properly efficient solution and Geoffrion properly efficient solution for set-valued optimization problems. In Part 3, generalized inmonicity is introduced and its relationship with generalized invexity is established. Several examples are given to show that these generalized inmonicities are proper generalization of the corresponding generalized monotonicities. Moreover, some examples are also presented to illustrate the properly inclusive relations among the generalized inmonicities. Finally in Part 4, several second order symmetric duality models are provided for single-objective and multi-objective nonlinear programming problems. Weak and strong duality theorems are established under first order or second order generalized convexity assumptions. Our study extends some of known results in the recent literature. It is worth noting that special cases of our models and results can be reduced to the corresponding first order cases, but most of second order symmetric models presented in the literature do not possess this nice property.