Generalization of preinvex and B-vex functions

A class of functions called B-preinvex functions is introduced by relaxing the definitions of preinvex and B-vex functions. Examples are given to show that there exist functions which are B-preinvex but not preinvex or B-vex or quasipreinvex. Some of the properties of B-preinvex functions are obtained.The contribution of this author was made partially at University of Manitoba and partially at St. Lawrence University during the summer of 1990. Support for the visit to St. Lawrence University was made possible under the GTE/SLU Visiting Scholars Grant.     

B-vex functions

A class of functions, called b-vex functions, is introduced by relaxing the definition of convexity of a function. Both the differentiable and nondifferentiable cases are presented. Members of this class satisfy most of the basic properties of convex functions. This class forms a subset of the sets of both semistrictly quasiconvex as well as quasiconvex functions, but are not necessarily included in the class of preinvex functions.The first author is thankful to the Natural Science and Engineering Research Council of Canada for financial support through Grant A-5319. The authors are also grateful to an anonymous referee for the constructive criticism of the first version of the paper, to Dr. Collen Knickerbocker of St. Lawrence University, and to Mrs. Meena K. Bector for their help in sharpening Examples 2.1 and 2.2, respectively.     

Generalization of preinvex and B-vex fuzzy mappings

A class of fuzzy mappings, called B-preinvex, and strictly B-preinvex fuzzy mappings, is introduced by relaxing the definition of preinvexity of a fuzzy mapping. Similarly, based on a notion of differentiability of fuzzy mappings different from the usual one, the class of pseudo-B-vex, B-invex, and pseudo-B-invex fuzzy mappings is defined as a generalization of pseudo-convex, invex, and pseudo-invex fuzzy mappings. We prove that a strictly B-vex, or B-preinvex fuzzy mapping has at most one global minimum point, and that the class of B-vex fuzzy mappings forms a subset of the class of quasi-convex fuzzy mappings. B-vex (resp. B-preinvex) fuzzy mappings satisfy most of the basic properties of convex (resp. preinvex) fuzzy mappings. In addition characterizations and sufficient optimality conditions are obtained for B-vex and B-preinvex fuzzy mappings.     

Semi-B-Preinvex Functions

In this note, a class of functions, called semi-B-preinvex function, which are a generalization of the semipreinvex functions and the B-vex functions, is introduced. Examples are given to show that there exist functions which are semi-B-preinvex functions, but are neither semipreinvex nor B-vex. A property of the semi-B-preinvex functions is obtained.Communicated by F. GiannessiThis research was partially supported by a Science Committee Project, Research Foundation of Chongqing, Grant 8409. The authors are thankful to the referees and Prof. F. Giannessi for suggestions and comments which helped to give the present form to this paper.     

具有广义B-凸函数的非光滑多目标规划的最优性与向量Lagrange鞍点理论

利用广义B-凸函数等概念，讨论了一类非光滑多目标规划，给出了广义最优性充分条件和Mond-Weir型对偶结果，讨论了向量Lagrange乘子性质并证明了向量值鞍点定理。     

Lipschitz B-Vex Functions and Nonsmooth Programming

In this paper, the equivalence between the class of B-vex functions and that of quasiconvex functions is proved. Necessary and sufficient conditions, under which a locally Lipschitz function is B-vex, are established in terms of the Clarke subdifferential. Regularity of locally Lipschitz B-vex functions is discussed. Furthermore, under appropriate conditions, a necessary optimality condition of the Slater type and a sufficient optimality condition are obtained for a nonsmooth programming problem involving B-vex functions.     

Optimality conditions and duality in subdifferentiable multiobjective fractional programming

Fritz John and Kuhn-Tucker necessary and sufficient conditions for a Pareto optimum of a subdifferentiable multiobjective fractional programming problem are derived without recourse to an equivalent convex program or parametric transformation. A dual problem is introduced and, under convexity assumptions, duality theorems are proved. Furthermore, a Lagrange multiplier theorem is established, a vector-valued ratio-type Lagrangian is introduced, and vector-valued saddle-point results are presented.The authors are thankful to the referees and Professor P. L. Yu for their many useful comments and suggestions which have improved the presentation of the paper.The first author is thankful to the Natural Science and Engineering Research Council of Canada for financial support through Grant No. A-5319. The authors are also thankful to the Dean's Office, Faculty of Management, University of Manitoba, for the financial support provided for the third author's visit to the Faculty.     

广义B-凸模糊映射及其应用

提出B-E-凸模糊映射的概念,并讨论了其一些性质,得到了这一类广义B-凸模糊映射的若干刻划定理,还讨论了这一类广义B-凸模糊映射在模糊优化中的应用,并得到了和最优解有关的一些结果。     

Sufficient optimality conditions and duality for a quasiconvex programming problem

Under differentiability assumptions, Fritz John Sufficient optimality conditions are proved for a nonlinear programming problem in which the objective function is assumed to be quasiconvex and the constraint functions are assumed to quasiconcave/strictly pseudoconcave. Duality theorems are proved for Mond-Weir type duality under the above generalized convexity assumptions.The first author is thankful to the Natural Science and Engineering Research Council of Canada for financial support through Grant No. A-5319. The authors are thankful to Professor B. Mond for suggestions that improved the original draft of the paper.     

Generalized fractional programming duality: A parametric approach

Using a parametric approach, duality is presented for a minimax fractional programming problem that involves several ratios in the objective function.The first author is thankful to Natural Science and Engineering Research Council of Canada for financial support through Grant A-5319, and the authors are thankful to the anonymous referees for useful suggestions.